Finite Group Databasebeta

← #33108598

Subgroups — #33108598

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988827C211yesmaximal, characteristic
2C2×#17988679C211yesmaximal, characteristic, complemented
3C2×#107×D73C211yesmaximal, characteristic
4#22612941C211yesmaximal, characteristic
5#22612938C211yesmaximal, characteristic
6#22610791C211yesmaximal, characteristic, complemented
7#22610663C211yesmaximal, characteristic, complemented
8#22610573C211yesmaximal, characteristic, complemented
9#22610237C211yesmaximal, characteristic
10#22610235C211yesmaximal, characteristic
11#22610187C211yesmaximal, characteristic, complemented
12#22610182C211yesmaximal, characteristic, complemented
13C4×#14556524C211yesmaximal, characteristic, complemented
14#22609701C211yesmaximal, characteristic
15C73×#1928C211yesmaximal, characteristic, complemented
16C22×#11924018C2211yescharacteristic
17C22×C8×D73C2211yescharacteristic
18C2×#14556571C2211yescharacteristic
19C2×#14556567C2211yescharacteristic
20C2×Q8×Dic73C2211yescharacteristic
21C2×#14556556C2211yescharacteristic
22C2×C4⋊C4×D73C2211yescharacteristic
23C2×C4⋊C4×D73C2211yescharacteristic
24#17988827C418yescomplemented
25#17988679—24no—
26C2×#14556525C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556524C2211yescharacteristic
30#107×D73—24no—
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556515C2211yescharacteristic
34C2×#14556514C2211yescharacteristic
35C2×#14556503C2211yescharacteristic
36C2×C4×#10596831C2211yescharacteristic
37C2×C8×Dic73C2211yescharacteristic
38#17988493C2211yescharacteristic
39#17988486C2211yescharacteristic
40#17988485—21no—
41#17988484—21no—
42#17988483C2211yescharacteristic
43#17988472—21no—
44#17988469—21no—
45#17988461—21no—
46#17988461—21no—
47C4×C4⋊Dic73C2211yescharacteristic
48C4×#11924005—21no—
49C4×#11924005C2211yescharacteristic
50#17988434—21no—
51#17988350C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×#107×C73C2211yescharacteristic
56C2×#103×C73C2211yescharacteristic
57C2×#103×C73C2211yescharacteristic
58#391×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#336×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×C4×D73—21no—
64C22×C4×D73—21no—
65C22×C4×D73C2311yescharacteristic
66C2×#11924018—22no—
67C2×#11924018—22no—
68C2×#11924018C2×C414yescomplemented
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71C2×C8×D73—22no—
72C2×C8×D73—22no—
73C2×C8×D73C2×C414yes—
74#14556571—22no—
75#14556571C2×C414yes—
76#14556567—22no—
77#14556567C2×C414yes—
78Q8×Dic73—22no—
79Q8×Dic73—41no—
80#14556556—22no—
81#14556556—41no—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73—21no—
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—21no—
92C2×#11924007—21no—
93C4⋊C4×D73—22no—
94C4⋊C4×D73—22no—
95C4⋊C4×D73—41no—
96C4⋊C4×D73—41no—
97C2×#11924005—21no—
98C2×#11924005—21no—
99C2×#11924005D411yescharacteristic
100C2×#11924005D411yescharacteristic
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103C2×C4×Dic73—21no—
104C2×C4×Dic73C2311yescharacteristic
105#14556525—22no—
106#14556524—22no—
107#14556524C2×C414yescomplemented
108#14556524D414yes—
109#14556516—22no—
110#14556516C2×C414yescomplemented
111#14556515—41no—
112#14556514—41no—
113#14556503—22no—
114C4×#10596831—22no—
115C8×Dic73—22no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120#14556498—21no—
121#14556498D411yescharacteristic
122#14556498D411yescharacteristic
123C22×Q8×C73C2311yescharacteristic
