Finite Group Databasebeta

← #29665291

Subgroups — #29665291

Order 15616.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic
2C2×#16839962C211yesmaximal, characteristic
3#21454900C211yesmaximal, characteristic, complemented
4#21445237C214yesmaximal, complemented
5#21445169C214yesmaximal, complemented
6#21444591C211yesmaximal, characteristic, complemented
7#21444488C211yesmaximal, characteristic
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3078C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C2×#14519774C2211yescharacteristic
12#16840136C2212yes—
13#16840110C2212yes—
14#16839962—22no—
15#16839962—22no—
16#16839962C2214yes—
17#16839912—22no—
18Q8×#92418—22no—
19Q8×#92418C2214yescomplemented
20#16839815—22no—
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519697—21no—
25#16839512C2214yescomplemented
26#16839511C2212yescomplemented
27#16839508C2212yescomplemented
28#16839505C2214yes—
29C61×#533C2211yescharacteristic
30C2×C4×Q8×C61C2211yescharacteristic
31C61×#479—22no—
32C61×#477C2212yescomplemented
33C61×#475C2212yescomplemented
34C61×#428—21no—
35C61×#423C2211yescharacteristic
36C61×#419—21no—
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C2×#11758201—24no—
41C2×#11758201—24no—
42C2×#11758201—44no—
43C22×#92418—21no—
44#14519774—21no—
45#14519774—21no—
46#14519774—21no—
47#14519774C2312yes—
48C4⋊#92418—21no—
49C4⋊#92418—24no—
50C4⋊#92418D412yes—
51C4⋊#92418C2312yes—
52C4⋊#92418C2312yes—
53#14519697—24no—
54#14519697—24no—
55C22⋊#92418—21no—
56C22⋊#92418D412yes—
57C4×#92418—21no—
58C4×#92418—24no—
59C4×#92418C2312yes—
60C22×Q8×C61D411yescharacteristic
61C2×M16×C61—21no—
62C4⋊Q8×C61C2311yescharacteristic
63C4⋊Q8×C61C2311yescharacteristic
64#125×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×Q8×C61D412yes—
72C4×Q8×C61C2312yes—
73#117×C61—21no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C42×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#107×C61—22no—
79#106×C61—22no—
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82#103×C61—22no—
83#103×C61—22no—
84C23×D61—41no—
85C2×Q8×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61SD1614yescomplemented
90C2×Q8×C61C2×D412yes—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61#4111yescharacteristic
96C2×D244—22no—
97C2×D244—22no—
98C2×D244—22no—
99C2×D244—42no—
100C2×D244—42no—
101C2×D244—41no—
102C2×D244—41no—
103C2×D244—41no—
104C2×D244—82no—
105C2×D244—82no—
106M16×C61—42no—
107C2×C8×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61C2×D412yes—
118C22⋊C4×C61—41no—
119C42×C61—22no—
120C42×C61—22no—
121C42×C61—41no—
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C42×C61C2411yescharacteristic
126#11758201—48no—
127#11758201—84no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—42no—
131C2×#92418C2×D414yes—
132C23×C61#12111yescharacteristic
133C22×D61—42no—
134C22×D61—82no—
135C22×D61—82no—
136C22×D61—82no—
137Q8×C61—22no—
138Q8×C61—24no—
139Q8×C61—42no—
140Q8×C61—81no—
141Q8×C61C2×SD1614yes—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61C2×SD1612yes—
154C2×C4×C61#13512yes—
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158D244—42no—
159D244—42no—
160D244—44no—
161D244—82no—
162D244—82no—
163D244—82no—
164D244—82no—
165C8×C61—42no—
166#92418—24no—
167#92418—42no—
168#92418—42no—
169#19396—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#46512yes—
173C22×C61C2×#12111yescharacteristic
174C2×D61—82no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—164no—
178C4×C61—21no—
179C4×C61—24no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—42no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#46512yes—
186C4×C61C22×SD1611yescharacteristic
187C4×C61C2×#13511yescharacteristic
188C4×C61#57712yes—
189#3208—611no—
190#3078D6111yescharacteristic, complemented
191C2×#460—611no—
192#2982—611no—
193#1678—614no—
194#1666—614no—
195#1570—611no—
196#1545—611no—
197#1542—611no—
198C2×C61—22no—
199C2×C61#304311yescharacteristic
200C2×C61C2×#46511yescharacteristic
201C2×C61#306811yescharacteristic
202D61—162no—
203C22×SD16—1221no—
