Finite Group Databasebeta

← #29663848

Subgroups — #29663848

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456379C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic
3C2×Q8×#92418C211yesmaximal, characteristic, complemented
4#21445247C212yesmaximal, complemented
5#21445237C212yesmaximal, complemented
6#21445203C212yesmaximal, complemented
7#21445169C212yesmaximal, complemented
8#21444574C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic
11C61×#3078C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840312C2211yescharacteristic
14#16840158C2211yescharacteristic
15#16840136C2211yescharacteristic
16#16840112C2211yescharacteristic
17#16840110C2211yescharacteristic
18#16839993—21no—
19#16839993—21no—
20#16839984—22no—
21#16839972C2212yes—
22#16839962C2212yes—
23Q8×#92418—21no—
24Q8×#92418—21no—
25Q8×#92418C2212yes—
26Q8×#92418C2212yes—
27C4×#11758201—21no—
28C4×#11758201—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418—21no—
31C2×C4⋊#92418C2211yescharacteristic
32#16839761—21no—
33C2×C4×#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839512C2212yes—
36#16839512C2212yes—
37#16839511C2212yes—
38#16839509C2212yes—
39#16839508C2212yes—
40#16839505C2212yes—
41C61×#533C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#479—22no—
44C61×#477C2212yes—
45C61×#475C2212yes—
46C61×#428—21no—
47C61×#423C2211yescharacteristic
48C61×#419—21no—
49C2×#11758222—21no—
50#14519808—44no—
51C2×#11758202—22no—
52C2×#11758201—22no—
53C2×#11758201—21no—
54C2×#11758201—21no—
55C2×#11758194—21no—
56C22×#92418—21no—
57C22×#92418D411yescharacteristic
58C22×#92418D411yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66#14519708—21no—
67#14519708—21no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418—22no—
71C4⋊#92418—22no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74#14519701—21no—
75#14519701—21no—
76#14519700—22no—
77#14519697—22no—
78#14519697—21no—
79#14519697—21no—
80C4×#92418—22no—
81C4×#92418—22no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×Q8×C61D411yescharacteristic
87C2×M16×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61D412yes—
98C4×Q8×C61C2312yes—
99#117×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105#106×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#103×C61—22no—
109#103×C61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61SD1614yescomplemented
115C2×Q8×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—41no—
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129M16×C61—42no—
130C2×C8×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61C2×D412yes—
141C22⋊C4×C61—41no—
142C42×C61—22no—
143C42×C61—22no—
144C42×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149#11758202—44no—
150#11758201—42no—
151#11758201—44no—
152#11758201—42no—
153#11758198—41no—
154#11758198—41no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157#11758194—42no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—41no—
162C2×#92418—42no—
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C23×C61#12111yescharacteristic
168C22×D61—41no—
169Q8×C61—22no—
170Q8×C61—24no—
171Q8×C61—41no—
172Q8×C61—41no—
173Q8×C61—81no—
174Q8×C61C2×SD1614yes—
175C2×C4×C61—22no—
176C2×C4×C61—24no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12111yescharacteristic
187C2×C4×C61C2×SD1612yes—
188C2×C4×C61#13512yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C22⋊D61—82no—
193C2×Dic61—41no—
194D244—41no—
195D244—41no—
196D244—41no—
197D244—41no—
198D244—81no—
199C4×D61—82no—
200Dic122—41no—
201Dic122—41no—
202C8×C61—42no—
203#92418—21no—
204#92418—21no—
205#92418—21no—
206#92418—42no—
207#92418—42no—
208#92418#13712yescomplemented
209#19260—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61#46512yes—
213C22×C61C2×#12111yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—24no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61#46512yes—
225C4×C61C22×SD1611yescharacteristic
226C4×C61C2×#13511yescharacteristic
227C4×C61C2×#13711yescharacteristic
228C4×C61#57511yescharacteristic
229Dic61—81no—
230#3167—611no—
231#3078D6111yescharacteristic, complemented
232#3037—611no—
233C2×C8×Q8—611no—
234#1680—612no—
235#1678—612no—
236#1672—612no—
237#1666—612no—
238#1566—611no—
239#1550—611no—
240#1547—611no—
241C2×C61—22no—
242C2×C61#303711yescharacteristic
243C2×C61C2×#46511yescharacteristic
244C2×C61#306711yescharacteristic
245D61—161no—
246C2×#137—1221no—
247#533C2×D6111yescharacteristic