124C22×C8×C73C2311yescharacteristic
125C22×C8×C73C2311yescharacteristic
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131#107×C73—22no—
132#103×C73—22no—
133#103×C73C2×C414yescomplemented
134#95×C73—21no—
135#95×C73—21no—
136#95×C73—21no—
137C23×D73C2×D411yescharacteristic
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146—21no—
143C2×Dic146C22×C412yes—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—21no—
148C22×Dic73C2×D411yescharacteristic
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C2×C4×D73—21no—
152C2×C4×D73—21no—
153C2×C4×D73—21no—
154C2×C4×D73—21no—
155C2×C4×D73—42no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73—42no—
159C2×C4×D73—81no—
160C2×C4×D73—81no—
161C2×C4×D73C22×C412yes—
162C2×C4×D73#4112yes—
163#11924018—28no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831C22×C412yes—
168C2×#10596831C2×D412yes—
169C2×#10596831#4112yes—
170C8×D73—28no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—24no—
174C4⋊Dic73—21no—
175C4⋊Dic73—21no—
176C4⋊Dic73—42no—
177C4⋊Dic73—81no—
178C4⋊Dic73C22×C412yes—
179C4⋊Dic73C2×D412yes—
180#11924007—24no—
181#11924007—42no—
182#11924007—42no—
183#11924007—42no—
184#11924007—81no—
185#11924005—24no—
186#11924005—24no—
187#11924005—22no—
188#11924005Q1614yescomplemented
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—42no—
192C4×Dic73—42no—
193C4×Dic73—81no—
194C2×Q8×C73—21no—
195C2×Q8×C73—21no—
196C2×Q8×C73—21no—
197C2×Q8×C73—21no—
198C2×Q8×C73—21no—
199C2×Q8×C73C22×C412yes—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—21no—
203C22×C4×C73—21no—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2411yescharacteristic
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73C22×C412yes—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73C22×C412yes—
218C4⋊C4×C73C2×D412yes—
219C42×C73—21no—
220C42×C73—21no—
221C42×C73—81no—
222C23×C73C22×D411yescharacteristic
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73C4×D412yes—
228C22×D73C2×Q1612yes—
229C22×D73#13512yes—
230Q8×C73—22no—
231Q8×C73—22no—
232Q8×C73—41no—
233Q8×C73—41no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—22no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—41no—
243C2×C4×C73—41no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73C2×Q1612yes—
252C2×C4×C73C23×C411yescharacteristic
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73C2×#4111yescharacteristic
256C2×C4×C73H(5,𝔽2)11yescharacteristic
257C2×C4×C732−1+411yescharacteristic
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—22no—
263C2×Dic73—22no—
264C2×Dic73—21no—
265C2×Dic73—22no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73C4×D412yes—
278C2×Dic73C2×Q1612yes—
279C4×D73—22no—
280C4×D73—22no—
281C4×D73—22no—
282C4×D73—22no—
283C4×D73—84no—
284C4×D73—84no—
285Dic146—22no—
286Dic146—22no—
287Dic146—41no—
288Dic146—41no—
289C8×C73—22no—
290C8×C73—41no—
291#10596831—41no—
292#10596831C4×D414yes—
293C22×C73C2×C4×D411yescharacteristic
294C22×C73D4211yescharacteristic
295C22×C73#54711yescharacteristic
296C22×C73C22×Q1611yescharacteristic
297C22×C73C2×#13511yescharacteristic
298C22×C73#57611yescharacteristic
299C22×C73#57811yescharacteristic
300C2×D73—22no—
301C2×D73—22no—
302C2×D73—22no—
303C2×D73—22no—
304C2×D73—22no—
305C2×D73—22no—
306C2×D73C4×Q1614yescomplemented
307C4×C73—22no—
308C4×C73—41no—
309C4×C73—42no—
310C4×C73—42no—
311C4×C73—42no—
312C4×C73—81no—
313C4×C73C2×C4×D411yescharacteristic
314C4×C73#52011yescharacteristic
