Finite Group Databasebeta

← #33104354

Subgroups — #33104354

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988689C211yesmaximal, characteristic
2C2×#17988689C211yesmaximal, characteristic
3C2×C4×#11924017C211yesmaximal, characteristic
4C4⋊C4×Dic146C211yesmaximal, characteristic, complemented
5#22612842C211yesmaximal, characteristic
6#22610669C211yesmaximal, characteristic, complemented
7#22610664C211yesmaximal, characteristic, complemented
8#22610202C211yesmaximal, characteristic, complemented
9#22610195C211yesmaximal, characteristic
10#22610188C211yesmaximal, characteristic, complemented
11#22610181C211yesmaximal, characteristic
12#22610099C211yesmaximal, characteristic, complemented
13#22609947C211yesmaximal, characteristic, complemented
14#22609932C211yesmaximal, characteristic
15C73×#1771C211yesmaximal, characteristic, complemented
16C22×#11924017C2211yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C2×C4×Dic146C2211yescharacteristic
20C2×C4×D292C2211yescharacteristic
21C2×C4×D292C2211yescharacteristic
22C2×C4×D292C2211yescharacteristic
23#17988689—24no—
24#17988689C418yescomplemented
25C4×#11924017—24no—
26C2×#14556525C2211yescharacteristic
27C2×#14556525C2211yescharacteristic
28C2×#14556525C2211yescharacteristic
29C2×#14556523C2211yescharacteristic
30C2×#14556523C2211yescharacteristic
31C2×#14556516C2211yescharacteristic
32C2×#14556516C2211yescharacteristic
33C2×#14556516C2211yescharacteristic
34C2×#14556513C2211yescharacteristic
35C2×#14556513C2211yescharacteristic
36C2×#14556513C2211yescharacteristic
37#17988485C2211yescharacteristic
38#17988485C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988484—21no—
42#17988484—21no—
43#17988458—21no—
44#17988458—21no—
45#17988457C2211yescharacteristic
46#17988456—21no—
47#17988453C2211yescharacteristic
48#17988446—21no—
49C4⋊C4×Dic73—21no—
50#95×D73—21no—
51#17988350C2211yescharacteristic
52#17988350C2211yescharacteristic
53#17988350C2211yescharacteristic
54#17988350C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×#97×C73C2211yescharacteristic
57C2×#97×C73C2211yescharacteristic
58C2×C4×C8×C73C2211yescharacteristic
59C4×C4⋊C4×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61#336×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C22×D292C2311yescharacteristic
64C22×C4×D73—21no—
65C22×C4×D73—21no—
66C22×C4×D73—21no—
67C2×#11924017C2×C414yes—
68C2×#11924017D414yes—
69C2×#11924017Q814yes—
70C4×Dic146—22no—
71C4×Dic146—22no—
72C4×Dic146—22no—
73C4×Dic146—41no—
74C4×Dic146—41no—
75C4×Dic146C2×C414yescomplemented
76C4×D292—22no—
77C4×D292—22no—
78C4×D292—22no—
79C4×D292—41no—
80C4×D292—41no—
81C4×D292C2×C414yes—
82C2×C4⋊Dic73—21no—
83C2×C4⋊Dic73—21no—
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C4⋊Dic73C2311yescharacteristic
89C2×C4⋊Dic73C2311yescharacteristic
90C2×#11924007—21no—
91C2×#11924007—21no—
92C2×#11924007—21no—
93C2×#11924007—21no—
94C2×#11924007—21no—
95C2×#11924007—21no—
96C2×#11924007—21no—
97C2×#11924005—21no—
98C2×#11924005—21no—
99C2×#11924005—21no—
100C2×C4×Dic73—21no—
101C2×C4×Dic73—21no—
102C2×C4×Dic73—21no—
103#14556525—22no—
104#14556525—41no—
105#14556525C2×C414yescomplemented
106#14556523—22no—
107#14556523—41no—
108#14556516—22no—
109#14556516—22no—
110#14556516C2×C414yescomplemented
111#14556513—22no—
112#14556513—22no—
113#14556513C2×C414yes—
114#14556498—21no—
115#14556498—21no—
116#14556498—21no—
117#14556498—21no—
118#14556498—21no—
119#14556498—21no—