248#532—611no—
249C2×C4×Q8C2×D6111yescharacteristic
250#498—611no—
251#492—611no—
252#487—611no—
253#486—611no—
254#479—22no—
255#477C2×D6112yes—
256#475C2×D6112yes—
257#465—1221no—
258#465—1221no—
259#464—1222no—
260#462—612no—
261#460—612no—
262C8×Q8—612no—
263C8×Q8—612no—
264C8×Q8—1221no—
265C8×Q8—1221no—
266C4×SD16—1221no—
267C4×SD16—1221no—
268#428—21no—
269#423—611no—
270#423C2×D6111yescharacteristic
271C2×C4⋊C8—611no—
272C2×C4⋊C8—1221no—
273#419—21no—
274#417—1221no—
275C2×C4×C8—611no—
276C2×C4×C8—1221no—
277#334—612no—
278#334—612no—
279#333—612no—
280#332—612no—
281#332—612no—
282#331—612no—
283C61#1926011yescharacteristic, complemented, Sylow
284C2×#41—1221no—
285C22×Q8C22⋊D6111yescharacteristic
286#137—2444no—
287C2×Q16—1222no—
288C2×SD16—1221no—
289C2×SD16—1221no—
290C2×SD16—1222no—
291C2×M16—21no—
292C2×M16—1221no—
293C22×C8—611no—
294C22×C8—611no—
295C22×C8—1221no—
296C4⋊Q8—611no—
297C4⋊Q8C22×D6111yescharacteristic
298C4⋊Q8C22×D6111yescharacteristic
299#127—611no—
300#125—21no—
301#124—1221no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1221no—
305#121—1221no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8C22⋊D6112yes—
311C4×Q8C22×D6112yes—
312C4×D4—1221no—
313C4×D4—1221no—
314#117—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C42—21no—
318C2×C42C22×D6111yescharacteristic
319#107—22no—
320#106—22no—
321#106—1221no—
322#106—1221no—
323C4⋊C8—21no—
324C4⋊C8—612no—
325C4⋊C8—612no—
326C4⋊C8—1222no—
327C4⋊C8—1222no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331#103—22no—
332#103—22no—
333#103—1221no—
334#103—1221no—
335#103—1222no—
336#102—1221no—
337#102—1221no—
338#102—1222no—
339C4×C8—611no—
340C4×C8—611no—
341C4×C8—611no—
342C4×C8—611no—
343C4×C8—1222no—
344C4×C8—1222no—
345#41—2442no—
346#41—2442no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—1221no—
352C2×Q8#1175820114yescomplemented
353C2×Q8C2×C22⋊D6112yes—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—2441no—
361C22×C4D4×D6111yescharacteristic
362C22×C4#1175822411yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364Q16—2444no—
365SD16—2442no—
366SD16—2442no—
367SD16—2444no—
368M16—42no—
369M16—2442no—
370C2×C8—22no—
371C2×C8—611no—
372C2×C8—611no—
373C2×C8—611no—
374C2×C8—611no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8—1222no—
378C2×C8—2441no—
379C2×C8—2442no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4C2×C22⋊D6112yes—
392C22⋊C4—41no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42D4×D6111yescharacteristic
399C42#1175822411yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—2441no—
403C23#1451974911yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—41no—
407Q8—41no—
408Q8—81no—
409Q8—2441no—
410Q8—2441no—
411Q8C2×#1175820114yes—
412D4—4881no—
413D4—4882no—
414D4—2441no—
415D4—2441no—
416D4—2441no—
417D4—2441no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—4882no—
428C2×C4—2441no—
429C2×C4#1451974911yescharacteristic
430C2×C4#1451974911yescharacteristic
431C2×C4#1451974911yescharacteristic
432C2×C4C2×#1175820112yes—
433C2×C4#1451979712yes—
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4C22×C22⋊D6111yescharacteristic
436C2×C4C2×#1175822411yescharacteristic
437C8—42no—
438C8—612no—
439C8—1221no—
440C8—1221no—
441C8—1221no—
442C8—2442no—
443C8—2442no—
444C22—21no—
445C22—21no—
446C22—4881no—
447C22—4881no—
448C22—4881no—
449C22#1683999312yes—
450C22C2×#1451974911yescharacteristic
451C4—24no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4—4881no—
459C4#1683999312yes—
460C4C22×#1175820111yescharacteristic
461C4C2×#1451979711yescharacteristic
462C4#137×D6111yescharacteristic
463C4#1684056111yescharacteristic
464C2—22no—
465C2—9761no—
466C2#2145528811yescharacteristic
467C2C2×#1683999311yescharacteristic
468C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3078C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3078D61
hypocenterC61#19260
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. C2×Q8×#92418C2semidirect
4. #21445247C2semidirect
5. #21445237C2semidirect
6. #21445203C2semidirect
7. #21445169C2semidirect
8. #21444574C2semidirect
9. #21444509C2semidirect
11. C61×#3078C2semidirect
114. C2×Q8×C61SD16semidirect
208. #92418#137semidirect
231. #3078D61semidirect
283. C61#19260semidirect
352. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214563792
#214552912
C2×Q8×#924182
#214452472
#214452372
#214452032
#214451692
#214445742
#214445092
#214444972
C61×#30782
#1926061