Finite Group Databasebeta

← #33098292

Subgroups — #33098292

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619721C211yesmaximal, characteristic
2#22618865C211yesmaximal, characteristic, complemented
3#22618573C211yesmaximal, characteristic
4#22610874C211yesmaximal, characteristic, complemented
5#22610707C211yesmaximal, characteristic, complemented
6#22610679C212yesmaximal, complemented
7#22610635C212yesmaximal
8#22610580C212yesmaximal
9#22610531C212yesmaximal, complemented
10#22609952C211yesmaximal, characteristic
11C8×C4⋊C4×C73C211yesmaximal, characteristic, complemented
12#17989144C2211yescharacteristic
13C4×#11924028C2211yescharacteristic
14#17988895C412yescomplemented
15#17988888C412yescomplemented
16#17988883C411yescharacteristic, complemented
17#17988878C411yescharacteristic, complemented
18#17988873C411yescharacteristic, complemented
19#17988868C411yescharacteristic, complemented
20#17988717—21no—
21#17988717—21no—
22C8×Dic146—21no—
23C8×Dic146—21no—
24#17988646—21no—
25#17988646—21no—
26#17988638—21no—
27#17988638—21no—
28C2×#14556527C2212yes—
29#17988625C2212yes—
30#17988613C2211yescharacteristic
31#17988613C2212yes—
32#17988613C2211yescharacteristic
33#17988611C2211yescharacteristic
34C2×#14556519C2212yes—
35#17988584C2212yes—
36#17988553C2211yescharacteristic
37#17988553C2211yescharacteristic
38#17988542C2212yes—
39C4×#11924012C2211yescharacteristic
40#17988352C2212yes—
41#17988347C2212yes—
42#17988347C2212yes—
43#17988347C2212yes—
44C2×C4⋊C8×C73C2211yescharacteristic
45C2×C4×C8×C73C2212yes—
46C2×C4×C8×C73C2211yescharacteristic
47C4×C4⋊C4×C73C2211yescharacteristic
48#336×C73C2212yes—
49C2×#11924028C2311yescharacteristic
50C2×#11924012C2311yescharacteristic
51C2×#11924012C2312yes—
52C2×#11924012C2311yescharacteristic
53#14556591C2×C411yescharacteristic
54#14556591C2×C411yescharacteristic
55#14556583C2×C412yes—
56#14556570—21no—
57#14556570—21no—
58C4×Dic146—21no—
59C4×Dic146—21no—
60C4×Dic146C2×C411yescharacteristic
61C4×Dic146C2×C411yescharacteristic
62C4×D292C2×C411yescharacteristic
63C4×D292C2×C411yescharacteristic
64C4×C22⋊D73—21no—
65C4×C22⋊D73—21no—
66#14556550C2311yescharacteristic
67#14556550C2312yes—
68#14556548—21no—
69C42×D73—21no—
70#14556527—42no—
71#14556525C2×C412yes—
72#14556525C2×C412yes—
73#14556523C2×C412yes—
74#14556523C2×C412yes—
75C4⋊#10596831—21no—
76C4⋊#10596831—21no—
77#14556520—21no—
78#14556520—21no—
79#14556520—21no—
80#14556520—21no—
81#14556520—22no—
82#14556520—22no—
83#14556519—42no—
84#14556516C2×C412yes—
85#14556516C2×C412yes—
86#14556513C2×C412yes—
87#14556513C2×C412yes—
88#14556503—21no—
89#14556503—21no—
90#14556503—21no—
91#14556503—21no—
92#14556502—21no—
93#14556501—22no—
94C4×#10596831—21no—
95C8×Dic73—22no—
96C22×C8×C73C2312yes—
97C22×C8×C73C2311yescharacteristic
98C22×C8×C73C2311yescharacteristic
99C2×C4⋊C4×C73—21no—
100C2×C42×C73C2312yes—
101C2×C42×C73C2311yescharacteristic
102C4⋊C8×C73C2×C411yescharacteristic
