Finite Group Databasebeta

← #29833365

Subgroups — #29833365

Order 15616.

660 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
1C2×#16840204C211yesmaximal, characteristic
2#21454694C211yesmaximal, characteristic
3#21454682C211yesmaximal, characteristic, complemented
4#21447687C211yesmaximal, characteristic, complemented
5#21447686C211yesmaximal, characteristic, complemented
6#21447685C211yesmaximal, characteristic, complemented
7Q32×Dic61C211yesmaximal, characteristic, complemented
8C4×#14519723C211yesmaximal, characteristic, complemented
9C4×#14519720C211yesmaximal, characteristic, complemented
10#21447670C211yesmaximal, characteristic, complemented
11#21447606C211yesmaximal, characteristic, complemented
12#21447525C211yesmaximal, characteristic
13#358×D61C211yesmaximal, characteristic, complemented
14#21447417C211yesmaximal, characteristic, complemented
15C61×#2176C211yesmaximal, characteristic, complemented
16C2×#14519800C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18#16840244—21no—
19#16840238—21no—
20#16840204C418yescomplemented
21C2×#14519723C2211yescharacteristic
22C2×#14519722C2211yescharacteristic
23C2×#14519720C2211yescharacteristic
24C2×#14519716C2211yescharacteristic
25C2×C16×D61C2211yescharacteristic
26#16839867—21no—
27#16839866—21no—
28#16839865—21no—
29#16839860—21no—
30C4×Dic244C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32Q16×Dic61C2211yescharacteristic
33#16839856C2211yescharacteristic
34C4×D488C2211yescharacteristic
35#16839844C2211yescharacteristic
36#16839799C2211yescharacteristic
37#107×D61C2211yescharacteristic
38#16839769—21no—
39#16839762—21no—
40#16839716C2211yescharacteristic
41#16839589C2211yescharacteristic
42#16839588C2211yescharacteristic
43#16839587C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839566C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839564C2211yescharacteristic
49#16839562C2211yescharacteristic
50#16839537C2211yescharacteristic
51C4×#11758185C2211yescharacteristic
52C16×Dic61C2211yescharacteristic
53C2×Q32×C61C2211yescharacteristic
54C4×Q16×C61C2211yescharacteristic
55C4×Q16×C61C2211yescharacteristic
56C61×#367C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C61×#346C2211yescharacteristic
60C2×#11758226—21no—
61C2×Q8×D61—21no—
62#14519800—22no—
63#14519800C2×C414yes—
64C2×Dic244C2311yescharacteristic
65C2×#11758202—21no—
66Q16×D61—22no—
67Q16×D61C2×C414yescomplemented
68C2×#11758201—21no—
69C2×D488C2311yescharacteristic
70C2×#11758189D411yescharacteristic
71C2×C8×D61C2311yescharacteristic
72Q8×Dic61—21no—
73Q8×Dic61—21no—
74C4×Dic122—21no—
75C4×Dic122—41no—
76#14519742—21no—
77#14519742—41no—
78C4×D244—21no—
79C4×D244—41no—
80#14519739—21no—
81#14519739—41no—
82#14519735—21no—
83#14519735—41no—
84#14519734—41no—
85#14519734—41no—
86#14519734D411yescharacteristic
87C4⋊C4×D61—21no—
88C4⋊C4×D61—41no—
89#14519723C2×C414yescomplemented
90#14519722C2×C414yescomplemented
91#14519720C2×C414yescomplemented
92#14519716C2×C414yescomplemented
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95C16×D61—22no—
96#14519711C2311yescharacteristic
97#14519711C2311yescharacteristic
98#14519710—21no—
99#14519708—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—21no—
103#14519700—21no—
104#14519699—21no—
105#14519697—21no—
106#14519688D411yescharacteristic
107#14519687D411yescharacteristic
108C8×Dic61C2311yescharacteristic
109C2×Q16×C61C2311yescharacteristic
110C2×Q16×C61C2311yescharacteristic
111C4×Q8×C61—21no—
112C4×Q8×C61—21no—
113Q32×C61C2×C414yescomplemented
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#107×C61C2311yescharacteristic
118#103×C61—21no—
119#103×C61—21no—
120C4×C8×C61C2311yescharacteristic
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123#11758226—24no—
124#11758226—42no—
125Q8×D61—24no—
126Q8×D61—42no—
127C2×D244—21no—
