Finite Group Databasebeta

← #29663845

Subgroups — #29663845

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#21456378C211yesmaximal, characteristic
2#21455624C211yesmaximal, characteristic, complemented
3#21455291C211yesmaximal, characteristic
4#21445249C212yesmaximal, complemented
5#21445235C212yesmaximal, complemented
6#21445201C212yesmaximal, complemented
7#21445167C212yesmaximal, complemented
8#21444575C211yesmaximal, characteristic, complemented
9#21444507C211yesmaximal, characteristic, complemented
10#21444494C211yesmaximal, characteristic
11C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
12C2×#14519808—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15#16840160C2211yescharacteristic
16#16840135C2211yescharacteristic
17#16840113C2211yescharacteristic
18#16840109C2211yescharacteristic
19#16840069—22no—
20#16840057C2212yes—
21#16840047C2212yes—
22#16839991—21no—
23#16839991—21no—
24#16839984—22no—
25#16839972C2212yes—
26#16839962C2212yes—
27C4×#11758190—21no—
28C4×#11758190—21no—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839786C2211yescharacteristic
32#16839771—21no—
33#16839761—21no—
34#16839513C2212yes—
35#16839513C2212yes—
36#16839510C2212yes—
37#16839509C2212yes—
38#16839507C2212yes—
39#16839505C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C8×Q8×C61—21no—
42C8×Q8×C61—21no—
43C8×Q8×C61C2212yes—
44C8×Q8×C61C2212yes—
45C2×C4⋊C8×C61—21no—
46C2×C4⋊C8×C61C2211yescharacteristic
47C2×C4×C8×C61—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50#14519808—44no—
51C2×#11758202—22no—
52C2×#11758190—21no—
53C2×#11758190—21no—
54C2×#11758201—22no—
55C2×#11758194—21no—
56C2×#11758194—21no—
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519769—21no—
62#14519765—21no—
63#14519756—21no—
64#14519756—21no—
65#14519751—21no—
66#14519751—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×D244—21no—
70C4×D244—21no—
71#14519736—21no—
72#14519710—22no—
73#14519709—21no—
74#14519709—21no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79C4⋊#92418C2312yes—
80#14519702—21no—
81#14519702—21no—
82#14519701—22no—
83#14519701—22no—
84#14519700—22no—
85#14519699—21no—
86#14519699—21no—
87#14519697—22no—
88C22×Q8×C61D411yescharacteristic
89C22×C8×C61—21no—
90C22×C8×C61D411yescharacteristic
91C22×C8×C61D411yescharacteristic
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61D412yes—
95C4×Q8×C61C2312yes—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C42×C61—21no—
99C2×C42×C61C2311yescharacteristic
100C4⋊C8×C61—22no—
101C4⋊C8×C61—22no—
102C4⋊C8×C61—21no—
103C4⋊C8×C61C2312yes—
104C4×C8×C61—22no—
105C4×C8×C61—22no—
106C4×C8×C61D411yescharacteristic
107C4×C8×C61D411yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61SD1614yescomplemented
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—41no—
120#11758222—42no—
121#11758222—42no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C2×C8×C61—41no—
132C2×C8×C61—42no—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D411yescharacteristic
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D411yescharacteristic
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61C2×D412yes—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D411yescharacteristic
144C42×C61C2×D411yescharacteristic
145C42×C61#4111yescharacteristic
146C42×C61C2411yescharacteristic
147C22⋊Dic61—41no—
148#11758202—44no—
149#11758201—44no—
150#11758198—41no—
151#11758198—41no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158#11758196—41no—
159#11758196—41no—
160C4×Dic61—41no—
161#11758194—42no—
162#11758194—42no—
163C2×#92418—22no—
164C2×#92418—22no—
165#11758190—42no—
166#11758190—42no—
167C23×C61#12111yescharacteristic
168C22×D61—41no—
169Q8×C61—22no—
170Q8×C61—24no—
171Q8×C61C2×SD1614yes—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61#12111yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61C2×SD1612yes—
183C2×C4×C61#13512yes—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C22⋊D61—82no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191D244—41no—
192D244—41no—
193D244—41no—
194D244—41no—
195D244—81no—
196C4×D61—82no—
197Dic122—41no—
198Dic122—41no—
199Dic122—41no—
200Dic122—41no—
201Dic122—81no—
202C8×C61—21no—
203C8×C61—21no—
204C8×C61—21no—
205C8×C61—42no—
206C8×C61#13712yescomplemented
207#92418—42no—
208#92418—42no—
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#46512yes—
223C4×C61C22×SD1611yescharacteristic
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×#13711yescharacteristic
226C4×C61#57511yescharacteristic
227Dic61—81no—
228Dic61—81no—
229Dic61—81no—
230#3167—611no—
231#3078—611no—
232#3037—611no—
233C2×C8×Q8D6111yescharacteristic, complemented