315C4×C73#54811yescharacteristic
316C4×C73#55311yescharacteristic
317Dic73—22no—
318Dic73—22no—
319Dic73—42no—
320Dic73—42no—
321Dic73—42no—
322Dic73—41no—
323Dic73—81no—
324#23024—731nomaximal, Sylow
325C2×C73#236111yescharacteristic
326C2×C73C2×C4×Q1611yescharacteristic
327C2×C73#295111yescharacteristic
328C2×C73#328011yescharacteristic
329C2×C73D4×Q1611yescharacteristic
330C2×C73#333611yescharacteristic
331C2×C73#333811yescharacteristic
332D73—24no—
333C2×#467—731no—
334C2×C4×Q16—731no—
335C22×#107—731no—
336#2360—731no—
337#2360—731no—
338#1969—731no—
339#1942—731no—
340#1928—731no—
341#1928D7311yescharacteristic, complemented
342#1872—731no—
343#1872—731no—
344#1863—731no—
345#1863—731no—
346C4×#107—731no—
347C8×C22⋊C4—731no—
348C73#2302411yescharacteristic, complemented, Sylow
349C22×Q16—731no—
350C23×C8—731no—
351C2×C22⋊Q8—731no—
352C2×C22⋊Q8—731no—
353C2×C4×Q8—731no—
354C2×C4×Q8—731no—
355C22×C4⋊C4—731no—
356C22×C4⋊C4—731no—
357#467—738no—
358C4×Q16—1464no—
359C2×#107—731no—
360C2×#107—731no—
361C2×#107—731no—
362C2×#107—731no—
363C2×#107—1464no—
364C2×#107C2×D7311yescharacteristic
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103C2×D7311yescharacteristic
370C2×#103C2×D7311yescharacteristic
371C2×C22⋊C8—731no—
372C2×C4×C8—731no—
373C2×C4×C8—731no—
374#391—731no—
375#391C2×D7311yescharacteristic
376#389—731no—
377#389—731no—
378#389—1461no—
379#389—1461no—
380#387—1461no—
381#387—1461no—
382#385—1461no—
383#385—1461no—
384C4×C4⋊C4—731no—
385C4×C4⋊C4C2×D7311yescharacteristic
386C4×C22⋊C4—731no—
387C4×C22⋊C4—1461no—
388C4×C22⋊C4—1461no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340C2×D7311yescharacteristic
393#336—731no—
394#336C2×D7311yescharacteristic
395C22×Q8—731no—
396C22×Q8C22×D7311yescharacteristic
397C23×C4—731no—
398C23×C4—1461no—
399C23×C4—1461no—
400C2×Q16—734no—
401C2×Q16—1462no—
402C2×Q16—1462no—
403C22×C8—731no—
404C22×C8—734no—
405C22×C8—731no—
406C22×C8—1462no—
407C22×C8—1462no—
408C22×C8C22×D7311yescharacteristic
409C22×C8C22×D7311yescharacteristic
410C22⋊Q8—734no—
411C22⋊Q8—734no—
412C22⋊Q8—1462no—
413C22⋊Q8—1462no—
414C4×Q8—2921no—
415C4×Q8—2921no—
416C4×Q8—1462no—
417C4×Q8—1462no—
418C2×C4⋊C4—2921no—
419C2×C4⋊C4—2921no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—1462no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1462no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4C22×D7311yescharacteristic
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C22⋊C4—731no—
437C2×C22⋊C4—731no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C42—21no—
441C2×C42—731no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42C22×D7311yescharacteristic
446#107—22no—
447#107—734no—
448#107—734no—
449#107—1462no—
450#107—1462no—
451#103—22no—
452#103—2921no—
453#103—2921no—
454#103—734no—
455#103—1462no—
456#103C4×D7314yescomplemented
457C22⋊C8—1462no—
458C4×C8—1462no—
459C4×C8—1462no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—731no—
464#95—731no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470C24—731no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—732no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8—1461no—
481C2×Q8—1461no—
482C2×Q8C2×C4×D7312yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—2922no—
490C22×C4—2922no—
491C22×C4—5841no—
492C22×C4—5841no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—732no—
496C22×C4—732no—
497C22×C4—731no—
