Finite Group Databasebeta

← #33242831

Subgroups — #33242831

Order 18688.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622300C211yesmaximal, characteristic, complemented
2#22621618C211yesmaximal, characteristic
3#22612163C211yesmaximal, characteristic, complemented
4#22612129C211yesmaximal, characteristic, complemented
5#22612123C211yesmaximal, characteristic, complemented
6#22612089C211yesmaximal, characteristic, complemented
7#22612027C211yesmaximal, characteristic, complemented
8#22612022C211yesmaximal, characteristic, complemented
9#22611994C211yesmaximal, characteristic, complemented
10#22611873C211yesmaximal, characteristic, complemented
11#22611862C211yesmaximal, characteristic, complemented
12C4×#14556536C211yesmaximal, characteristic, complemented
13#22611639C211yesmaximal, characteristic, complemented
14C2×#17988389C211yesmaximal, characteristic, complemented
15D4×C16×C73C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17C22×D584C2211yescharacteristic
18#17989277—21no—
19#17989233—21no—
20C2×Dic584C2211yescharacteristic, complemented
21C2×#14556536C2211yescharacteristic
22C2×#14556536C2211yescharacteristic
23#17988989C2211yescharacteristic, complemented
24#17988971C2211yescharacteristic
25#17988929—21no—
26#17988918—21no—
27#17988895—21no—
28#17988828C2211yescharacteristic
29#17988828C2211yescharacteristic
30#17988828C2211yescharacteristic, complemented
31#17988823C2211yescharacteristic
32#17988803—21no—
33#17988740—21no—
34#17988734—21no—
35C4×Dic292C2211yescharacteristic, complemented
36C4×D584C2211yescharacteristic
37#17988625C2211yescharacteristic, complemented
38C2×#14556525C2211yescharacteristic
39#17988415C2211yescharacteristic, complemented
40#17988415C2211yescharacteristic, complemented
41#17988413C2211yescharacteristic, complemented
42#17988392C2211yescharacteristic, complemented
43#17988392C2211yescharacteristic, complemented
44#17988392C2211yescharacteristic, complemented
45#17988392C2211yescharacteristic, complemented
46#17988389C2211yescharacteristic
47#17988389C2211yescharacteristic
48#17988389C2211yescharacteristic
49#17988389C2211yescharacteristic
50#17988389C2211yescharacteristic
51#17988389C2211yescharacteristic
52C22×C16×C73C2211yescharacteristic
53C22×C16×C73C2211yescharacteristic, complemented
54C8×D4×C73C2211yescharacteristic
55C4⋊C16×C73C2211yescharacteristic, complemented
56C22⋊C16×C73C2211yescharacteristic
57C22⋊C16×C73C2211yescharacteristic, complemented
58C4×C16×C73C2211yescharacteristic, complemented
59C2×#11924028—21no—
60C22×D292—21no—
61#14556613—22no—
62#14556613—41no—
63C2×Dic292C2311yescharacteristic
64C2×Dic292C2311yescharacteristic
65C2×#11924017—21no—
66C2×#11924017—21no—
67C2×D584—22no—
68C2×D584—21no—
69C2×D584—41no—
70C2×D584C2311yescharacteristic
71C2×D584C2311yescharacteristic
72#14556584—41no—
73#14556583—21no—
74#14556581—41no—
75#14556579—21no—
76#14556576—41no—
77#14556572—41no—
78#14556570—21no—
79#14556570—21no—
80#14556570—21no—
81#14556565—21no—
82#14556564—41no—
83#14556559—41no—
84C4×Dic146—21no—
85C4×D292—21no—
86#14556550—21no—
87#14556547—41no—
88C2×C4⋊Dic73—21no—
89C22⋊C4×D73—41no—
90Dic584—41no—
91#14556536—22no—
92#14556536—41no—
93#14556525—21no—
94#14556525C2311yescharacteristic
95#14556525C2311yescharacteristic
