Finite Group Databasebeta

← #33242823

Subgroups — #33242823

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#22621618C211yesmaximal, characteristic
2#22612165C211yesmaximal, characteristic, complemented
3#22612128C211yesmaximal, characteristic, complemented
4#22612122C211yesmaximal, characteristic, complemented
5#22612085C211yesmaximal, characteristic, complemented
6#22612029C211yesmaximal, characteristic, complemented
7#22612020C211yesmaximal, characteristic, complemented
8#22611993C211yesmaximal, characteristic, complemented
9#22611869C211yesmaximal, characteristic, complemented
10#22611859C211yesmaximal, characteristic, complemented
11C4×#14556534C211yesmaximal, characteristic, complemented
12D4×#11924003C211yesmaximal, characteristic, complemented
13#22611636C211yesmaximal, characteristic, complemented
14C2×#17988387C211yesmaximal, characteristic, complemented
15C73×#3336C211yesmaximal, characteristic, complemented
16C2×#14556613C2211yescharacteristic
17C22×D584C2211yescharacteristic
18#17989233—21no—
19C2×#14556537C2211yescharacteristic, complemented
20C2×#14556534C2211yescharacteristic
21C2×#14556534C2211yescharacteristic
22C22×#11924003C2211yescharacteristic
23C22×#11924003C2211yescharacteristic, complemented
24#17988971C2211yescharacteristic
25#17988828C2211yescharacteristic
26#17988823C2211yescharacteristic
27#17988803—21no—
28#17988740—21no—
29#17988734—21no—
30C4×D584C2211yescharacteristic
31#17988414C2211yescharacteristic, complemented
32#17988414C2211yescharacteristic, complemented
33#17988412C2211yescharacteristic, complemented
34C4⋊#11924003C2211yescharacteristic, complemented
35#17988391C2211yescharacteristic, complemented
36#17988390C2211yescharacteristic, complemented
37#17988390C2211yescharacteristic, complemented
38#17988390C2211yescharacteristic, complemented
39#17988387C2211yescharacteristic
40#17988387C2211yescharacteristic
41#17988387C2211yescharacteristic
42#17988387C2211yescharacteristic
43#17988387C2211yescharacteristic
44#17988387C2211yescharacteristic
45C22⋊#11924003C2211yescharacteristic
46C22⋊#11924003C2211yescharacteristic, complemented
47C4×#11924003C2211yescharacteristic, complemented
48#552×C73—21no—
49#499×C73C2211yescharacteristic, complemented
50#485×C73—21no—
51#483×C73—21no—
52#479×C73—21no—
53#467×C73C2211yescharacteristic
54#467×C73C2211yescharacteristic, complemented
55C4×Q16×C73C2211yescharacteristic, complemented
56C8×D4×C73C2211yescharacteristic
57#427×C73C2211yescharacteristic, complemented
58C2×#107×C73C2211yescharacteristic
59C2×#11924028—21no—
60C22×D292—21no—
61#14556613—22no—
62#14556613—41no—
63C2×Dic292C2311yescharacteristic
64C2×#11924017—21no—
65C2×#11924017—21no—
66C2×D584—22no—
67C2×D584—21no—
68C2×D584—41no—
69C2×D584C2311yescharacteristic
70C2×D584C2311yescharacteristic
71#14556579—21no—
72#14556576—41no—
73#14556570—21no—
74#14556565—21no—
75#14556564—41no—
76#14556559—41no—
77C4×D292—21no—
78C22⋊C4×D73—41no—
79#14556537—41no—
80#14556534—22no—
81#14556534—41no—
82C2×#11924003—41no—
83C2×#11924003—41no—
84C2×#11924003C2311yescharacteristic
85C2×#11924003C2311yescharacteristic
86C2×#11924003C2311yescharacteristic
87C2×#11924003C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556516—21no—
90#14556513—21no—
91#14556513—21no—
92C2×Q16×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94C22×C8×C73C2311yescharacteristic
95#126×C73—41no—
96#125×C73—21no—
97#125×C73—41no—
98#123×C73—41no—
99C22⋊Q8×C73—21no—
100C22⋊Q8×C73—21no—
101C4×Q8×C73—21no—
102C4×D4×C73D411yescharacteristic
103#117×C73—21no—
104#117×C73—41no—
105C2×C4⋊C4×C73—21no—
106#107×C73—21no—
107#107×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#106×C73—21no—
111#106×C73—21no—
112C4⋊C8×C73D411yescharacteristic
113#103×C73—21no—
114#103×C73—21no—
115#103×C73—21no—
116C22⋊C8×C73D411yescharacteristic
117C22⋊C8×C73D411yescharacteristic
