Finite Group Databasebeta

← #33104271

Subgroups — #33104271

Order 18688.

1018 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988687C211yesmaximal, characteristic
2C2×#17988684C211yesmaximal, characteristic
3C2×#17988678C211yesmaximal, characteristic
4#22612845C211yesmaximal, characteristic, complemented
5D4×#11924007C211yesmaximal, characteristic
6#22610673C211yesmaximal, characteristic, complemented
7#22610671C211yesmaximal, characteristic, complemented
8#22610188C211yesmaximal, characteristic, complemented
9#22610182C211yesmaximal, characteristic, complemented
10#22610180C211yesmaximal, characteristic
11#22610089C211yesmaximal, characteristic, complemented
12#22609935C211yesmaximal, characteristic, complemented
13#22609929C211yesmaximal, characteristic
14#22609797C211yesmaximal, characteristic, complemented
15C73×#1862C211yesmaximal, characteristic
16C22×#11924016C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×#14556556C2211yescharacteristic
19C2×#14556556C2211yescharacteristic
20C2×D4×Dic73C2211yescharacteristic
21C2×D4×Dic73C2211yescharacteristic
22C22×#11924007—21no—
23#17988687—24no—
24#17988684—24no—
25#17988678C418yescomplemented
26C2×#14556525C2211yescharacteristic
27C2×#14556524C2211yescharacteristic
28C2×#14556524C2211yescharacteristic
29C2×#14556522C2211yescharacteristic
30C2×#14556520C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556514C2211yescharacteristic
34C2×#14556512C2211yescharacteristic
35C2×#14556502C2211yescharacteristic
36C2×#14556501C2211yescharacteristic
37C2×C8×Dic73C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988484—21no—
41#17988483C2211yescharacteristic
42#17988482—21no—
43#17988460—21no—
44#17988459—21no—
45#17988456—21no—
46#17988456C2211yescharacteristic
47#17988456C2211yescharacteristic
48#17988445—21no—
49C4×#11924007C2211yescharacteristic
50#17988438—21no—
51#17988350C2211yescharacteristic
52#17988349C2211yescharacteristic
53#17988349C2211yescharacteristic
54#17988341C2211yescharacteristic
55C2×C4×D4×C73C2211yescharacteristic
56C2×#107×C73C2211yescharacteristic
57C2×#102×C73C2211yescharacteristic
58C2×#102×C73C2211yescharacteristic
59#389×C73C2211yescharacteristic
60#340×C73C2211yescharacteristic
61#336×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C23×Dic73—21no—
64C23×Dic73—21no—
65C2×#11924016C2×C414yes—
66C2×#11924016D414yes—
67C2×#11924016Q814yes—
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C4×Dic146—22no—
71C4×Dic146—41no—
72#14556556—22no—
73#14556556—22no—
74#14556556—41no—
75#14556556C2×C414yescomplemented
76D4×Dic73—22no—
77D4×Dic73—22no—
78D4×Dic73—41no—
79D4×Dic73C2×C414yes—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×#11924007—21no—
92C2×#11924007—21no—
93C2×#11924007—44no—
94C2×#11924007D411yescharacteristic
95C2×#11924007D411yescharacteristic
96C2×C22⋊Dic73—21no—
97C2×C22⋊Dic73—21no—
98C2×C4×Dic73—21no—
99C2×C4×Dic73—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103#14556525—41no—
104#14556524—22no—
105#14556524—22no—
106#14556522C2×C414yescomplemented
107#14556520—22no—
108#14556516—22no—
109#14556516C2×C414yescomplemented
110#14556514—22no—
111#14556512—22no—
112#14556502—22no—
113#14556501—22no—
114C8×Dic73C2×C414yescomplemented
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498D411yescharacteristic
119#14556498D411yescharacteristic
120C22×D4×C73C2311yescharacteristic
121C23×C4×C73—21no—
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C4×D4×C73—22no—
125C4×D4×C73—41no—