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#533—611no—
248#532—611no—
249C2×C4×Q8C2×D6111yescharacteristic
250#498—611no—
251#492—611no—
252#487—611no—
253#486—611no—
254#479—1222no—
255#477—612no—
256#475—612no—
257#465—1221no—
258#465—1221no—
259#464—1222no—
260#462—612no—
261#460—612no—
262C8×Q8—21no—
263C8×Q8—21no—
264C8×Q8C2×D6112yes—
265C8×Q8C2×D6112yes—
266C4×SD16—1221no—
267C4×SD16—1221no—
268#428—1221no—
269#423—611no—
270#423—611no—
271C2×C4⋊C8—21no—
272C2×C4⋊C8C2×D6111yescharacteristic
273#419—1221no—
274#417—1221no—
275C2×C4×C8—21no—
276C2×C4×C8C2×D6111yescharacteristic
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×Q8D24411yescharacteristic
286#137—2444no—
287C2×Q16—1222no—
288C2×SD16—1222no—
289C2×SD16—1221no—
290C2×SD16—1221no—
291C2×M16—1221no—
292C2×M16—1221no—
293C22×C8—21no—
294C22×C8C22⋊D6111yescharacteristic
295C22×C8C22⋊D6111yescharacteristic
296C4⋊Q8—611no—
297C4⋊Q8—611no—
298C4⋊Q8—611no—
299#127—611no—
300#125—1221no—
301#124—1221no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304#121—1221no—
305#121—1221no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×Q8D24412yes—
311C4×Q8C22×D6112yes—
312C4×D4—1221no—
313C4×D4—1221no—
314#117—1221no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4D24411yescharacteristic
317C2×C42—21no—
318C2×C42C22×D6111yescharacteristic
319#107—1222no—
320#106—1221no—
321#106—1221no—
322#106—1222no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—21no—
326C4⋊C8—612no—
327C4⋊C8—612no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330C4⋊C8C22×D6112yes—
331#103—1222no—
332#103—1222no—
333#103—1221no—
334#103—1221no—
335#103—1222no—
336#102—1221no—
337#102—1221no—
338#102—1222no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8C22⋊D6111yescharacteristic
342C4×C8C22⋊D6111yescharacteristic
343C4×C8C22×D6111yescharacteristic
344C4×C8C22×D6111yescharacteristic
345#41—2442no—
346#41—2442no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×Q8#1175819014yescomplemented
353C2×Q8C2×D24412yes—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—2441no—
361C22×C4C2×D24411yescharacteristic
362C22×C4#1175822211yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364Q16—2444no—
365SD16—2442no—
366SD16—2444no—
367SD16—2442no—
368M16—2442no—
369M16—2442no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—1222no—
375C2×C8—1222no—
376C2×C8C2×C22⋊D6111yescharacteristic
377C2×C8C2×C22⋊D6111yescharacteristic
378C2×C8C2×C22⋊D6111yescharacteristic
379C2×C8C2×C22⋊D6111yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4C2×D24412yes—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C42—22no—
396C42—22no—
397C42—2441no—
398C42C2×D24411yescharacteristic
399C42#1175822211yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—2441no—
403C23#1451975111yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—4881no—
407Q8—2441no—
408Q8—2441no—
409Q8—2441no—
410Q8—2441no—
411Q8C2×#1175819014yes—
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—4882no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4#1451975111yescharacteristic
430C2×C4#1451975111yescharacteristic
431C2×C4#1451975111yescharacteristic
432C2×C4C2×#1175819012yes—
433C2×C4#1451979912yes—
434C2×C4C22×C22⋊D6111yescharacteristic
435C2×C4C22×D24411yescharacteristic
436C2×C4C2×#1175822211yescharacteristic
437C8—21no—
438C8—21no—
439C8—21no—
440C8—42no—
441C8—2442no—
442C8—2442no—
443C8#1451980812yescomplemented
444C22—21no—
445C22—21no—
446C22—4881no—
447C22—4881no—
448C22—4881no—
449C22#1683999112yes—
450C22C2×#1451975111yescharacteristic
451C4—24no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—4881no—
457C4—4881no—
458C4—4881no—
459C4#1683999112yes—
460C4C22×#1175819011yescharacteristic
461C4C2×#1451979911yescharacteristic
462C4C2×#1451980811yescharacteristic
463C4#1684056011yescharacteristic
464C2—22no—
465C2—9761no—
466C2#2145529111yescharacteristic
467C2C2×#1683999111yescharacteristic
468C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C8×Q8×C61C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×C8×Q8D61
hypocenterC61#19260
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #21455624C2semidirect
4. #21445249C2semidirect
5. #21445235C2semidirect
6. #21445201C2semidirect
7. #21445167C2semidirect
8. #21444575C2semidirect
9. #21444507C2semidirect
11. C2×C8×Q8×C61C2semidirect
112. C2×Q8×C61SD16semidirect
206. C8×C61#137semidirect
233. C2×C8×Q8D61semidirect
283. C61#19260semidirect
352. C2×Q8#11758190semidirect
443. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214556242
#214552912
#214452492
#214452352
#214452012
#214451672
#214445752
#214445072
#214444942
C2×C8×Q8×C612
#1926061