103C4⋊C8×C73C2×C411yescharacteristic
104C4⋊C8×C73C2×C412yes—
105C4×C8×C73—42no—
106C4×C8×C73D411yescharacteristic
107C4×C8×C73D411yescharacteristic
108C4×C8×C73D411yescharacteristic
109C4×C8×C73D411yescharacteristic
110#95×C73—21no—
111#11924028—21no—
112#11924028—21no—
113#11924028—21no—
114#11924028—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146C22×C411yescharacteristic
118C2×D292C22×C411yescharacteristic
119C2×C22⋊D73—21no—
120C2×C4×D73—21no—
121C2×C4×D73—21no—
122C2×C4×D73—21no—
123#11924012—22no—
124#11924012—21no—
125#11924012—21no—
126#11924012—21no—
127#11924012—22no—
128#11924012—21no—
129C2×#10596831—22no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73C22×C412yes—
135C4⋊Dic73C22×C412yes—
136C4⋊Dic73C22×C411yescharacteristic
137C4⋊Dic73C22×C411yescharacteristic
138#11924007—21no—
139#11924007—21no—
140#11924007—22no—
141#11924007—22no—
142C22⋊Dic73—21no—
143C22⋊Dic73—22no—
144#11924005—21no—
145#11924005—21no—
146C4×Dic73—21no—
147C4×Dic73—21no—
148C4×Dic73—22no—
149C4×Dic73—21no—
150C22×C4×C73—22no—
151C22×C4×C73#4111yescharacteristic
152C22×C4×C73#4111yescharacteristic
153C22×C4×C73C2411yescharacteristic
154C2×C8×C73—21no—
155C2×C8×C73—22no—
156C2×C8×C73—22no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73—21no—
160C2×C8×C73C22×C412yes—
161C2×C8×C73C22×C412yes—
162C2×C8×C73C2×D411yescharacteristic
163C2×C8×C73C2×D411yescharacteristic
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C4⋊C4×C73—24no—
167C42×C73—22no—
168C42×C73—22no—
169C42×C73C22×C411yescharacteristic
170C42×C73C22×C411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C23×C73C2×#4111yescharacteristic
174C22×D73—21no—
175C2×C4×C73—22no—
176C2×C4×C73—24no—
177C2×C4×C73—41no—
178C2×C4×C73—42no—
179C2×C4×C73—42no—
180C2×C4×C73C4×D411yescharacteristic
181C2×C4×C73C4×D411yescharacteristic
182C2×C4×C73C4×D411yescharacteristic
183C2×C4×C73C4×D411yescharacteristic
184C2×C4×C73C23×C411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73H(5,𝔽2)11yescharacteristic
187C2×C4×C732−1+411yescharacteristic
188C2×C4×C732−1+411yescharacteristic
189C2×C4×C732−1+411yescharacteristic
190C22⋊D73—41no—
191C22⋊D73—41no—
192C2×Dic73—21no—
193C2×Dic73—21no—
194C2×Dic73—21no—
195C2×Dic73—22no—
196C2×Dic73—22no—
197D292—21no—
198D292—21no—
199C4×D73—42no—
200C4×D73—41no—
201C4×D73—41no—
202C4×D73—42no—
203Dic146—21no—
204Dic146—21no—
205Dic146—42no—
206Dic146—42no—
207C8×C73—22no—
208C8×C73—22no—
209C8×C73—21no—
210C8×C73—21no—
211C8×C73—21no—
212C8×C73—21no—
213#10596831—41no—
214#10596831—42no—
215#10596831—41no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73C2×C4×D411yescharacteristic
219C22×C73#54111yescharacteristic
220C22×C73#54211yescharacteristic
221C2×D73—41no—
222C2×D73—41no—
223C2×D73—41no—
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—44no—
228C4×C73#44312yes—
229C4×C73#51911yescharacteristic
230C4×C73#52011yescharacteristic
