Finite Group Databasebeta

← #29834720

Subgroups — #29834720

Order 15616.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456359C211yesmaximal, characteristic, complemented
2#21454695C211yesmaximal, characteristic
3#21447985C212yesmaximal, complemented
4#21447974C212yesmaximal, complemented
5#21447969C212yesmaximal, complemented
6#21447954C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8C2×#16839588C211yesmaximal, characteristic, complemented
9#21447577C211yesmaximal, characteristic, complemented
10#21447531C211yesmaximal, characteristic
11C61×#2138C211yesmaximal, characteristic
12#16840390—21no—
13#16840245—21no—
14C2×#14519717C2211yescharacteristic
15#16840169—22no—
16#16840166C2212yes—
17#16840163C2212yes—
18#16839863—22no—
19Q16×Dic61C2212yes—
20D8×Dic61C2212yes—
21#16839805—21no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24#16839763—21no—
25#16839717C2211yescharacteristic
26#16839589C2212yes—
27#16839588C2212yes—
28#16839588C2212yes—
29#16839587C2212yes—
30#16839580C2212yes—
31#16839580C2212yes—
32#16839566C2212yes—
33#16839563C2212yes—
34C2×#135×C61C2211yescharacteristic
35C22×C16×C61C2211yescharacteristic
36C61×#427C2211yescharacteristic
37C61×#358C2211yescharacteristic
38C61×#358C2211yescharacteristic
39C61×#357C2211yescharacteristic
40C61×#357C2211yescharacteristic
41C2×#11758194D411yescharacteristic
42#14519777—41no—
43#14519776—22no—
44#14519769—41no—
45#14519768—22no—
46#14519754—42no—
47Q8×Dic61—22no—
48D4×Dic61—22no—
49D4×Dic61—42no—
50#14519736—41no—
51#14519736D411yescharacteristic
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—41no—
55#14519717—22no—
56C2×#11758185C2312yes—
57#14519711—21no—
58#14519711—21no—
59#14519711C2312yes—
60#14519711C2312yes—
61#14519710—22no—
62#14519708—22no—
63#14519701—22no—
64#14519698—22no—
65#14519687D412yes—
66C8×Dic61C2312yes—
67C2×#41×C61—21no—
68#135×C61—22no—
69#135×C61Q814yescomplemented
70C2×Q16×C61C2311yescharacteristic
71C2×SD16×C61—21no—
72C2×D8×C61C2311yescharacteristic
73C22×C8×C61C2311yescharacteristic
74#117×C61—21no—
75C2×C16×C61—22no—
76C2×C16×C61C2311yescharacteristic
77C2×C16×C61C2311yescharacteristic
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#41×C61—24no—
82#41×C61—42no—
83C2×Q8×C61—21no—
84C2×D4×C61—21no—
85C2×D4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C22×Dic61—41no—
89C22×Dic61—41no—
90C22×Dic61—42no—
91C2×Dic122—42no—
92Q16×C61—21no—
93Q16×C61C2×Q812yes—
94SD16×C61—24no—
95D8×C61—21no—
96D8×C61C2×Q812yes—
97C2×C8×C61C2×D411yescharacteristic
98C2×C8×C61C2×Q812yes—
99C2×C8×C61#4112yes—
100C2×C8×C61C2411yescharacteristic
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C22⋊Dic61—42no—
106C22⋊Dic61D812yes—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—42no—
112C4⋊Dic61—42no—
113C4⋊Dic61D811yescharacteristic
114C4⋊Dic61D811yescharacteristic
115#11758196—42no—
116#11758196—42no—
117C4×Dic61—42no—
118C4×Dic61C2×D412yes—
119#11758194—22no—
120C2×#92418C2×D412yes—
121C16×C61—21no—
122C16×C61—21no—
123#11758185—22no—
124C23×C61—41no—
125C23×C61C2×D811yescharacteristic
126Q8×C61—22no—
127Q8×C61—41no—
128D4×C61—22no—
129D4×C61—44no—
130D4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—44no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61C22⋊Q812yes—
136C2×C4×C61C2×D811yescharacteristic
137C2×C4×C61#13712yes—
138C2×C4×C61C22×D411yescharacteristic
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—162no—
142C2×Dic61—161no—
143C2×Dic61—161no—
144C2×Dic61C2×D812yes—
145C2×Dic61C2×D812yes—
146Dic122—162no—
147C8×C61C22⋊Q812yes—
148C8×C61C22×Q811yescharacteristic
149C8×C61C2×#4111yescharacteristic
150#92418—22no—
151#33633—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—42no—
155C22×C61—81no—
156C22×C61—81no—
157C22×C61#47412yes—
158C22×C61C22×D811yescharacteristic
159C4×C61—42no—
160C4×C61—81no—
161C4×C61—81no—
162C4×C61#47412yes—
163C4×C61C2×C22⋊Q811yescharacteristic
164C4×C61C2×#13711yescharacteristic
165Dic61—22no—
166Dic61—22no—
167Dic61—82no—
168Dic61—82no—
169#3164—611no—