96#14556525C2311yescharacteristic
97#14556525C2311yescharacteristic
98#14556523—21no—
99#14556523—21no—
100#14556516—21no—
101#14556516—21no—
102#14556516—21no—
103#14556516—21no—
104#14556513—21no—
105#14556513—21no—
106C22×C8×C73C2311yescharacteristic
107C22×C8×C73C2311yescharacteristic
108C4×D4×C73D411yescharacteristic
109C2×C16×C73—41no—
110C2×C16×C73—41no—
111C2×C16×C73C2311yescharacteristic
112C2×C16×C73C2311yescharacteristic
113C2×C16×C73C2311yescharacteristic
114C2×C16×C73C2311yescharacteristic
115C4⋊C8×C73D411yescharacteristic
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119C23×D73—41no—
120#11924028—42no—
121#11924028—81no—
122C2×Dic146—21no—
123C2×Dic146—21no—
124C2×D292—21no—
125C2×D292—21no—
126C2×D292—42no—
127C2×D292—41no—
128C2×D292—81no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C22×Dic73—41no—
132C2×C4×D73—41no—
133C2×C4×D73—41no—
134Dic292—21no—
135Dic292—21no—
136Dic292—21no—
137Dic292—21no—
138#11924017—42no—
139#11924017—81no—
140D584—21no—
141D584—21no—
142D584—21no—
143D584—42no—
144D584#4112yes—
145C4⋊Dic73—21no—
146C4⋊Dic73—21no—
147C4⋊Dic73—21no—
148C4⋊Dic73—21no—
149C4⋊Dic73—41no—
150#11924007—41no—
151#11924007—41no—
152#11924007—41no—
153#11924007—41no—
154#11924007—41no—
155#11924007—41no—
156C22⋊Dic73—41no—
157C22⋊Dic73—41no—
158C22⋊Dic73—41no—
159C22⋊Dic73—41no—
160#11924005—41no—
161#11924005—41no—
162#11924005—41no—
163#11924005—41no—
164C4×Dic73—41no—
165C4×Dic73—41no—
166C2×D4×C73D811yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C2×C8×C73—21no—
170C2×C8×C73—21no—
171C2×C8×C73—21no—
172C2×C8×C73—21no—
173C2×C8×C73C2×D411yescharacteristic
174C2×C8×C73C2×D411yescharacteristic
175C2×C8×C73C2×D411yescharacteristic
176C2×C8×C73C2411yescharacteristic
177C4⋊C4×C73D811yescharacteristic
178C22⋊C4×C73D811yescharacteristic
179C22⋊C4×C73D811yescharacteristic
180C42×C73C2×D411yescharacteristic
181C16×C73—21no—
182C16×C73—21no—
183C16×C73—21no—
184C16×C73—21no—
185C23×C73C2×D811yescharacteristic
186C23×C73C2×D811yescharacteristic
187C22×D73—41no—
188C22×D73—41no—
189C22×D73—82no—
190C22×D73—82no—
191C22×D73—161no—
192D4×C73—22no—
193C2×C4×C73—21no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73C2×D811yescharacteristic
198C2×C4×C73C2×D811yescharacteristic
199C2×C4×C73C2×D811yescharacteristic
200C2×C4×C73C2×D811yescharacteristic
201C2×C4×C73C22×D411yescharacteristic
202C22⋊D73—82no—
203C22⋊D73—161no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73—41no—
208C2×Dic73—41no—
209C2×Dic73—41no—
210C2×Dic73—82no—
211D292—22no—
212D292—41no—
213D292—41no—
214D292—41no—
215D292—82no—
216C4×D73—82no—
217C4×D73—161no—
218Dic146—41no—
219Dic146—41no—
220Dic146—41no—
221Dic146—41no—
222C8×C73—21no—
223C8×C73—21no—
224C8×C73—21no—
225C8×C73C2×#4111yescharacteristic
226C8×C73H(5,𝔽2)11yescharacteristic
227C22×C73—21no—
228C22×C73—21no—
229C22×C73—21no—
230C22×C73—21no—
231C22×C73—21no—
232C22×C73#50812yes—
233C22×C73C22×D811yescharacteristic
234C2×D73—42no—
235C2×D73—81no—
236C2×D73—81no—
237C2×D73—81no—
238C2×D73—81no—
239C2×D73—81no—
240C2×D73—164no—
241C4×C73—21no—
242C4×C73—21no—
243C4×C73—21no—
244C4×C73—21no—
245C4×C73#54611yescharacteristic
246C4×C73#57711yescharacteristic
247Dic73—81no—
248Dic73—81no—
249Dic73—81no—
250Dic73—81no—
251Dic73—81no—
252Dic73—81no—