118C4×C8×C73C2311yescharacteristic
119C23×D73—41no—
120#11924028—42no—
121#11924028—81no—
122C2×Dic146—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292—42no—
126C2×D292—41no—
127C2×D292—81no—
128C2×C22⋊D73—41no—
129C2×C22⋊D73—41no—
130C2×C4×D73—41no—
131C2×C4×D73—41no—
132Dic292—21no—
133Dic292—21no—
134#11924017—42no—
135#11924017—81no—
136D584—21no—
137D584—21no—
138D584—21no—
139D584—42no—
140D584#4112yes—
141C4⋊Dic73—21no—
142#11924007—41no—
143C22⋊Dic73—41no—
144#11924005—41no—
145#11924005—41no—
146#11924005—41no—
147#11924005—41no—
148#11924003—21no—
149#11924003—21no—
150#11924003—21no—
151#11924003—21no—
152C2×Q8×C73—21no—
153C2×D4×C73D811yescharacteristic
154C22×C4×C73—41no—
155C22×C4×C73C2×D411yescharacteristic
156C22×C4×C73C2×D411yescharacteristic
157Q16×C73—21no—
158Q16×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73—21no—
161C2×C8×C73—21no—
162C2×C8×C73—21no—
163C2×C8×C73C2×D411yescharacteristic
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73C2411yescharacteristic
167C4⋊C4×C73—21no—
168C4⋊C4×C73—21no—
169C4⋊C4×C73—21no—
170C4⋊C4×C73—41no—
171C4⋊C4×C73—41no—
172C4⋊C4×C73—41no—
173C4⋊C4×C73—41no—
174C4⋊C4×C73—41no—
175C4⋊C4×C73—41no—
176C4⋊C4×C73D811yescharacteristic
177C22⋊C4×C73—41no—
178C22⋊C4×C73—41no—
179C22⋊C4×C73—41no—
180C22⋊C4×C73D811yescharacteristic
181C22⋊C4×C73D811yescharacteristic
182C42×C73—41no—
183C42×C73—41no—
184C42×C73C2×D411yescharacteristic
185C23×C73C2×D811yescharacteristic
186C23×C73C2×D811yescharacteristic
187C22×D73—41no—
188C22×D73—41no—
189C22×D73—82no—
190C22×D73—82no—
191C22×D73—161no—
192Q8×C73—41no—
193Q8×C73—41no—
194D4×C73—22no—
195C2×C4×C73—21no—
196C2×C4×C73—21no—
197C2×C4×C73—21no—
198C2×C4×C73—21no—
199C2×C4×C73—41no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73—41no—
203C2×C4×C73—82no—
204C2×C4×C73C2×D811yescharacteristic
205C2×C4×C73C2×D811yescharacteristic
206C2×C4×C73C2×D811yescharacteristic
207C2×C4×C73C2×D811yescharacteristic
208C2×C4×C73C22×D411yescharacteristic
209C22⋊D73—82no—
210C22⋊D73—161no—
211C2×Dic73—41no—
212C2×Dic73—41no—
213D292—22no—
214D292—41no—
215D292—41no—
216D292—41no—
217D292—82no—
218C4×D73—82no—
219C4×D73—161no—
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—81no—
246C4×C73—81no—
247C4×C73—81no—
248C4×C73—81no—
249C4×C73#54611yescharacteristic
250C4×C73#57711yescharacteristic
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#3336D7311yescharacteristic, complemented
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×C16—731no—
274#2138—731no—
275C2×#357—731no—
276C73#3386311yescharacteristic, complemented, Sylow
277C2×#135—731no—
278C22×D8—731no—
279#552—21no—
280#546—1461no—
281C2×Q32—731no—
282C2×SD32—731no—
283C2×SD32—731no—
284C22×C16—731no—
285C22×C16—731no—
286#499C2×D7311yescharacteristic, complemented
287#495—731no—
288#485—21no—
289#483—21no—
290#479—21no—
291#467—731no—
292#467C2×D7311yescharacteristic
293#467C2×D7311yescharacteristic, complemented
294#466—731no—
295#463—1461no—
296#450—1461no—
297#449—1461no—
298C4×Q16C2×D7311yescharacteristic, complemented
299C4×D8—731no—
300C8×D4C2×D7311yescharacteristic
301#427C2×D7311yescharacteristic, complemented
302C2×#107C2×D7311yescharacteristic
303#367—731no—
304#367—731no—
305#366—731no—
306C4⋊C16—731no—
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⋊C16—731no—
318C22⋊C16—731no—
319C4×C16—731no—
320C2×#41—1461no—
321C22×D4—1461no—
322#135—2921no—
323#135—1462no—
324C2×Q16—731no—
325C2×Q16C22×D7311yescharacteristic
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—41no—
336#125—21no—
337#125—41no—
338#124—1461no—
339#123—41no—
340#123—2921no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—1461no—
344#121—2921no—
345#121—1461no—
346#120—2921no—
347C4×Q8—21no—
348C4×D4—1461no—
349C4×D4C22⋊D7311yescharacteristic
350#117—21no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C22⋊C4—2921no—