128C2×D244—41no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61—41no—
133C2×C4×D61—41no—
134C2×C4×D61C2×D411yescharacteristic
135C2×Dic122—21no—
136C2×Dic122—41no—
137Q16×C61—21no—
138Q16×C61—21no—
139Q16×C61C22×C412yes—
140Q16×C61C22×C412yes—
141C2×C8×C61C2×D411yescharacteristic
142C2×C8×C61C2411yescharacteristic
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C42×C61—41no—
148C42×C61—41no—
149C42×C61C2×D411yescharacteristic
150#11758202—24no—
151#11758201—24no—
152#11758198—41no—
153#11758198—41no—
154#11758198—41no—
155#11758198D811yescharacteristic
156#11758198D811yescharacteristic
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—41no—
160C4⋊Dic61—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164#11758196D811yescharacteristic
165#11758196D811yescharacteristic
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61—41no—
170C4×Dic61—41no—
171C4×Dic61C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173C2×#92418C2×D411yescharacteristic
174Dic244—21no—
175Dic244C22×C412yes—
176D488—21no—
177D488C22×C412yes—
178#11758189—22no—
179C8×D61C22×C412yes—
180C8×D61#4112yes—
181C16×C61—21no—
182C16×C61C22×C412yes—
183#11758185—21no—
184#11758185C22×C412yes—
185C22×D61—41no—
186C22×D61C2×D811yescharacteristic
187Q8×C61—22no—
188Q8×C61—22no—
189Q8×C61—41no—
190Q8×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203C2×Dic61C2×D811yescharacteristic
204D244—22no—
205D244—44no—
206D244—41no—
207C4×D61—44no—
208C4×D61—44no—
209C4×D61—161no—
210C4×D61—161no—
211C4×D61—161no—
212C4×D61C4×D412yes—
213C4×D61#13512yes—
214Dic122—22no—
215Dic122—41no—
216Dic122—44no—
217C8×C61—21no—
218C8×C61C23×C411yescharacteristic
219C8×C61C2×#4111yescharacteristic
220#92418—21no—
221#92418C4×D412yes—
222#33455—611nomaximal, Sylow
223C22×C61C22×D811yescharacteristic
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—42no—
227C2×D61—81no—
228C2×D61—81no—
229C2×D61C4×D812yes—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—81no—
235C4×C61—81no—
236C4×C61C2×C4×D411yescharacteristic
237C4×C61C2×#13511yescharacteristic
238Dic61—21no—
239Dic61—21no—
240Dic61—42no—
241Dic61—81no—
242Dic61—81no—
243Dic61C4×D812yes—
244C2×#510—611no—
245#2951—611no—
246#2950—611no—
247#2176—611no—
248#2176D6111yescharacteristic, complemented
249#2175—611no—
250#2175—611no—
251C4×Q32—611no—
252C4×Q32—611no—
253C4×SD32—611no—
254C4×SD32—611no—
255#2159—611no—
256#2138—611no—
257C2×#358—611no—
258#2108—611no—
259C2×C61C2×C4×D811yescharacteristic
260C2×C61#341611yescharacteristic
261C2×C61#341711yescharacteristic
262D61—22no—
263D61—82no—
264C2×#135—611no—
265C22×Q16—611no—
266#520—1221no—
267#519—1221no—
268#510—618no—
269C2×Q32—611no—
270C2×Q32C2×D6111yescharacteristic
271C2×SD32—611no—
272C2×SD32—611no—
273C2×M32—611no—
274C22×C16—611no—
275#441—1221no—
276#441—1221no—
277#440—1221no—
278#440—1221no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×Q16—611no—
282C4×Q16—611no—
283C4×Q16C2×D6111yescharacteristic
284C4×Q16C2×D6111yescharacteristic
285C4×D8—611no—
286C4×D8—611no—
287#427—611no—
288C2×#107—611no—
289#419—1221no—
290#417—1221no—
291#405—611no—
292#367—611no—
293#367C2×D6111yescharacteristic
294#366—611no—
295#366—611no—
296#358—611no—
297#358—611no—
298#358—611no—
299#358—611no—
300#358C2×D6111yescharacteristic
301#358C2×D6111yescharacteristic
302#357—611no—
303#357—611no—
304#346—611no—
305#346C2×D6111yescharacteristic
306C4×C16—611no—
307C4×C16—611no—
308C61#3345511yescharacteristic, complemented, Sylow
309C2×#41—1221no—
310C22×Q8—1221no—
311#135—614no—
312#135—1222no—
313C2×Q16—614no—
314C2×Q16—611no—
315C2×Q16—1222no—
316C2×Q16—1221no—
317C2×Q16C22×D6111yescharacteristic
318C2×Q16C22×D6111yescharacteristic
319C2×SD16—1221no—
320C2×D8—611no—
321C2×M16—611no—
322C22×C8—611no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×Q8—1221no—
328C4×Q8—1221no—
329C4×Q8—2441no—
