Finite Group Databasebeta

← #29665295

Subgroups — #29665295

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456744C211yesmaximal, characteristic
2#21455291C211yesmaximal, characteristic
3#21454900C211yesmaximal, characteristic, complemented
4#21445247C214yesmaximal, complemented
5#21445169C214yesmaximal, complemented
6#21444589C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic
9C2×C61×#475C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11#16840312C2211yescharacteristic
12#16840136C2212yes—
13#16840112C2212yes—
14#16839984—22no—
15#16839972C2212yes—
16#16839964—22no—
17#16839962C2212yes—
18#16839901—22no—
19Q8×#92418—22no—
20Q8×#92418C2214yes—
21#16839815—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839786C2211yescharacteristic
25#16839761—21no—
26#16839512C2214yes—
27#16839509C2212yes—
28#16839508C2214yes—
29#16839505C2212yes—
30C2×C4⋊Q8×C61C2211yescharacteristic
31C2×C4×Q8×C61C2211yescharacteristic
32C61×#475—22no—
33C61×#475C2214yes—
34C2×#106×C61—21no—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#103×C61—21no—
37C2×#11758222—21no—
38#14519808—44no—
39C2×#11758202—22no—
40C2×#11758201—22no—
41C2×#11758201—24no—
42C2×#11758194—21no—
43#14519777C2311yescharacteristic
44#14519774C2311yescharacteristic
45#14519765—21no—
46#14519751—21no—
47#14519751—21no—
48C4×D244—21no—
49C4×D244—21no—
50C4⋊#92418—21no—
51C4⋊#92418—24no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519701—24no—
56#14519700—22no—
57#14519697—22no—
58C22⋊#92418—21no—
59C22⋊#92418D412yes—
60C4×#92418—21no—
61C4×#92418—24no—
62C4×#92418C2312yes—
63C22×Q8×C61—21no—
64C22×Q8×C61D411yescharacteristic
65C22×C8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61C2312yes—
70C4×Q8×C61—21no—
71C4×Q8×C61—21no—
72C4×Q8×C61D412yes—
73C4×Q8×C61C2312yes—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61D411yescharacteristic
78C2×C42×C61—21no—
79C2×C42×C61C2311yescharacteristic
80#106×C61—24no—
81C4⋊C8×C61—21no—
82C4⋊C8×C61C2312yes—
83#103×C61—24no—
84C2×Q8×C61—21no—
85C2×Q8×C61—22no—
86C2×Q8×C61—22no—
87C2×Q8×C61—41no—
88C2×Q8×C61—82no—
89C2×Q8×C61SD1614yescomplemented
90C2×Q8×C61C2×D412yes—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C2×C22⋊D61—41no—
98#11758222—42no—
99#11758222—42no—
100C2×D244—21no—
101C2×D244—21no—
102C2×D244—21no—
103C2×C4×D61—41no—
104C2×Dic122—21no—
105C2×C8×C61—22no—
106C2×C8×C61—42no—
107C4⋊C4×C61—22no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61C2×D412yes—
115C42×C61—22no—
116C42×C61—22no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758202—44no—
122#11758201—44no—
123#11758201—84no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418C2×D414yes—
132C23×C61#12111yescharacteristic
133C22×D61—41no—
134Q8×C61—22no—
135Q8×C61—24no—
136Q8×C61—44no—
137Q8×C61—82no—
138Q8×C61C2×SD1614yes—
139C2×C4×C61—22no—
140C2×C4×C61—24no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—81no—
146C2×C4×C61—82no—
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61C2×SD1612yes—
151C2×C4×C61#13512yes—
152C2×C4×C61C22×D411yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×C4×C61C2×#4111yescharacteristic
155C22⋊D61—82no—
156C2×Dic61—41no—
157D244—41no—
158D244—41no—
159D244—42no—
160D244—81no—
161C4×D61—82no—
162Dic122—41no—
163Dic122—41no—
164C8×C61—42no—
165#92418—24no—
166#92418—42no—
167#92418—42no—
168#19397—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61#46512yes—
172C22×C61C2×#12111yescharacteristic
173C2×D61—81no—
174C2×D61—81no—
175C2×D61—81no—
176C4×C61—21no—
177C4×C61—24no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—42no—
181C4×C61—82no—
182C4×C61#46512yes—
183C4×C61C22×SD1611yescharacteristic
184C4×C61C2×#13511yescharacteristic
185C4×C61#57712yes—
186Dic61—81no—
187#3206—611no—
188C2×#475D6111yescharacteristic, complemented
189#3037—611no—
190#2982—611no—
191#1680—614no—
192#1666—614no—
193#1570—611no—
194#1549—611no—
195#1540—611no—
196C2×C61—22no—
197C2×C61#304311yescharacteristic
