Finite Group Databasebeta

← #33069985

Subgroups — #33069985

Order 18688.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619724C211yesmaximal, characteristic, complemented
2#22619391C211yesmaximal, characteristic
3C2×#17988666C211yesmaximal, characteristic, complemented
4#22609475C212yesmaximal, complemented
5#22609461C212yesmaximal, complemented
6#22609427C212yesmaximal, complemented
7#22609393C212yesmaximal, complemented
8#22608690C211yesmaximal, characteristic, complemented
9#22608621C211yesmaximal, characteristic, complemented
10#22608608C211yesmaximal, characteristic, complemented
11C73×#3167C211yesmaximal, characteristic, complemented
12C2×#14556616—21no—
13C22×#11924011—21no—
14C2×C4×D292C2211yescharacteristic
15#17988884—22no—
16#17988874C2212yes—
17#17988864C2212yes—
18#17988799—22no—
19#17988789C2212yes—
20#17988779C2212yes—
21#17988666—21no—
22#17988666—21no—
23#17988666C2212yes—
24#17988666C2212yes—
25#17988630—21no—
26#17988611C2211yescharacteristic
27#17988611C2211yescharacteristic
28C2×C4⋊#10596831C2211yescharacteristic
29#17988591—21no—
30#17988588—21no—
31C2×#14556503—21no—
32#17988340C2212yescomplemented
33#17988340C2212yescomplemented
34#17988336C2212yescomplemented
35#17988333C2212yescomplemented
36#17988333C2212yescomplemented
37#17988331C2212yescomplemented
38#533×C73C2211yescharacteristic
39#532×C73C2211yescharacteristic
40#498×C73C2211yescharacteristic, complemented
41#492×C73C2211yescharacteristic, complemented
42#487×C73C2211yescharacteristic, complemented
43#486×C73C2211yescharacteristic, complemented
44#465×C73—21no—
45#465×C73—21no—
46C4×SD16×C73—21no—
47C4×SD16×C73—21no—
48C2×C4×C8×C73C2211yescharacteristic
49C22×D292D411yescharacteristic
50C22×C4×D73—21no—
51#14556616—44no—
52C2×#11924015—22no—
53C2×#11924013—22no—
54C2×#11924012—21no—
55C2×#11924012—21no—
56C2×#11924011—22no—
57C2×#11924011—22no—
58C2×#11924011—44no—
59C22×#10596831—21no—
60C4×D292—21no—
61C4×D292—21no—
62C4×D292D412yes—
63C4×D292C2312yes—
64C2×C4⋊Dic73D411yescharacteristic
65C2×#11924005—21no—
66#14556524—22no—
67#14556522—22no—
68C4⋊#10596831—21no—
69C4⋊#10596831—21no—
70C4⋊#10596831—21no—
71C4⋊#10596831C2312yes—
72C4⋊#10596831C2312yes—
73C4⋊#10596831C2312yes—
74#14556515—22no—
75#14556512—22no—
76#14556511—22no—
77#14556511—22no—
78#14556503—22no—
79#14556503—22no—
80C2×#41×C73—21no—
81C2×SD16×C73—21no—
82C2×SD16×C73—21no—
83C22×C8×C73D411yescharacteristic
84C22×C8×C73D411yescharacteristic
85C4⋊Q8×C73C2311yescharacteristic
86C4⋊Q8×C73C2311yescharacteristic
87C4⋊Q8×C73C2311yescharacteristic
88#127×C73C2311yescharacteristic
89#125×C73—21no—
90#124×C73—21no—
91C22⋊Q8×C73—21no—
92C22⋊Q8×C73—21no—
93#121×C73—21no—
94#121×C73—21no—
95C4×Q8×C73—21no—
96C4×Q8×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73—21no—
99#117×C73—21no—
100C2×C42×C73C2311yescharacteristic
101#106×C73—21no—
102#106×C73—21no—
103#103×C73—21no—
104#103×C73—21no—
105#102×C73—21no—
106#102×C73—21no—