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C22×C8×C73C2311yescharacteristic
124C2×C4⋊C4×C73D411yescharacteristic
125C2×C4⋊C4×C73D411yescharacteristic
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2311yescharacteristic
129C4⋊C8×C73—22no—
130#97×C73—22no—
131#97×C73C2×C414yescomplemented
132C4×C8×C73—22no—
133#95×C73D411yescharacteristic
134#95×C73D411yescharacteristic
135C23×D73—21no—
136C2×Dic146—21no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146C22×C412yes—
140C2×Dic146C2×D412yes—
141C2×Dic146C2×Q812yes—
142C2×D292—21no—
143C2×D292—21no—
144C2×D292—21no—
145C2×D292C22×C412yes—
146C2×D292C2×D412yes—
147C2×D292C2×Q812yes—
148C22×Dic73—21no—
149C22×Dic73—21no—
150C22×Dic73—21no—
151C22×Dic73—21no—
152C22×Dic73—21no—
153C22×Dic73—21no—
154C22×Dic73—21no—
155C2×C4×D73—24no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73—42no—
159C2×C4×D73—42no—
160C2×C4×D73—81no—
161C2×C4×D73—81no—
162C2×C4×D73—81no—
163C2×C4×D73—81no—
164#11924017C4⋊C4116yescomplemented
165C4⋊Dic73—21no—
166C4⋊Dic73—21no—
167C4⋊Dic73—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73—21no—
171C4⋊Dic73—21no—
172C4⋊Dic73—21no—
173C4⋊Dic73—21no—
174C4⋊Dic73—21no—
175C4⋊Dic73—42no—
176C4⋊Dic73—81no—
177C4⋊Dic73C22×C412yes—
178C4⋊Dic73C22×C412yes—
179#11924007—24no—
180#11924007—42no—
181#11924007—42no—
182#11924007—81no—
183#11924007—81no—
184#11924007—81no—
185#11924007—81no—
186#11924005—24no—
187#11924005—42no—
188#11924005—42no—
189C4×Dic73—24no—
190C4×Dic73—42no—
191C4×Dic73—42no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2×D411yescharacteristic
195C22×C4×C73C2×D411yescharacteristic
196C22×C4×C73C2×D411yescharacteristic
197C22×C4×C73C2×D411yescharacteristic
198C22×C4×C73C2411yescharacteristic
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73—21no—
207C2×C8×C73C22×C412yes—
208C2×C8×C73C22×C412yes—
209C2×C8×C73C2×D412yes—
210C2×C8×C73C2×Q812yes—
211C4⋊C4×C73—22no—
212C4⋊C4×C73—41no—
213C42×C73—21no—
214C42×C73—21no—
215C42×C73—21no—
216C42×C73—21no—
217C42×C73C22×C412yes—
218C42×C73#4112yes—
219C23×C73C22×D411yescharacteristic
220C22×D73—22no—
221C22×D73—22no—
222C22×D73—22no—
223C22×D73—41no—
224C22×D73—41no—
225C22×D73—41no—
226C22×D73—41no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
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—21no—
242C2×C4×C73C4×D412yes—
243C2×C4×C73C4×D412yes—
244C2×C4×C73#13712yes—
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×Q811yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×C4×C73H(5,𝔽2)11yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—22no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272D292—22no—
273D292—22no—
274D292—22no—
275D292C2×C4⋊C414yes—
276C4×D73—84no—
277C4×D73—84no—
278C4×D73—84no—
279Dic146—22no—
280Dic146—22no—
281Dic146—22no—
282Dic146C2×C4⋊C414yes—
283C8×C73—22no—
284C8×C73—41no—
285C8×C73—41no—
286C8×C73C2×C4⋊C414yes—
287C22×C73C2×C4×D411yescharacteristic
288C22×C73D4211yescharacteristic
289C22×C73D4×Q811yescharacteristic
290C22×C73C2×#13711yescharacteristic
291C22×C73#57511yescharacteristic
292C22×C73#57611yescharacteristic
293C22×C73#57711yescharacteristic
294C2×D73—24no—
295C2×D73—42no—
296C2×D73—42no—
297C2×D73—42no—
298C2×D73—42no—
299C2×D73—42no—
300C2×D73—42no—
301C4×C73—22no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—41no—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73C22×C4⋊C411yescharacteristic
308C4×C73#52011yescharacteristic