126C2×C4⋊C4×C73—21no—
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C22⋊C4×C73—21no—
131C2×C42×C73C2311yescharacteristic
132#107×C73—41no—
133#102×C73—22no—
134#102×C73C2×C414yes—
135#95×C73—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146C22×C412yes—
140C2×Dic146C2×D412yes—
141C2×Dic146C2×Q812yes—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C22×Dic73—24no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73—42no—
148C22×Dic73—42no—
149C22×Dic73—42no—
150C22×Dic73—42no—
151C22×Dic73—81no—
152C22×Dic73C2×D411yescharacteristic
153C22×Dic73C2×D411yescharacteristic
154C22×Dic73C2×D411yescharacteristic
155C22×Dic73C2×D411yescharacteristic
156#11924016C4⋊C4116yescomplemented
157C2×#10596831—21no—
158C2×#10596831—21no—
159C2×#10596831—21no—
160C2×#10596831C22×C412yes—
161C2×#10596831C2×D412yes—
162C2×#10596831C2×Q812yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—81no—
169C4⋊Dic73C22×C412yes—
170#11924007—24no—
171#11924007—22no—
172#11924007—22no—
173#11924007—42no—
174#11924007—42no—
175#11924007—42no—
176#11924007—42no—
177#11924007—42no—
178#11924007—42no—
179#11924007—81no—
180#11924007—81no—
181C22⋊Dic73—24no—
182C22⋊Dic73—42no—
183C4×Dic73—21no—
184C4×Dic73—21no—
185C4×Dic73—24no—
186C4×Dic73—42no—
187C4×Dic73—42no—
188C4×Dic73C22×C412yes—
189C4×Dic73#4112yes—
190C24×C73—21no—
191C2×D4×C73—21no—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73C22×C412yes—
195C2×D4×C73C2×D412yes—
196C2×D4×C73C2×Q812yes—
197C22×C4×C73—21no—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—42no—
201C22×C4×C73—42no—
202C22×C4×C73—81no—
203C22×C4×C73C2×D411yescharacteristic
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2411yescharacteristic
206C2×C8×C73—21no—
207C2×C8×C73—21no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73C22×C412yes—
212C4⋊C4×C73—21no—
213C4⋊C4×C73—21no—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—81no—
218C4⋊C4×C73C22×C412yes—
219C22⋊C4×C73—42no—
220C42×C73—21no—
221C42×C73—21no—
222C23×C73—22no—
223C23×C73—22no—
224C23×C73—22no—
225C23×C73—41no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73C22×D411yescharacteristic
230D4×C73—22no—
231D4×C73—22no—
232D4×C73—22no—
233D4×C73C2×C4⋊C414yes—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—41no—
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—84no—
250C2×C4×C73C23×C411yescharacteristic
251C2×C4×C73C22×D411yescharacteristic
252C2×C4×C73C22×Q811yescharacteristic
253C2×C4×C73C2×#4111yescharacteristic
254C2×C4×C73H(5,𝔽2)11yescharacteristic
255C2×C4×C732−1+411yescharacteristic
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—22no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—22no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—48no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—84no—
278C2×Dic73C4×D412yes—
279C2×Dic73C4×D412yes—
280C2×Dic73AGL(1,ℤ/8)12yes—
281C2×Dic73#13712yes—
282Dic146—22no—
283Dic146—22no—
284Dic146—22no—
285Dic146C2×C4⋊C414yescomplemented
286C8×C73—22no—
287C8×C73—41no—
288#10596831—41no—
289#10596831C2×C4⋊C414yescomplemented
290C22×C73—24no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—42no—
297C22×C73C2×C4×D411yescharacteristic
298C22×C73D4211yescharacteristic
299C22×C73D4×Q811yescharacteristic
300C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C73C2×#13711yescharacteristic
302C22×C73#57611yescharacteristic
303C22×C73#57811yescharacteristic
304C4×C73—42no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—81no—