107C4×C8×C73D411yescharacteristic
108C4×C8×C73D411yescharacteristic
109C4×C8×C73C2311yescharacteristic
110C4×C8×C73C2311yescharacteristic
111C23×D73—21no—
112C2×D292—22no—
113C2×D292—22no—
114C2×D292—21no—
115C2×D292C2×D412yes—
116C22×Dic73—21no—
117C2×C4×D73—22no—
118C2×C4×D73—22no—
119C2×C4×D73—42no—
120C2×C4×D73—42no—
121#11924015—44no—
122#11924013—44no—
123#11924012—42no—
124#11924012—42no—
125#11924011—44no—
126#11924011—44no—
127C2×#10596831—22no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—42no—
131C4⋊Dic73—21no—
132C4⋊Dic73C2×D412yes—
133#11924005—22no—
134#11924005—22no—
135#41×C73—42no—
136#41×C73—42no—
137C2×Q8×C73—21no—
138C2×Q8×C73—21no—
139C2×Q8×C73—21no—
140C2×D4×C73—21no—
141C2×D4×C73—21no—
142C2×D4×C73—21no—
143C2×D4×C73—41no—
144C22×C4×C73—41no—
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C22×C4×C73#4111yescharacteristic
148SD16×C73—42no—
149SD16×C73—42no—
150C2×C8×C73—22no—
151C2×C8×C73—41no—
152C2×C8×C73C2×D411yescharacteristic
153C2×C8×C73C2×D411yescharacteristic
154C2×C8×C73C2×D411yescharacteristic
155C2×C8×C73C2×D411yescharacteristic
156C4⋊C4×C73—21no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—41no—
163C4⋊C4×C73—41no—
164C22⋊C4×C73—41no—
165C22⋊C4×C73—41no—
166C22⋊C4×C73—41no—
167C42×C73—41no—
168C42×C73C2×D411yescharacteristic
169C42×C73C2×D411yescharacteristic
170C42×C73#4111yescharacteristic
171C42×C73C2411yescharacteristic
172C23×C73—41no—
173C23×C73#12111yescharacteristic
174C22×D73—22no—
175C22×D73—42no—
176C22×D73—42no—
177C22×D73—42no—
178Q8×C73—41no—
179Q8×C73—41no—
180Q8×C73—41no—
181Q8×C73—41no—
182Q8×C73—81no—
183D4×C73—41no—
184D4×C73—41no—
185D4×C73—41no—
186D4×C73—41no—
187D4×C73—81no—
188D4×C73—82no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—41no—
192C2×C4×C73—41no—
193C2×C4×C73—41no—
194C2×C4×C73—41no—
195C2×C4×C73—82no—
196C2×C4×C73#12111yescharacteristic
197C2×C4×C73#12111yescharacteristic
198C2×C4×C73#12111yescharacteristic
199C2×C4×C73AGL(1,ℤ/8)12yes—
200C2×C4×C73#13712yes—
201C2×C4×C73C22×D411yescharacteristic
202C2×C4×C73C22×D411yescharacteristic
203C2×C4×C73C2×#4111yescharacteristic
204C2×Dic73—22no—
205C2×Dic73—42no—
206D292—22no—
207D292—22no—
208D292—42no—
209C4×D73—44no—
210C4×D73—42no—
211C4×D73—42no—
212C8×C73—21no—
213C8×C73—21no—
214C8×C73—21no—
215C8×C73AGL(1,ℤ/8)12yescomplemented
216#10596831—42no—
217#10596831—42no—
218#10596831—42no—
219C22×C73—21no—
220C22×C73—21no—
221C22×C73—81no—
222C22×C73—81no—
223C22×C73—81no—
224C22×C73#46812yes—
225C22×C73C2×#12111yescharacteristic
226C2×D73—42no—
227C2×D73—42no—
228C2×D73—42no—
229C2×D73—84no—
230C4×C73—21no—
231C4×C73—21no—
232C4×C73—21no—
233C4×C73—81no—
234C4×C73—81no—
235C4×C73—81no—
236C4×C73#46812yes—
237C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
239C4×C73C2×#13711yescharacteristic
240C4×C73#57511yescharacteristic
241Dic73—42no—
242#19279—731nomaximal, Sylow
243C2×C73—22no—
244C2×C73—161no—
245C2×C73#303611yescharacteristic