170#2951—611no—
171#2229—612no—
172#2228—612no—
173#2227—612no—
174#2224—612no—
175#2159—611no—
176C2×#366—611no—
177#2151—611no—
178#2139—611no—
179#2138D6111yescharacteristic
180C2×C61—22no—
181C2×C61—82no—
182C2×C61C2×#47411yescharacteristic
183C2×C61#341511yescharacteristic
184C2×C61#341711yescharacteristic
185C2×#135C2×D6111yescharacteristic
186#544—1221no—
187#520—1221no—
188C2×M32—611no—
189C22×C16C2×D6111yescharacteristic
190#501—1222no—
191#500—612no—
192#499—612no—
193#440—1222no—
194C4×Q16—612no—
195C4×D8—612no—
196#428—1221no—
197#427C2×D6111yescharacteristic
198C2×#107—611no—
199C2×#107—611no—
200#417—1221no—
201#405—611no—
202#367—612no—
203#366—612no—
204#366—612no—
205#366—612no—
206C4⋊C16—612no—
207C4⋊C16—612no—
208#358—612no—
209#358C2×D6111yescharacteristic
210#358C2×D6111yescharacteristic
211#357—612no—
212#357C2×D6111yescharacteristic
213#357C2×D6111yescharacteristic
214C61#3363311yescharacteristic, complemented, Sylow
215C2×#41—21no—
216#135—22no—
217#135Dic12214yescomplemented
218C2×Q16C22×D6111yescharacteristic
219C2×SD16—21no—
220C2×D8C22×D6111yescharacteristic
221C2×M16—611no—
222C22×C8C22×D6111yescharacteristic
223C4⋊Q8—1222no—
224C4⋊Q8—2441no—
225#125—1222no—
226#125—2441no—
227C22⋊Q8—2442no—
228C4×Q8—1222no—
229C4×D4—1222no—
230C4×D4—2442no—
231#117—21no—
232#117—611no—
233#117—2441no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4—2441no—
237M32—1222no—
238C2×C16—22no—
239C2×C16—612no—
240C2×C16C22×D6111yescharacteristic
241C2×C16C22×D6111yescharacteristic
242#107—612no—
243#107—612no—
244#107—1222no—
245#107—1221no—
246#107—1221no—
247#107C22×D6111yescharacteristic
248#107C22×D6111yescharacteristic
249#106—21no—
250#106—1222no—
251#103—1222no—
252#102—1222no—
253#97—612no—
254C4×C8—612no—
255#41—24no—
256#41—42no—
257C2×Q8—21no—
258C2×Q8—2442no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—2442no—
263C22×C4—2441no—
264C22×C4—2441no—
265C22×C4D4×D6111yescharacteristic
266Q16—21no—
267Q16C2×Dic12212yes—
268SD16—24no—
269D8—21no—
270D8C2×Dic12212yes—
271M16—1222no—
272C2×C8—612no—
273C2×C8C2×Dic12212yes—
274C2×C8D4×D6111yescharacteristic
275C2×C8#1175822412yes—
276C2×C8C23×D6111yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—611no—
280C4⋊C4—611no—
281C4⋊C4—1222no—
282C4⋊C4—1222no—
283C4⋊C4—2442no—
284C4⋊C4—2442no—
285C4⋊C4—2442no—
286C4⋊C4—2442no—
287C4⋊C4—2441no—
288C4⋊C4—2441no—
289C22⋊C4—41no—
290C22⋊C4—612no—
291C22⋊C4—2442no—
292C42—41no—
293C42—612no—
294C42—2442no—
295C16—21no—
296C16—21no—
297C16—1222no—
298C23—41no—
299C23D8×D6111yescharacteristic
300Q8—22no—
301Q8—41no—
302Q8—9762no—
303D4—22no—
304D4—44no—
305D4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—9762no—
313C2×C4—9761no—
314C2×C4—9761no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4#1451975412yes—
318C2×C4D8×D6111yescharacteristic
319C2×C4#1451980612yes—
320C2×C4C2×D4×D6111yescharacteristic
321C8—1222no—
322C8#1451975412yes—
323C8C22×Dic12211yescharacteristic
324C8C2×#1175822411yescharacteristic
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#1684004012yes—
331C22C2×D8×D6111yescharacteristic
332C4—42no—
333C4—81no—
334C4—81no—
335C4—1222no—
336C4—1222no—
337C4—4882no—
338C4—4882no—
339C4#1684004012yes—
340C4C2×#1451975411yescharacteristic
341C4C2×#1451980611yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#1684004011yescharacteristic
345C2#2145895711yescharacteristic
346C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radicalC2×#135×C61C22
hypercenter#2138D61
hypocenterC61#33633
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21456359C2semidirect
3. #21447985C2semidirect
4. #21447974C2semidirect
5. #21447969C2semidirect
6. #21447954C2semidirect
7. #21447608C2semidirect
8. C2×#16839588C2semidirect
9. #21447577C2semidirect
69. #135×C61Q8semidirect
214. C61#33633semidirect
217. #135Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563592
#214546952
#214479852
#214479742
#214479692
#214479542
#214476082
C2×#168395882
#214475772
#214475312
C61×#21382
#3363361