231C4×C73#53611yescharacteristic
232C4×C73#53711yescharacteristic
233Dic73—41no—
234Dic73—42no—
235Dic73—41no—
236Dic73—42no—
237Dic73—41no—
238#22130—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73#232311yescharacteristic
242C2×C73C2×#44311yescharacteristic
243C2×C73#295511yescharacteristic
244D73—81no—
245#3076—731no—
246#2960—731no—
247#2922—731no—
248#1986—731no—
249#1950—731no—
250#1945—732no—
251#1937—732no—
252#1929—732no—
253#1922—732no—
254#1806—731no—
255C8×C4⋊C4D7311yescharacteristic, complemented
256C73#2213011yescharacteristic, complemented, Sylow
257#533—731no—
258C4×#41—731no—
259#479—732no—
260#478—732no—
261#477—731no—
262#476—731no—
263#475—731no—
264#474—731no—
265#446—1461no—
266#446—1461no—
267C8×Q8—1461no—
268C8×Q8—1461no—
269#433—1461no—
270#433—1461no—
271#431—1461no—
272#431—1461no—
273C2×#108—732no—
274#427—732no—
275#424—731no—
276#424—731no—
277#424—732no—
278#423—731no—
279C2×C4⋊C8C2×D7311yescharacteristic
280C2×#104—732no—
281#416—732no—
282#408—731no—
283#408—731no—
284#405—732no—
285C4×M16—731no—
286C2×C4×C8C2×D7312yes—
287C2×C4×C8C2×D7311yescharacteristic
288C4×C4⋊C4C2×D7311yescharacteristic
289#341—732no—
290#339—732no—
291#339—732no—
292#339—732no—
293#336C2×D7312yes—
294C2×#41—731no—
295C2×M16—731no—
296C2×M16—731no—
297C2×M16—732no—
298C22×C8C22×D7312yes—
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C4⋊Q8—731no—
302C4⋊Q8—731no—
303#125—732no—
304C22⋊Q8—1461no—
305C22⋊Q8—1461no—
306C4×Q8—731no—
307C4×Q8—731no—
308C4×Q8—1461no—
309C4×Q8—1461no—
310C4×D4—731no—
311C4×D4—731no—
312C4×D4—1461no—
313C4×D4—1461no—
314#117—731no—
315#117—732no—
316#117—1461no—
317C2×C4⋊C4—21no—
318C2×C42—1461no—
319C2×C42C22×D7312yes—
320C2×C42C22×D7311yescharacteristic
321#108—2922no—
322#107—732no—
323#107—732no—
324#106—732no—
325#106—732no—
326C4⋊C8—1461no—
327C4⋊C8—1461no—
328C4⋊C8—1461no—
329C4⋊C8—1461no—
330C4⋊C8—1462no—
331C4⋊C8—1462no—
332C4⋊C8—1461no—
333C4⋊C8—1461no—
334C4⋊C8C4×D7311yescharacteristic
335C4⋊C8C4×D7311yescharacteristic
336C4⋊C8C4×D7312yes—
337#104—2922no—
338#103—732no—
339#103—732no—
340#102—732no—
341#102—732no—
342C22⋊C8—1461no—
343C22⋊C8—1461no—
344C22⋊C8—1461no—
345C22⋊C8—1461no—
346#97—1462no—
347#97—1461no—
348C4×C8—42no—
349C4×C8—1462no—
350C4×C8—1461no—
351C4×C8D29211yescharacteristic
352C4×C8D29211yescharacteristic
353C4×C8D29211yescharacteristic
354C4×C8D29211yescharacteristic
355#95—21no—
356#41—1461no—
357#41—1461no—
358#41—1461no—
359#41—1461no—
360C2×Q8—731no—
361C2×Q8—1461no—
362C2×Q8—1461no—
363C2×D4—731no—
364C2×D4—1461no—
365C22×C4—22no—
366C22×C4—1461no—
367C22×C4—1461no—
368C22×C4—1461no—
369C22×C4#1192402811yescharacteristic
370C22×C4#1192402811yescharacteristic
371C22×C4C23×D7311yescharacteristic
372M16—1462no—
373M16—1461no—
374M16—1461no—
375M16—1461no—
376M16—1461no—
377M16—1462no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—1461no—
385C2×C8—1461no—
386C2×C8—1462no—
387C2×C8C2×C4×D7312yes—