498C22×C4—1461no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4—1461no—
504C22×C4—1461no—
505C22×C4—1461no—
506C22×C4D4×D7311yescharacteristic
507C22×C4D4×D7311yescharacteristic
508C22×C4C23×D7311yescharacteristic
509Q16—1468no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—732no—
516C2×C8—732no—
517C2×C8—732no—
518C2×C8—1468no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8C2×C4×D7312yes—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—2922no—
528C4⋊C4—2922no—
529C4⋊C4—2922no—
530C4⋊C4—2922no—
531C4⋊C4—5841no—
532C4⋊C4—5841no—
533C4⋊C4—732no—
534C4⋊C4—732no—
535C4⋊C4—1464no—
536C4⋊C4—1464no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4C2×C4×D7312yes—
542C4⋊C4D4×D7312yes—
543C22⋊C4—734no—
544C22⋊C4—1462no—
545C22⋊C4—1464no—
546C22⋊C4—1464no—
547C42—21no—
548C42—21no—
549C42—81no—
550C42—2922no—
551C42—2922no—
552C42—5841no—
553C42—1461no—
554C42—1461no—
555C23—732no—
556C23—732no—
557C23—732no—
558C23—1461no—
559C23—1461no—
560C23—1461no—
561C23—1461no—
562C23C2×D4×D7311yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—2921no—
568Q8—2921no—
569Q8—1462no—
570Q8—1462no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—5844no—
598C2×C4—5844no—
599C2×C4—732no—
600C2×C4—732no—
601C2×C4—1461no—
602C2×C4—1462no—
603C2×C4—1461no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1462no—
612C2×C4—1462no—
613C2×C4—1462no—
614C2×C4—1462no—
615C2×C4Q16×D7312yes—
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C2×D4×D7311yescharacteristic
618C2×C4#41×D7311yescharacteristic
619C2×C4#41×D7311yescharacteristic
620C2×C4#1455663711yescharacteristic
621C2×C4#1455663811yescharacteristic
622C8—22no—
623C8—41no—
624C8—2921no—
625C8—734no—
626C22—734no—
627C22—1462no—
628C22—1462no—
629C22—1462no—
630C22—1462no—
631C22—1462no—
632C22—1462no—
633C22C4×D4×D7311yescharacteristic
634C22#1798922111yescharacteristic
635C22#1798923511yescharacteristic
636C22C2×Q16×D7311yescharacteristic
637C22#135×D7311yescharacteristic
638C22#1798939111yescharacteristic
639C22#1798940311yescharacteristic
640C4—22no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—2922no—
647C4—2921no—
648C4—2922no—
649C4—2922no—
650C4—5841no—
651C4—1462no—
652C4—1462no—
653C4C4×D4×D7311yescharacteristic
654C4#1798906711yescharacteristic
655C4#1798924511yescharacteristic
656C4#1798928811yescharacteristic
657C2—1464no—
658C2#2261294411yescharacteristic
659C2C4×Q16×D7311yescharacteristic
660C2#2261880311yescharacteristic
661C2#2262161111yescharacteristic
662C2#2262167811yescharacteristic
663C2#2262229211yescharacteristic
664C2#2262231811yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1928C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1928D73
hypocenterC73#23024
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#17988679C2semidirect
6. #22610791C2semidirect
7. #22610663C2semidirect
8. #22610573C2semidirect
11. #22610187C2semidirect
12. #22610182C2semidirect
13. C4×#14556524C2semidirect
15. C73×#1928C2semidirect
24. #17988827C4semidirect
68. C2×#11924018C2×C4semidirect
107. #14556524C2×C4semidirect
110. #14556516C2×C4semidirect
133. #103×C73C2×C4semidirect
188. #11924005Q16semidirect
306. C2×D73C4×Q16semidirect
341. #1928D73semidirect
348. C73#23024semidirect
456. #103C4×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179888272
C2×#179886792
C2×#107×D732
#226129412
#226129382
#226107912
#226106632
#226105732
#226102372
#226102352
#226101872
#226101822
C4×#145565242
#226097012
C73×#19282
#2302473