309C4×C73#54711yescharacteristic
310C4×C73#55411yescharacteristic
311Dic73—24no—
312Dic73—42no—
313Dic73—42no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317Dic73—81no—
318#22678—731nomaximal, Sylow
319C2×C73D4×C4⋊C411yescharacteristic
320C2×C73C4×#13711yescharacteristic
321C2×C73#294611yescharacteristic
322C2×C73#326611yescharacteristic
323C2×C73#326811yescharacteristic
324C2×C73#338511yescharacteristic
325C2×C73#338811yescharacteristic
326D73—44no—
327C2×#440—731no—
328C2×#440—731no—
329C2×C4×SD16—731no—
330Q8×C4⋊C4—731no—
331#2349—731no—
332#1943—731no—
333#1942—731no—
334#1865—731no—
335#1864—731no—
336#1863—731no—
337#1862—731no—
338#1847—731no—
339#1805—731no—
340#1802—731no—
341#1771D7311yescharacteristic, complemented
342C73#2267811yescharacteristic, complemented, Sylow
343C22×SD16—731no—
344C2×C4×Q8—731no—
345C2×C4×Q8—731no—
346C2×C4×Q8—731no—
347C2×C4×D4—731no—
348C2×C4×D4—731no—
349C2×C4×D4—731no—
350#440—738no—
351#440—1464no—
352C4×SD16—1464no—
353C2×#107—731no—
354C2×#107—731no—
355C2×#107—731no—
356C2×#106—731no—
357C2×#106—731no—
358C2×C4⋊C8C2×D7311yescharacteristic
359C2×#103—731no—
360C2×#103—731no—
361C2×#103—731no—
362C2×#102—731no—
363C2×#102—731no—
364C2×#102—731no—
365C2×#97C2×D7311yescharacteristic
366C2×#97C2×D7311yescharacteristic
367C2×C4×C8C2×D7311yescharacteristic
368#389—731no—
369#389—731no—
370#389—731no—
371#389—731no—
372#389—1461no—
373#389—1461no—
374#385—1461no—
375#385—1461no—
376#384—731no—
377#384—1461no—
378#382—731no—
379#380—1461no—
380C4×C4⋊C4—1461no—
381C4×C4⋊C4C2×D7311yescharacteristic
382C2×#95—1461no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#336C2×D7311yescharacteristic
388#336C2×D7311yescharacteristic
389C22×Q8—731no—
390C22×D4—731no—
391C23×C4—1461no—
392C23×C4—1461no—
393C23×C4—1461no—
394C2×SD16—734no—
395C2×SD16—734no—
396C2×SD16—734no—
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C22×C8C22×D7311yescharacteristic
400C22×C8C22×D7311yescharacteristic
401C4×Q8—2921no—
402C4×Q8—2921no—
403C4×Q8—734no—
404C4×Q8—1462no—
405C4×Q8—1462no—
406C4×Q8—1462no—
407C4×D4—2921no—
408C4×D4—2921no—
409C4×D4—734no—
410C4×D4—1462no—
411C4×D4—1462no—
412C4×D4—1462no—
413C2×C4⋊C4—731no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—1461no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4D29211yescharacteristic
429C2×C4⋊C4D29211yescharacteristic
430C2×C22⋊C4—1461no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—1461no—
434C2×C42—1461no—
435C2×C42—1461no—
436C2×C42C22×D7311yescharacteristic
437C2×C42C22×D7311yescharacteristic
438C2×C42C22×D7311yescharacteristic
439#107—2921no—
440#107—734no—
441#107—1462no—
442#106—2921no—
443#106—1462no—
444C4⋊C8—22no—
445#103—734no—
446#103—1462no—
447#103—1462no—
448#102—734no—
449#102—1462no—
450#102—1462no—
451#97—22no—
452#97C4×D7314yescomplemented
453C4×C8—22no—
454#95—1461no—
455#95—1461no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95D29211yescharacteristic
461#95D29211yescharacteristic
462C24—1461no—
463C2×Q8—732no—
464C2×Q8—732no—
465C2×Q8—732no—
466C2×Q8—1461no—
467C2×Q8—1461no—
468C2×Q8—1461no—
469C2×D4—732no—
470C2×D4—732no—
471C2×D4—732no—
472C2×D4—1461no—
473C2×D4—1461no—
474C2×D4—1461no—
475C22×C4—2922no—
476C22×C4—2922no—
477C22×C4—2922no—
478C22×C4—2922no—
479C22×C4—5841no—
480C22×C4—5841no—
481C22×C4—5841no—
482C22×C4—5841no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1464no—
490C22×C4—1461no—
491C22×C4C2×D29211yescharacteristic