253#33863—731nomaximal, Sylow
254C2×C73—22no—
255C2×C73—41no—
256C2×C73#329411yescharacteristic
257C2×C73C2×#50811yescharacteristic
258C2×C73#341611yescharacteristic
259D73—82no—
260D73—161no—
261#3336—731no—
262#3280—731no—
263#2242—731no—
264#2235—731no—
265#2234—731no—
266#2228—731no—
267#2217—731no—
268#2216—731no—
269#2211—731no—
270#2189—731no—
271#2187—731no—
272C4×SD32—731no—
273D4×C16D7311yescharacteristic, complemented
274#2138—731no—
275C2×#357—731no—
276C73#3386311yescharacteristic, complemented, Sylow
277C2×#135—731no—
278C22×D8—731no—
279#552—1461no—
280#546—1461no—
281C2×Q32—731no—
282C2×SD32—731no—
283C2×SD32—731no—
284C22×C16C2×D7311yescharacteristic
285C22×C16C2×D7311yescharacteristic, complemented
286#499—731no—
287#495—731no—
288#485—1461no—
289#483—1461no—
290#479—1461no—
291#467—731no—
292#467—731no—
293#467—731no—
294#466—731no—
295#463—1461no—
296#450—1461no—
297#449—1461no—
298C4×Q16—731no—
299C4×D8—731no—
300C8×D4C2×D7311yescharacteristic
301#427—731no—
302C2×#107—731no—
303#367—731no—
304#367—731no—
305#366—731no—
306C4⋊C16C2×D7311yescharacteristic, complemented
307#358—731no—
308#358—731no—
309#358—731no—
310#358—731no—
311#357—731no—
312#357—731no—
313#357—731no—
314#357—731no—
315#357—731no—
316#357—731no—
317C22⋊C16C2×D7311yescharacteristic
318C22⋊C16C2×D7311yescharacteristic, complemented
319C4×C16C2×D7311yescharacteristic, complemented
320C2×#41—1461no—
321C22×D4—1461no—
322#135—2921no—
323#135—1462no—
324C2×Q16—731no—
325C2×Q16—731no—
326C2×SD16—1461no—
327C2×SD16—1461no—
328C2×D8—2921no—
329C2×D8—731no—
330C2×D8—731no—
331C2×D8—1462no—
332C2×D8—1461no—
333C22×C8C22×D7311yescharacteristic
334C22×C8C22×D7311yescharacteristic
335#126—2921no—
336#125—2921no—
337#125—1461no—
338#124—1461no—
339#123—2921no—
340#123—2921no—
341C22⋊Q8—1461no—
342C22⋊Q8—1461no—
343C22⋊Q8—1461no—
344#121—2921no—
345#121—1461no—
346#120—2921no—
347C4×Q8—1461no—
348C4×D4—1461no—
349C4×D4D29211yescharacteristic
350#117—2921no—
351#117—1461no—
352C2×C4⋊C4—1461no—
353C2×C22⋊C4—2921no—
354Q32—2921no—
355SD32—2921no—
356SD32—1462no—
357C2×C16—41no—
358C2×C16—41no—
359C2×C16C22×D7311yescharacteristic
360C2×C16C22×D7311yescharacteristic
361C2×C16C22×D7311yescharacteristic
362C2×C16C22×D7311yescharacteristic
363#107—731no—
364#107—731no—
365#107—731no—
366#107—731no—
367#107—1461no—
368#106—1461no—
369#106—1461no—
370C4⋊C8D29211yescharacteristic
371#103—1461no—
372#103—1461no—
373#103—1461no—
374#103—1461no—
375#102—1461no—
376#102—1461no—
377C22⋊C8D29211yescharacteristic
378C22⋊C8D29211yescharacteristic
379C4×C8C22×D7311yescharacteristic
380C24—2921no—
381#41—2922no—
382#41—5841no—
383C2×Q8—1461no—
384C2×Q8—1461no—
385C2×D4—2922no—
386C2×D4—2921no—
387C2×D4—2921no—
388C2×D4—2921no—
389C2×D4—5841no—
390C2×D4—1461no—
391C2×D4—1461no—
392C2×D4D58411yescharacteristic
393C22×C4—2921no—
394C22×C4—2921no—
395C22×C4—2921no—
396C22×C4C2×D29211yescharacteristic
397C22×C4C2×D29211yescharacteristic
398Q16—1461no—
399Q16—1461no—
400Q16—1461no—
401Q16—1461no—
402SD16—2922no—
403SD16—5841no—
404D8—2922no—
405D8—732no—
406D8—1461no—
407D8—1461no—
408D8—1461no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D29211yescharacteristic
414C2×C8C2×D29211yescharacteristic
415C2×C8C2×D29211yescharacteristic
416C2×C8C23×D7311yescharacteristic