354Q32—2921no—
355SD32—2921no—
356SD32—1462no—
357C2×C16—2921no—
358C2×C16—2921no—
359C2×C16—731no—
360C2×C16—731no—
361C2×C16—731no—
362C2×C16—731no—
363#107—21no—
364#107—731no—
365#107C22×D7311yescharacteristic
366#107C22×D7311yescharacteristic
367#107C22×D7311yescharacteristic
368#106—21no—
369#106—21no—
370C4⋊C8C22⋊D7311yescharacteristic
371#103—21no—
372#103—21no—
373#103—21no—
374#103—1461no—
375#102—1461no—
376#102—1461no—
377C22⋊C8C22⋊D7311yescharacteristic
378C22⋊C8C22⋊D7311yescharacteristic
379C4×C8C22×D7311yescharacteristic
380C24—2921no—
381#41—2922no—
382#41—5841no—
383C2×Q8—21no—
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×D4#1192401311yescharacteristic
393C22×C4—41no—
394C22×C4—2921no—
395C22×C4—2921no—
396C22×C4C2×C22⋊D7311yescharacteristic
397C22×C4C2×C22⋊D7311yescharacteristic
398Q16—21no—
399Q16—21no—
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×C22⋊D7311yescharacteristic
414C2×C8C2×C22⋊D7311yescharacteristic
415C2×C8C2×C22⋊D7311yescharacteristic
416C2×C8C23×D7311yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—2921no—
427C4⋊C4—1461no—
428C4⋊C4#1192401311yescharacteristic
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2921no—
433C22⋊C4—2921no—
434C22⋊C4—2921no—
435C22⋊C4—2921no—
436C22⋊C4—2921no—
437C22⋊C4#1192401311yescharacteristic
438C22⋊C4#1192401311yescharacteristic
439C42—41no—
440C42—41no—
441C42C2×C22⋊D7311yescharacteristic
442C16—1461no—
443C16—1461no—
444C16—1461no—
445C16—1461no—
446C23—2921no—
447C23—2921no—
448C23—5842no—
449C23—5842no—
450C23—11681no—
451C23C2×#1192401311yescharacteristic
452C23C2×#1192401311yescharacteristic
453Q8—41no—
454Q8—41no—
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—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—2921no—
475C2×C4—2921no—
476C2×C4—5842no—
477C2×C4—11681no—
478C2×C4C2×#1192401311yescharacteristic
479C2×C4C2×#1192401311yescharacteristic
480C2×C4C2×#1192401311yescharacteristic
481C2×C4C2×#1192401311yescharacteristic
482C2×C4C22×C22⋊D7311yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D7311yescharacteristic
487C8#1455663711yescharacteristic
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#1798901712yes—
501C22C22×#1192401311yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—5841no—
511C4—5841no—
512C4#1798922511yescharacteristic
513C4#1798939511yescharacteristic
514C2—22no—
515C2—41no—
516C2—5842no—
517C2—11681no—
518C2#2262173811yescharacteristic
519C2C2×#1798901711yescharacteristic
520C2#2262307511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3336C2
socleC22×C73C22×D8
prime radical#22621618C2
hypercenter#3336D73
hypocenterC73#33863
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #22612165C2semidirect
3. #22612128C2semidirect
4. #22612122C2semidirect
5. #22612085C2semidirect
6. #22612029C2semidirect
7. #22612020C2semidirect
8. #22611993C2semidirect
9. #22611869C2semidirect
10. #22611859C2semidirect
11. C4×#14556534C2semidirect
12. D4×#11924003C2semidirect
13. #22611636C2semidirect
14. C2×#17988387C2semidirect
15. C73×#3336C2semidirect
19. C2×#14556537C22semidirect
23. C22×#11924003C22semidirect
31. #17988414C22semidirect
32. #17988414C22semidirect
33. #17988412C22semidirect
34. C4⋊#11924003C22semidirect
35. #17988391C22semidirect
36. #17988390C22semidirect
37. #17988390C22semidirect
38. #17988390C22semidirect
46. C22⋊#11924003C22semidirect
47. C4×#11924003C22semidirect
49. #499×C73C22semidirect
54. #467×C73C22semidirect
55. C4×Q16×C73C22semidirect
57. #427×C73C22semidirect
261. #3336D73semidirect
276. C73#33863semidirect
286. #499C2×D73semidirect
293. #467C2×D73semidirect
298. C4×Q16C2×D73semidirect
301. #427C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226216182
#226121652
#226121282
#226121222
#226120852
#226120292
#226120202
#226119932
#226118692
#226118592
C4×#145565342
D4×#119240032
#226116362
C2×#179883872
C73×#33362
#3386373