492C22×C4C2×D29211yescharacteristic
493C22×C4C2×D29211yescharacteristic
494C22×C4C2×D29211yescharacteristic
495C22×C4C2×D29211yescharacteristic
496C22×C4C2×D29211yescharacteristic
497C22×C4C23×D7311yescharacteristic
498SD16—7316no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8C2×C4×D7312yes—
508C2×C8C2×C4×D7312yes—
509C2×C8D4×D7312yes—
510C2×C8Q8×D7312yes—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—2922no—
514C4⋊C4—2922no—
515C4⋊C4—2922no—
516C4⋊C4—5841no—
517C4⋊C4—5841no—
518C4⋊C4—5841no—
519C4⋊C4—5841no—
520C4⋊C4—5841no—
521C4⋊C4—732no—
522C4⋊C4—732no—
523C4⋊C4—1461no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1464no—
532C4⋊C4—1461no—
533C4⋊C4—1461no—
534C22⋊C4—2922no—
535C22⋊C4—2922no—
536C22⋊C4—1464no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—2922no—
542C42—2922no—
543C42—1464no—
544C42C2×C4×D7312yes—
545C42#1192402812yes—
546C23—2921no—
547C23—2921no—
548C23—2921no—
549C23—2921no—
550C23—1462no—
551C23—1462no—
552C23—1462no—
553C23C22×D29211yescharacteristic
554Q8—734no—
555Q8—1462no—
556Q8—1462no—
557Q8—1462no—
558D4—734no—
559D4—1462no—
560D4—1462no—
561D4—1462no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—5844no—
594C2×C4—5844no—
595C2×C4—5844no—
596C2×C4—1462no—
597C2×C4—1462no—
598C2×C4—1462no—
599C2×C4—1462no—
600C2×C4—1462no—
601C2×C4C4×D29212yes—
602C2×C4C4×D29212yes—
603C2×C4#1455662412yes—
604C2×C4C22×C4×D7311yescharacteristic
605C2×C4C2×D4×D7311yescharacteristic
606C2×C4C2×Q8×D7311yescharacteristic
607C2×C4C2×#1192402811yescharacteristic
608C2×C4#1455663711yescharacteristic
609C2×C4#1455663811yescharacteristic
610C8—22no—
611C8—41no—
612C8—41no—
613C8C4⋊C4×D7314yes—
614C22—2922no—
615C22—2922no—
616C22—2922no—
617C22—2922no—
618C22—2922no—
619C22—2922no—
620C22—1464no—
621C22C2×C4×D29211yescharacteristic
622C22D4×D29211yescharacteristic
623C22Q8×D29211yescharacteristic
624C22C2×#1455662411yescharacteristic
625C22#1798938711yescharacteristic
626C22#1798939211yescharacteristic
627C22#1798939911yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—2922no—
635C4—2922no—
636C4—5841no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4—1464no—
641C4C2×C4⋊C4×D7311yescharacteristic
642C4#1798906711yescharacteristic
643C4#1798924211yescharacteristic
644C4#1798929711yescharacteristic
645C2—2924no—
646C2C4⋊C4×D29211yescharacteristic
647C2C4×#1455662411yescharacteristic
648C2#2261876511yescharacteristic
649C2#2262144611yescharacteristic
650C2#2262147611yescharacteristic
651C2#2262284311yescharacteristic
652C2#2262287811yescharacteristic
subgroupG/H
centerC23C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1771C2
socleC23×C73C22×D4
prime radicalC22×D292C23
cosocleC22×C4×C73C24
hypercenter#1771D73
hypocenterC73#22678
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic146C2semidirect
6. #22610669C2semidirect
7. #22610664C2semidirect
8. #22610202C2semidirect
10. #22610188C2semidirect
12. #22610099C2semidirect
13. #22609947C2semidirect
15. C73×#1771C2semidirect
24. #17988689C4semidirect
75. C4×Dic146C2×C4semidirect
105. #14556525C2×C4semidirect
110. #14556516C2×C4semidirect
131. #97×C73C2×C4semidirect
164. #11924017C4⋊C4semidirect
341. #1771D73semidirect
342. C73#22678semidirect
452. #97C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886892
C2×C4×#119240172
C4⋊C4×Dic1462
#226128422
#226106692
#226106642
#226102022
#226101952
#226101882
#226101812
#226100992
#226099472
#226099322
C73×#17712
#2267873