417C4⋊C4—2921no—
418C4⋊C4—2921no—
419C4⋊C4—2921no—
420C4⋊C4—2921no—
421C4⋊C4—2921no—
422C4⋊C4—2921no—
423C4⋊C4—2921no—
424C4⋊C4—1461no—
425C4⋊C4—1461no—
426C4⋊C4—1461no—
427C4⋊C4—1461no—
428C4⋊C4D58411yescharacteristic
429C22⋊C4—2921no—
430C22⋊C4—2921no—
431C22⋊C4—2921no—
432C22⋊C4—2921no—
433C22⋊C4—2921no—
434C22⋊C4—2921no—
435C22⋊C4—2921no—
436C22⋊C4—2921no—
437C22⋊C4D58411yescharacteristic
438C22⋊C4D58411yescharacteristic
439C42—2921no—
440C42—2921no—
441C42C2×D29211yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—2921no—
447C23—2921no—
448C23—5842no—
449C23—5842no—
450C23—11681no—
451C23C2×D58411yescharacteristic
452C23C2×D58411yescharacteristic
453Q8—2921no—
454Q8—2921no—
455Q8—2921no—
456Q8—2921no—
457D4—22no—
458D4—2921no—
459D4—2921no—
460D4—2921no—
461D4—5842no—
462D4—5842no—
463D4—11681no—
464D4—1462no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—2921no—
470C2×C4—2921no—
471C2×C4—2921no—
472C2×C4—2921no—
473C2×C4—2921no—
474C2×C4—2921no—
475C2×C4—5842no—
476C2×C4—5842no—
477C2×C4—11681no—
478C2×C4C2×D58411yescharacteristic
479C2×C4C2×D58411yescharacteristic
480C2×C4C2×D58411yescharacteristic
481C2×C4C2×D58411yescharacteristic
482C2×C4C22×D29211yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D7311yescharacteristic
487C8#1455663611yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—2922no—
494C22—5841no—
495C22—5841no—
496C22—5841no—
497C22—5841no—
498C22—5841no—
499C22—11684no—
500C22#1798901912yes—
501C22C22×D58411yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—5841no—
507C4—5841no—
508C4—5841no—
509C4—5841no—
510C4—5841no—
511C4—5841no—
512C4#1798923211yescharacteristic
513C4#1798939911yescharacteristic
514C2—22no—
515C2—41no—
516C2—5842no—
517C2—11681no—
518C2#2262174511yescharacteristic
519C2C2×#1798901911yescharacteristic
520C2#2262307911yescharacteristic
subgroupG/H
centerC22C22×D584
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingD4×C16×C73C2
socleC22×C73C22×D8
prime radical#22621618C2
hypercenterD4×C16D73
hypocenterC73#33863
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #22622300C2semidirect
3. #22612163C2semidirect
4. #22612129C2semidirect
5. #22612123C2semidirect
6. #22612089C2semidirect
7. #22612027C2semidirect
8. #22612022C2semidirect
9. #22611994C2semidirect
10. #22611873C2semidirect
11. #22611862C2semidirect
12. C4×#14556536C2semidirect
13. #22611639C2semidirect
14. C2×#17988389C2semidirect
15. D4×C16×C73C2semidirect
20. C2×Dic584C22semidirect
23. #17988989C22semidirect
30. #17988828C22semidirect
35. C4×Dic292C22semidirect
37. #17988625C22semidirect
39. #17988415C22semidirect
40. #17988415C22semidirect
41. #17988413C22semidirect
42. #17988392C22semidirect
43. #17988392C22semidirect
44. #17988392C22semidirect
45. #17988392C22semidirect
53. C22×C16×C73C22semidirect
55. C4⋊C16×C73C22semidirect
57. C22⋊C16×C73C22semidirect
58. C4×C16×C73C22semidirect
273. D4×C16D73semidirect
276. C73#33863semidirect
285. C22×C16C2×D73semidirect
306. C4⋊C16C2×D73semidirect
318. C22⋊C16C2×D73semidirect
319. C4×C16C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226223002
#226216182
#226121632
#226121292
#226121232
#226120892
#226120272
#226120222
#226119942
#226118732
#226118622
C4×#145565362
#226116392
C2×#179883892
D4×C16×C732
#3386373