198C2×C61C2×#46511yescharacteristic
199C2×C61#306911yescharacteristic
200D61—161no—
201C2×#137—1221no—
202#532—611no—
203C2×C4⋊Q8C2×D6111yescharacteristic
204C2×C4×Q8C2×D6111yescharacteristic
205#492—612no—
206#487—612no—
207#475—22no—
208#475C2×D6114yes—
209#464—1222no—
210#462—612no—
211#461—1222no—
212#460—612no—
213#448—1222no—
214C8×Q8—614no—
215C8×Q8—1222no—
216#431—611no—
217#431—611no—
218#431—1222no—
219C2×#106—21no—
220#423—611no—
221C2×C4⋊C8C2×D6111yescharacteristic
222#417—1221no—
223C2×#103—21no—
224#334—614no—
225#332—614no—
226#332—612no—
227#331—612no—
228C61#1939711yescharacteristic, complemented, Sylow
229C2×#41—1221no—
230C22×Q8—21no—
231C22×Q8C22⋊D6111yescharacteristic
232#137—2444no—
233C2×Q16—1222no—
234C2×SD16—1224no—
235C2×SD16—1222no—
236C2×M16—1221no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—611no—
242C4⋊Q8C22×D6112yes—
243#127—611no—
244#124—1221no—
245#121—1221no—
246#121—1221no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C22⋊D6112yes—
250C4×Q8C22×D6112yes—
251C4×D4—1221no—
252C4×D4—1221no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D6111yescharacteristic
257C2×C42—21no—
258C2×C42C22×D6111yescharacteristic
259#106—24no—
260C4⋊C8—21no—
261C4⋊C8—612no—
262C4⋊C8—612no—
263C4⋊C8—612no—
264C4⋊C8—1224no—
265C4⋊C8—1221no—
266C4⋊C8C22×D6112yes—
267#103—24no—
268#103—1224no—
269#103—1222no—
270#102—1222no—
271C22⋊C8—612no—
272C22⋊C8—1221no—
273C4×C8—612no—
274C4×C8—1224no—
275C4×C8—1221no—
276#41—2442no—
277#41—2442no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×Q8—1221no—
284C2×Q8#1175820114yescomplemented
285C2×Q8C2×C22⋊D6112yes—
286C2×D4—1221no—
287C2×D4—1221no—
288C2×D4—1221no—
289C2×D4—2441no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—2441no—
294C22×C4D4×D6111yescharacteristic
295C22×C4#1175822411yescharacteristic
296C22×C4C2×C22⋊D6111yescharacteristic
297Q16—2444no—
298SD16—4884no—
299SD16—2444no—
300M16—2442no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—614no—
304C2×C8—1222no—
305C2×C8—1222no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1221no—
314C4⋊C4—1221no—
315C4⋊C4C2×C22⋊D6112yes—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C42—22no—
319C42—22no—
320C42D4×D6111yescharacteristic
321C42#1175822411yescharacteristic
322C42C2×C22⋊D6111yescharacteristic
323C42C23×D6111yescharacteristic
324C23—2441no—
325C23#1451974911yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—44no—
329Q8—82no—
330Q8—2441no—
331Q8—2441no—
332Q8C2×#1175820114yes—
333D4—4881no—
334D4—4882no—
335D4—2441no—
336D4—2441no—
337D4—2442no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—4882no—
347C2×C4—2441no—
348C2×C4#1451974911yescharacteristic
349C2×C4#1451974911yescharacteristic
350C2×C4#1451974911yescharacteristic
351C2×C4C2×#1175820112yes—
352C2×C4#1451979712yes—
353C2×C4C2×D4×D6111yescharacteristic
354C2×C4C22×C22⋊D6111yescharacteristic
355C2×C4C2×#1175822411yescharacteristic
356C8—42no—
357C8—1224no—
358C8—2442no—
359C8—2442no—
360C22—21no—
361C22—21no—
362C22—4881no—
363C22—4881no—
364C22—4881no—
365C22#1683999312yes—
366C22C2×#1451974911yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—82no—
373C4—4881no—
374C4#1683999312yes—
375C4C22×#1175820111yescharacteristic
376C4C2×#1451979711yescharacteristic
377C4#1684056812yes—
378C2—22no—
379C2—9761no—
380C2#2145533711yescharacteristic
381C2C2×#1683999311yescharacteristic
382C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#475C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#475D61
hypocenterC61#19397
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #21454900C2semidirect
4. #21445247C2semidirect
5. #21445169C2semidirect
6. #21444589C2semidirect
7. #21444504C2semidirect
9. C2×C61×#475C2semidirect
89. C2×Q8×C61SD16semidirect
188. C2×#475D61semidirect
228. C61#19397semidirect
284. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567442
#214552912
#214549002
#214452472
#214451692
#214445892
#214445042
#214444692
C2×C61×#4752
#1939761