308C4×C73—81no—
309C4×C73C22×C4⋊C411yescharacteristic
310C4×C73#52011yescharacteristic
311C4×C73#54711yescharacteristic
312C4×C73#55411yescharacteristic
313Dic73—24no—
314Dic73—22no—
315Dic73—42no—
316Dic73—41no—
317Dic73—41no—
318Dic73—81no—
319Dic73—81no—
320Dic73#44014yescomplemented
321#22671—731nomaximal, Sylow
322C2×C73—44no—
323C2×C73D4×C4⋊C411yescharacteristic
324C2×C73C2×#44011yescharacteristic
325C2×C73#295111yescharacteristic
326C2×C73#327411yescharacteristic
327C2×C73#327611yescharacteristic
328C2×C73#338511yescharacteristic
329C2×C73#338611yescharacteristic
330C2×#440—731no—
331C2×#440—731no—
332C2×C4×SD16—731no—
333Q8×C4⋊C4—731no—
334D4×C4⋊C4—731no—
335#1944—731no—
336#1944—731no—
337#1863—731no—
338#1863—731no—
339#1862—731no—
340#1862D7311yescharacteristic
341#1845—731no—
342#1803—731no—
343#1802—731no—
344#1770—731no—
345C73#2267111yescharacteristic, complemented, Sylow
346C22×SD16—731no—
347C2×C4×Q8—731no—
348C2×C4×Q8—731no—
349C2×C4×Q8—731no—
350C2×C4×D4—731no—
351C2×C4×D4—731no—
352C2×C4×D4C2×D7311yescharacteristic
353C22×C4⋊C4—1461no—
354#440—1464no—
355#440—1464no—
356C4×SD16—738no—
357C2×#107—731no—
358C2×#107—731no—
359C2×#107—731no—
360C2×#107C2×D7311yescharacteristic
361C2×#106—731no—
362C2×C4⋊C8—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#102—731no—
367C2×#102C2×D7311yescharacteristic
368C2×#102C2×D7311yescharacteristic
369C2×#97—731no—
370C2×#97—731no—
371C2×C4×C8—731no—
372#389—731no—
373#389—731no—
374#389—731no—
375#389—1461no—
376#389—1461no—
377#389C2×D7311yescharacteristic
378#385—1461no—
379#385—1461no—
380#384—731no—
381#384—731no—
382#384—1461no—
383#380—1461no—
384C4×C4⋊C4—731no—
385C4×C4⋊C4—1461no—
386#340—731no—
387#340—731no—
388#340—731no—
389#340C2×D7311yescharacteristic
390#336—731no—
391#336C2×D7311yescharacteristic
392C22×Q8—731no—
393C22×D4C22×D7311yescharacteristic
394C23×C4—21no—
395C23×C4—1461no—
396C23×C4—1461no—
397C2×SD16—734no—
398C2×SD16—734no—
399C2×SD16—734no—
400C22×C8—731no—
401C22×C8—731no—
402C22×C8C22×D7311yescharacteristic
403C22×C8C22×D7311yescharacteristic
404C4×Q8—2921no—
405C4×Q8—2921no—
406C4×Q8—734no—
407C4×Q8—1462no—
408C4×Q8—1462no—
409C4×Q8—1462no—
410C4×D4—22no—
411C4×D4—41no—
412C4×D4—2921no—
413C4×D4—734no—
414C4×D4—1462no—
415C4×D4—1462no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—2924no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4—731no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
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—21no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C42—731no—
440C2×C42—731no—
441C2×C42—1461no—
442C2×C42—1461no—
443C2×C42—1461no—
444C2×C42C22×D7311yescharacteristic
445#107—41no—
446#107—2921no—
447#107—1462no—
448#107—1462no—
449#106—734no—
450C4⋊C8—1462no—
451#103—734no—
452#103—1462no—
453#103—1462no—
454#102—22no—
455#102—1462no—
456#102C4×D7314yes—
457#97—1462no—
458#97—1462no—
459C4×C8—734no—
460#95—21no—
461#95—731no—
462#95—731no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466C24—21no—
467C2×Q8—732no—
468C2×Q8—732no—
469C2×Q8—732no—
470C2×Q8—1461no—
471C2×Q8—1461no—
472C2×Q8—1461no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4C2×C4×D7312yes—
477C2×D4D4×D7312yes—
478C2×D4Q8×D7312yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—2922no—
486C22×C4—2922no—
487C22×C4—2922no—
488C22×C4—2922no—
489C22×C4—5841no—
490C22×C4—731no—
491C22×C4—731no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—1461no—