246C2×C73C2×#46811yescharacteristic
247C2×C73#306711yescharacteristic
248D73—82no—
249#3167D7311yescharacteristic, complemented
250#3077—731no—
251#3036—731no—
252C2×#436—731no—
253#1699—732no—
254#1697—732no—
255#1691—732no—
256#1685—732no—
257#1567—731no—
258#1550—731no—
259#1547—731no—
260C73#1927911yescharacteristic, complemented, Sylow
261C2×AGL(1,ℤ/8)—1461no—
262C22×M16—1461no—
263#533C2×D7311yescharacteristic
264#532C2×D7311yescharacteristic
265C2×C4×D4—731no—
266#498C2×D7311yescharacteristic, complemented
267#492C2×D7311yescharacteristic, complemented
268#487C2×D7311yescharacteristic, complemented
269#486C2×D7311yescharacteristic, complemented
270#478—1462no—
271#476—732no—
272#474—732no—
273#465—21no—
274#465—21no—
275#463—1462no—
276#461—732no—
277#459—732no—
278C4×SD16—21no—
279C4×SD16—21no—
280#436—732no—
281#436—732no—
282#436—1461no—
283#436—1461no—
284#428—1461no—
285#423—731no—
286#423—731no—
287C2×C4⋊C8—731no—
288#418—1461no—
289#417—1461no—
290C2×C22⋊C8—1461no—
291C2×C4×C8C2×D7311yescharacteristic
292#335—732no—
293#335—732no—
294#333—732no—
295#332—732no—
296#332—732no—
297#331—732no—
298C2×#41—21no—
299C22×D4—731no—
300C23×C4—1461no—
301AGL(1,ℤ/8)—2924no—
302C2×SD16—21no—
303C2×SD16—21no—
304C2×SD16—1462no—
305C2×D8—1462no—
306C2×M16—2924no—
307C2×M16—1461no—
308C2×M16—1461no—
309C2×M16—1462no—
310C2×M16—1462no—
311C22×C8—1461no—
312C22×C8C22⋊D7311yescharacteristic
313C22×C8C22⋊D7311yescharacteristic
314C4⋊Q8C22×D7311yescharacteristic
315C4⋊Q8C22×D7311yescharacteristic
316C4⋊Q8C22×D7311yescharacteristic
317#127C22×D7311yescharacteristic
318#125—21no—
319#124—21no—
320C22⋊Q8—21no—
321C22⋊Q8—21no—
322#121—21no—
323#121—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—21no—
327C4×D4—21no—
328C4×D4—732no—
329C4×D4—732no—
330C4×D4—1461no—
331C4×D4—1461no—
332#117—21no—
333C2×C4⋊C4—731no—
334C2×C22⋊C4—1461no—
335C2×C42C22×D7311yescharacteristic
336#107—1462no—
337#106—21no—
338#106—21no—
339#106—1462no—
340C4⋊C8—732no—
341C4⋊C8—732no—
342C4⋊C8—732no—
343C4⋊C8—1461no—
344C4⋊C8—1461no—
345C4⋊C8—1461no—
346#103—21no—
347#103—21no—
348#103—1462no—
349#102—21no—
350#102—21no—
351#102—1462no—
352#102—1462no—
353#102—1462no—
354C22⋊C8—1462no—
355C22⋊C8—1462no—
356C4×C8C22⋊D7311yescharacteristic
357C4×C8C22⋊D7311yescharacteristic
358C4×C8C22×D7311yescharacteristic
359C4×C8C22×D7311yescharacteristic
360C24—1461no—
361#41—42no—
362#41—42no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—732no—
371C2×D4—1461no—
372C2×D4—1462no—
373C2×D4—1462no—
374C22×C4—41no—
375C22×C4—2922no—
376C22×C4—2922no—
377C22×C4—1461no—
378C22×C4—1462no—
379C22×C4—1462no—
380C22×C4D4×D7311yescharacteristic
381C22×C4C2×C22⋊D7311yescharacteristic
382C22×C4#1192402711yescharacteristic
383SD16—42no—
384SD16—42no—
385SD16—2924no—
386D8—2924no—
387M16—2924no—
388M16—2922no—
389M16—2922no—
390M16—2924no—
391C2×C8—22no—
392C2×C8—41no—
393C2×C8—2922no—
394C2×C8—1462no—
395C2×C8—1462no—
396C2×C8—1462no—