330C4×Q8—2441no—
331C4×D4—1221no—
332C4×D4—1221no—
333C4×D4—2441no—
334C4×D4—2441no—
335#117—611no—
336#117—1221no—
337#117—2441no—
338#117—2441no—
339#117—2441no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—2441no—
342Q32—614no—
343Q32C4×D6114yescomplemented
344SD32—614no—
345SD32—614no—
346M32—614no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16—1222no—
350C2×C16C22×D6111yescharacteristic
351C2×C16C22×D6111yescharacteristic
352#107—611no—
353#107—611no—
354#107—1221no—
355#107C22×D6111yescharacteristic
356#107C22×D6111yescharacteristic
357#106—1221no—
358#103—21no—
359#103—21no—
360#103—1221no—
361#103—1221no—
362#103—1221no—
363#103—1221no—
364#102—1221no—
365#102—1221no—
366#97—611no—
367#97—611no—
368C4×C8—611no—
369C4×C8C22×D6111yescharacteristic
370#41—1224no—
371#41—2442no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—1224no—
375C2×Q8—1221no—
376C2×Q8—2441no—
377C2×Q8—2442no—
378C2×D4—1221no—
379C2×D4—2441no—
380C22×C4—611no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2441no—
386Q16—21no—
387Q16—21no—
388Q16—612no—
389Q16—1224no—
390Q16—1221no—
391Q16C2×C4×D6112yes—
392Q16C2×C4×D6112yes—
393SD16—1224no—
394D8—612no—
395D8—1221no—
396M16—1222no—
397C2×C8—611no—
398C2×C8—612no—
399C2×C8—612no—
400C2×C8—611no—
401C2×C8D4×D6111yescharacteristic
402C2×C8C23×D6111yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—611no—
408C4⋊C4—611no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4—2441no—
414C4⋊C4—2441no—
415C4⋊C4—2441no—
416C22⋊C4—611no—
417C22⋊C4—611no—
418C22⋊C4—2441no—
419C22⋊C4—2441no—
420C22⋊C4—2441no—
421C42—41no—
422C42—41no—
423C42—611no—
424C42—2441no—
425C42—2441no—
426C42—2441no—
427C42—2441no—
428C42—2441no—
429C42D4×D6111yescharacteristic
430C16—21no—
431C16—612no—
432C16—1221no—
433C16C2×C4×D6112yes—
434C23—611no—
435C23—2441no—
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—1222no—
441Q8—2444no—
442Q8—2441no—
443D4—1222no—
444D4—2444no—
445D4—2441no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—611no—
451C2×C4—612no—
452C2×C4—612no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—9761no—
456C2×C4—9761no—
457C2×C4—9761no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4D8×D6111yescharacteristic
464C2×C4D8×D6111yescharacteristic
465C2×C4C2×D4×D6111yescharacteristic
466C8—21no—
467C8—612no—
468C8—1221no—
469C8C22×C4×D6111yescharacteristic
470C8#41×D6111yescharacteristic
471C22—612no—
472C22—1221no—
473C22—1221no—
474C22—4881no—
475C22—4881no—
476C22—2442no—
477C22C2×D8×D6111yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—612no—
485C4—1221no—
486C4—1221no—
487C4—4881no—
488C4—4881no—
489C4—2442no—
490C4C4×D4×D6111yescharacteristic
491C4#135×D6111yescharacteristic
492C2—1222no—
493C2—4882no—
494C2C4×D8×D6111yescharacteristic
495C2#2145896411yescharacteristic
496C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2176C2
socleC22×C61C22×D8
prime radicalC2×#14519800C22
cosocleC2×C8×C61C24
hypercenter#2176D61
hypocenterC61#33455
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
3. #21454682C2semidirect
4. #21447687C2semidirect
5. #21447686C2semidirect
6. #21447685C2semidirect
7. Q32×Dic61C2semidirect
8. C4×#14519723C2semidirect
9. C4×#14519720C2semidirect
10. #21447670C2semidirect
11. #21447606C2semidirect
13. #358×D61C2semidirect
14. #21447417C2semidirect
15. C61×#2176C2semidirect
20. #16840204C4semidirect
67. Q16×D61C2×C4semidirect
89. #14519723C2×C4semidirect
90. #14519722C2×C4semidirect
91. #14519720C2×C4semidirect
92. #14519716C2×C4semidirect
113. Q32×C61C2×C4semidirect
248. #2176D61semidirect
308. C61#33455semidirect
343. Q32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402042
#214546942
#214546822
#214476872
#214476862
#214476852
Q32×Dic612
C4×#145197232
C4×#145197202
#214476702
#214476062
#214475252
#358×D612
#214474172
C61×#21762
#3345561