495C22×C4—1461no—
496C22×C4—1464no—
497C22×C4—1461no—
498C22×C4—1461no—
499C22×C4D4×D7311yescharacteristic
500C22×C4D4×D7311yescharacteristic
501C22×C4C23×D7311yescharacteristic
502SD16—7316no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—732no—
509C2×C8—732no—
510C2×C8—732no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8C2×C4×D7312yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—81no—
521C4⋊C4—2922no—
522C4⋊C4—2922no—
523C4⋊C4—2922no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—2922no—
527C4⋊C4—5841no—
528C4⋊C4—5841no—
529C4⋊C4—5841no—
530C4⋊C4—732no—
531C4⋊C4—1462no—
532C4⋊C4—1462no—
533C4⋊C4—1464no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1461no—
538C4⋊C4—1461no—
539C4⋊C4C2×C4×D7312yes—
540C22⋊C4—42no—
541C22⋊C4—2922no—
542C22⋊C4—1464no—
543C42—21no—
544C42—21no—
545C42—2922no—
546C42—2922no—
547C42—732no—
548C42—732no—
549C42—1464no—
550C42—1461no—
551C42—1461no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D7311yescharacteristic
560Q8—734no—
561Q8—1462no—
562Q8—1462no—
563Q8—1462no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4C4⋊C4×D7314yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2928no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—5844no—
594C2×C4—732no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1462no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1462no—
608C2×C4—1462no—
609C2×C4—1462no—
610C2×C4C22×C4×D7311yescharacteristic
611C2×C4C2×D4×D7311yescharacteristic
612C2×C4C2×Q8×D7311yescharacteristic
613C2×C4#41×D7311yescharacteristic
614C2×C4#1455663611yescharacteristic
615C2×C4#1455663811yescharacteristic
616C8—22no—
617C8—41no—
618C8—2921no—
619C8—734no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C4×D4×D7311yescharacteristic
628C22#1798921911yescharacteristic
629C22#1798925011yescharacteristic
630C22AGL(1,ℤ/8)×D7311yescharacteristic
631C22#137×D7311yescharacteristic
632C22#1798938811yescharacteristic
633C22#1798940111yescharacteristic
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—2922no—
640C4—2921no—
641C4—2921no—
642C4—5841no—
643C4—5841no—
644C4—734no—
645C4—1464no—
646C4—1462no—
647C4C2×C4⋊C4×D7311yescharacteristic
648C4#1798906611yescharacteristic
649C4#1798923511yescharacteristic
650C4#1798929111yescharacteristic
651C2—44no—
652C2#2261282911yescharacteristic
653C2#440×D7311yescharacteristic
654C2#2261880111yescharacteristic
655C2#2262154911yescharacteristic
656C2#2262157511yescharacteristic
657C2#2262283611yescharacteristic
658C2#2262284711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1862C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenter#1862D73
hypocenterC73#22671
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
4. #22612845C2semidirect
6. #22610673C2semidirect
7. #22610671C2semidirect
8. #22610188C2semidirect
9. #22610182C2semidirect
11. #22610089C2semidirect
12. #22609935C2semidirect
14. #22609797C2semidirect
25. #17988678C4semidirect
75. #14556556C2×C4semidirect
106. #14556522C2×C4semidirect
109. #14556516C2×C4semidirect
114. C8×Dic73C2×C4semidirect
156. #11924016C4⋊C4semidirect
285. Dic146C2×C4⋊C4semidirect
289. #10596831C2×C4⋊C4semidirect
320. Dic73#440semidirect
345. C73#22671semidirect

Maximal subgroups

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

subgroupindex
C2×#179886872
C2×#179886842
C2×#179886782
#226128452
D4×#119240072
#226106732
#226106712
#226101882
#226101822
#226101802
#226100892
#226099352
#226099292
#226097972
C73×#18622
#2267173