204#533C2×D6111yescharacteristic
205C2×#127—611no—
206C2×C4×Q8C2×D6111yescharacteristic
207#492—612no—
208#486—612no—
209#479—22no—
210#477C2×D6112yescomplemented
211#475C2×D6112yescomplemented
212#460—614no—
213#460—1222no—
214#460—1222no—
215#450—1222no—
216C8×Q8—614no—
217C8×Q8—1222no—
218#431—611no—
219#431—611no—
220#431—1222no—
221#428—21no—
222#423C2×D6111yescharacteristic
223C2×C4⋊C8—611no—
224#419—21no—
225C2×#102—1221no—
226#334—614no—
227#333—612no—
228#332—612no—
229#331—614no—
230C61#1939611yescharacteristic, complemented, Sylow
231C22×Q8C22⋊D6111yescharacteristic
232C22×D4—1221no—
233C22×D4—1221no—
234C22×D4—1221no—
235C2×SD16—1224no—
236C2×SD16—1224no—
237C2×SD16—2444no—
238C2×M16—21no—
239C22×C8—1221no—
240C4⋊Q8C22×D6111yescharacteristic
241C4⋊Q8C22×D6111yescharacteristic
242#127—612no—
243#127—1221no—
244#127—1221no—
245#127—1221no—
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D6112yes—
254C4×Q8C22×D6112yes—
255#117—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D6111yescharacteristic
258C2×C42—21no—
259C2×C42C22×D6111yescharacteristic
260#107—22no—
261#106—22no—
262C4⋊C8—21no—
263C4⋊C8—612no—
264C4⋊C8—612no—
265C4⋊C8—612no—
266C4⋊C8—1224no—
267C4⋊C8—1221no—
268C4⋊C8C22×D6112yes—
269#103—22no—
270#103—22no—
271#102—1224no—
272#102—1224no—
273C22⋊C8—612no—
274C22⋊C8—1221no—
275C4×C8—612no—
276C4×C8—1224no—
277C4×C8—1221no—
278C24—2441no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8#1175820114yescomplemented
284C2×Q8C2×C22⋊D6112yes—
285C2×D4—1222no—
286C2×D4—1222no—
287C2×D4—1222no—
288C2×D4—4882no—
289C2×D4—4882no—
290C2×D4—2441no—
291C2×D4—2441no—
292C2×D4—2441no—
293C2×D4—2442no—
294C2×D4—2442no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4D4×D6111yescharacteristic
298C22×C4#1175822411yescharacteristic
299C22×C4C2×C22⋊D6111yescharacteristic
300SD16—4884no—
301SD16—2448no—
302M16—42no—
303C2×C8—22no—
304C2×C8—614no—
305C2×C8—1222no—
306C2×C8—1222no—
307C2×C8—2442no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C2×C22⋊D6112yes—
318C22⋊C4—41no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42D4×D6111yescharacteristic
323C42#1175822411yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C42C23×D6111yescharacteristic
326C23—4882no—
327C23—4882no—
328C23—4882no—
329C23—2442no—
330C23#1451974911yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×#1175820114yes—
336D4—4882no—
337D4—4882no—
338D4—4882no—
339D4—4882no—
340D4—2442no—
341D4—2442no—
342D4—2444no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#1451974911yescharacteristic
352C2×C4#1451974911yescharacteristic
353C2×C4#1451974911yescharacteristic
354C2×C4C2×#1175820112yes—
355C2×C4#1451979712yes—
356C2×C4C2×D4×D6111yescharacteristic
357C2×C4C22×C22⋊D6111yescharacteristic
358C2×C4C2×#1175822411yescharacteristic
359C8—42no—
360C8—1224no—
361C8—2442no—
362C8—2442no—
363C22—21no—
364C22—21no—
365C22—9764no—
366C22—4882no—
367C22—4882no—
368C22—4882no—
369C22#1683999312yes—
370C22C2×#1451974911yescharacteristic
371C4—24no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1683999312yes—
379C4C22×#1175820111yescharacteristic
380C4C2×#1451979711yescharacteristic
381C4#1684057112yes—
382C2—22no—
383C2—9762no—
384C2#2145534211yescharacteristic
385C2C2×#1683999311yescharacteristic
386C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3078C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3078D61
hypocenterC61#19396
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. #21454900C2semidirect
4. #21445237C2semidirect
5. #21445169C2semidirect
6. #21444591C2semidirect
8. #21444476C2semidirect
9. C61×#3078C2semidirect
19. Q8×#92418C22semidirect
25. #16839512C22semidirect
26. #16839511C22semidirect
27. #16839508C22semidirect
32. C61×#477C22semidirect
33. C61×#475C22semidirect
89. C2×Q8×C61SD16semidirect
190. #3078D61semidirect
210. #477C2×D61semidirect
211. #475C2×D61semidirect
230. C61#19396semidirect
283. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168399622
#214549002
#214452372
#214451692
#214445912
#214444882
#214444762
C61×#30782
#1939661