388C2×C8C2×C4×D7312yes—
389C2×C8C2×D29211yescharacteristic
390C2×C8C2×D29211yescharacteristic
391C2×C8C2×D29211yescharacteristic
392C2×C8C2×D29211yescharacteristic
393C4⋊C4—24no—
394C4⋊C4—732no—
395C4⋊C4—732no—
396C4⋊C4—731no—
397C4⋊C4—731no—
398C4⋊C4—1461no—
399C4⋊C4—1461no—
400C4⋊C4—1462no—
401C4⋊C4—1462no—
402C4⋊C4—1461no—
403C4⋊C4—1461no—
404C22⋊C4—1461no—
405C22⋊C4—1462no—
406C22⋊C4—1461no—
407C22⋊C4—1461no—
408C42—22no—
409C42—22no—
410C42—1461no—
411C42—1461no—
412C42—1462no—
413C42—1461no—
414C42C2×C4×D7311yescharacteristic
415C42C2×C4×D7311yescharacteristic
416C42C2×D29211yescharacteristic
417C42C2×D29211yescharacteristic
418C23—1461no—
419C23C2×#1192402811yescharacteristic
420Q8—2922no—
421Q8—2922no—
422Q8—1461no—
423Q8—1461no—
424D4—2921no—
425D4—2921no—
426D4—1461no—
427D4—1461no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—2922no—
434C2×C4—2921no—
435C2×C4—2921no—
436C2×C4—2922no—
437C2×C4—1461no—
438C2×C4—1461no—
439C2×C4—1461no—
440C2×C4—1462no—
441C2×C4—1462no—
442C2×C4C4×D29211yescharacteristic
443C2×C4C4×D29211yescharacteristic
444C2×C4C4×D29211yescharacteristic
445C2×C4C4×D29211yescharacteristic
446C2×C4C22×C4×D7311yescharacteristic
447C2×C4C22×D29211yescharacteristic
448C2×C4#1455663611yescharacteristic
449C2×C4#1455663811yescharacteristic
450C2×C4#1455663811yescharacteristic
451C2×C4#1455663911yescharacteristic
452C8—21no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—22no—
457C8—22no—
458C8—2921no—
459C8—2921no—
460C8—2922no—
461C22—21no—
462C22—21no—
463C22—2921no—
464C22—2921no—
465C22—2921no—
466C22C2×C4×D29211yescharacteristic
467C22#1798919511yescharacteristic
468C22#1798920211yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—44no—
473C4—2921no—
474C4—2921no—
475C4—2922no—
476C4—2921no—
477C4—2922no—
478C4#1798870512yes—
479C4#1798906311yescharacteristic
480C4#1798906611yescharacteristic
481C4#1798916311yescharacteristic
482C4#1798916811yescharacteristic
483C2—21no—
484C2—21no—
485C2—5841no—
486C2#2261261411yescharacteristic
487C2C2×#1798870511yescharacteristic
488C2#2261883511yescharacteristic
subgroupG/H
centerC2×C4C22×D292
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC8×C4⋊C4×C73C2
socleC22×C73C2×C4×D4
prime radicalC2×#11924028C23
cosocleC22×C4×C73C24
hypercenterC8×C4⋊C4D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22618865C2semidirect
4. #22610874C2semidirect
5. #22610707C2semidirect
6. #22610679C2semidirect
9. #22610531C2semidirect
11. C8×C4⋊C4×C73C2semidirect
14. #17988895C4semidirect
15. #17988888C4semidirect
16. #17988883C4semidirect
17. #17988878C4semidirect
18. #17988873C4semidirect
19. #17988868C4semidirect
255. C8×C4⋊C4D73semidirect
256. C73#22130semidirect

Maximal subgroups

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

subgroupindex
#226197212
#226188652
#226185732
#226108742
#226107072
#226106792
#226106352
#226105802
#226105312
#226099522
C8×C4⋊C4×C732
#2213073