397C2×C8C2×C22⋊D7311yescharacteristic
398C2×C8C2×C22⋊D7311yescharacteristic
399C2×C8C2×C22⋊D7311yescharacteristic
400C2×C8C2×C22⋊D7311yescharacteristic
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—732no—
410C4⋊C4—1461no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—1462no—
415C22⋊C4—1462no—
416C42—41no—
417C42D4×D7311yescharacteristic
418C42C2×C22⋊D7311yescharacteristic
419C42#1192402711yescharacteristic
420C42C23×D7311yescharacteristic
421C23—41no—
422C23—2922no—
423C23—2922no—
424C23—2922no—
425C23—1462no—
426C23#1455656211yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—81no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—41no—
436D4—81no—
437D4—82no—
438D4—2922no—
439D4—1462no—
440D4—1462no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—2924no—
449C2×C4—2922no—
450C2×C4—2922no—
451C2×C4—2922no—
452C2×C4—1462no—
453C2×C4#1455656211yescharacteristic
454C2×C4#1455656211yescharacteristic
455C2×C4#1455656211yescharacteristic
456C2×C4#1455661812yes—
457C2×C4#1455662012yes—
458C2×C4C2×D4×D7311yescharacteristic
459C2×C4C22×C22⋊D7311yescharacteristic
460C2×C4C2×#1192402711yescharacteristic
461C8—21no—
462C8—21no—
463C8—21no—
464C8—2922no—
465C8—2922no—
466C8—2922no—
467C8#1455661612yescomplemented
468C22—21no—
469C22—21no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—2922no—
474C22—2922no—
475C22—2922no—
476C22—5844no—
477C22#1798883012yes—
478C22C2×#1455656211yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—2922no—
486C4#1798883012yes—
487C4C2×#1455661611yescharacteristic
488C4C2×#1455661811yescharacteristic
489C4C2×#1455662011yescharacteristic
490C4#1798938511yescharacteristic
491C2—22no—
492C2—161no—
493C2—5842no—
494C2#2261939111yescharacteristic
495C2C2×#1798883011yescharacteristic
496C2#2261964711yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3167C2
socleC22×C73C2×#121
prime radical#22619391C2
hypercenter#3167D73
hypocenterC73#19279
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #22619724C2semidirect
3. C2×#17988666C2semidirect
4. #22609475C2semidirect
5. #22609461C2semidirect
6. #22609427C2semidirect
7. #22609393C2semidirect
8. #22608690C2semidirect
9. #22608621C2semidirect
10. #22608608C2semidirect
11. C73×#3167C2semidirect
32. #17988340C22semidirect
33. #17988340C22semidirect
34. #17988336C22semidirect
35. #17988333C22semidirect
36. #17988333C22semidirect
37. #17988331C22semidirect
40. #498×C73C22semidirect
41. #492×C73C22semidirect
42. #487×C73C22semidirect
43. #486×C73C22semidirect
215. C8×C73AGL(1,ℤ/8)semidirect
249. #3167D73semidirect
260. C73#19279semidirect
266. #498C2×D73semidirect
267. #492C2×D73semidirect
268. #487C2×D73semidirect
269. #486C2×D73semidirect
467. C8#14556616semidirect

Maximal subgroups

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

subgroupindex
#226197242
#226193912
C2×#179886662
#226094752
#226094612
#226094272
#226093932
#226086902
#226086212
#226086082
C73×#31672
#1927973