Finite Group Databasebeta

← #29574867

Subgroups — #29574867

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758194C211yesmaximal, characteristic
2#21446764C211yesmaximal, characteristic
3C4×#14519701C211yesmaximal, characteristic
4#21444274C211yesmaximal, characteristic
5#21443721C211yesmaximal, characteristic
6#21443655C211yesmaximal, characteristic
7C61×#3104C211yesmaximal, characteristic
8#16839893—22no—
9#16839893—21no—
10Q8×#92418—21no—
11#16839786—21no—
12#16839786C2211yescharacteristic
13C2×#14519701—21no—
14C4×#11758194—21no—
15C4×#11758194C2211yescharacteristic
16C2×C4×#92418C2211yescharacteristic
17C4×C4⋊Dic61C2211yescharacteristic
18#16839524—21no—
19#16839521—21no—
20#16839521—21no—
21Q8⋊#92418—22no—
22C61×#533C2211yescharacteristic
23C2×C4×Q8×C61C2211yescharacteristic
24C61×#484—21no—
25C61×#477C411yescharacteristic, complemented
26C61×#477C411yescharacteristic, complemented
27C61×#462C411yescharacteristic, complemented
28C61×#462C411yescharacteristic, complemented
29C61×#451—21no—
30C4×Q16×C61—21no—
31C4×Q16×C61—21no—
32C61×#431C2211yescharacteristic
33C2×#11758194—21no—
34C2×#11758194D411yescharacteristic
35C2×#11758194D411yescharacteristic
36C22×#92418—21no—
37C2×C4⋊Dic61—21no—
38C2×C4×Dic61—21no—
39C4⋊#92418—22no—
40C4⋊#92418—22no—
41C4⋊#92418—21no—
42C4⋊#92418D412yes—
43#14519701—24no—
44#14519688—22no—
45#14519688—21no—
46C4×#92418—22no—
47C4×#92418—21no—
48C4×#92418—21no—
49C4×#92418—21no—
50#14519684—21no—
51C22×Q8×C61D411yescharacteristic
52C2×Q16×C61—21no—
53C2×Q16×C61—21no—
54C4⋊Q8×C61C2×C411yescharacteristic
55C4⋊Q8×C61C2×C411yescharacteristic
56#125×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×Q8×C61—21no—
61C4×Q8×C61—21no—
62C4×Q8×C61—21no—
63C4×Q8×C61C2×C411yescharacteristic
64C4×Q8×C61C2×C411yescharacteristic
65C4×Q8×C61D411yescharacteristic
66C4×Q8×C61D411yescharacteristic
67#117×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61D411yescharacteristic
70C2×C42×C61—21no—
71C2×C42×C61C2311yescharacteristic
72#107×C61—21no—
73#107×C61—21no—
74C4⋊C8×C61C2×C411yescharacteristic
75C4⋊C8×C61C2×C411yescharacteristic
76#103×C61—21no—
77#103×C61—21no—
78#103×C61—21no—
79#103×C61—21no—
80C22⋊C8×C61—21no—
81C4×C8×C61—21no—
82C2×Q8×C61—21no—
83C2×Q8×C61—22no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—41no—
87C2×Q8×C61C22⋊C411yescharacteristic
88C2×Q8×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C22×Dic61—41no—
95Q16×C61—42no—
96Q16×C61—42no—
97C2×C8×C61—21no—
98C2×C8×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61C22⋊C411yescharacteristic
111C4⋊C4×C61C22⋊C411yescharacteristic
112C22⋊C4×C61—41no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—21no—
117C42×C61—41no—
118C42×C61C22×C411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61C2×D411yescharacteristic
121C42×C61#4111yescharacteristic
122C4⋊Dic61—22no—
123C4⋊Dic61—41no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126#11758194—21no—
127#11758194—21no—
128#11758194—21no—
129#11758194—41no—
130#11758194—41no—
131#11758194Q1612yescomplemented
132C2×#92418—21no—
133C2×#92418—22no—
134C2×#92418—21no—
135C2×#92418—22no—
136C2×#92418—41no—
137C2×#92418—41no—
138C23×C61#12111yescharacteristic
139Q8×C61—21no—
140Q8×C61—21no—
141Q8×C61—21no—
142Q8×C61—42no—
143Q8×C61—42no—
144Q8×C61—41no—
145Q8×C61—41no—
146Q8×C61#10412yes—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—81no—
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12311yescharacteristic
164C2×C4×C61C2×Q1611yescharacteristic
165C2×C4×C61#13511yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×C4×C61#13711yescharacteristic
168C2×Dic61—42no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C8×C61—41no—
172C8×C61—41no—
173#92418—22no—
174#92418—42no—
175#92418—41no—
176#92418—42no—
177#8990—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61#38611yescharacteristic
181C22×C61#46211yescharacteristic
182C22×C61#46411yescharacteristic
183C4×C61—21no—
184C4×C61—22no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—41no—
190C4×C61—81no—
191C4×C61—81no—
192C4×C61C2×#10411yescharacteristic
193C4×C61#42111yescharacteristic
194C4×C61C4×Q1611yescharacteristic
195C4×C61#44111yescharacteristic
196C4×C61#45111yescharacteristic
197C4×C61#45211yescharacteristic
198Dic61—82no—
199#3104D6111yescharacteristic
200Q8×M16—611no—
201#1986—611no—
202C4×#103—611no—
203#1493—611no—
204#1303—611no—
205#1278—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61#187111yescharacteristic
209C2×C61#187811yescharacteristic
210C2×C61#188111yescharacteristic
211#533C2×D6111yescharacteristic
212C2×C4×Q8C2×D6111yescharacteristic
213#484—21no—
214#477Dic6111yescharacteristic, complemented
215#477Dic6111yescharacteristic, complemented
216#462Dic6111yescharacteristic, complemented
217#462Dic6111yescharacteristic, complemented
218#451—21no—
219#446—1222no—
220#446—1221no—
221C8×Q8—1221no—
222C4×Q16—21no—
223C4×Q16—21no—
224#431C2×D6111yescharacteristic
225#423—611no—
226#423—1221no—
227C2×#103—1221no—
228C4×M16—611no—
229C4×M16—1221no—
230C2×C4×C8—611no—
231C4×C4⋊C4—611no—
232#340—1221no—
233#339—1221no—
234#339—1221no—
235Q8⋊C8—1222no—
236C61#899011yescharacteristic, complemented, Sylow
237C22×Q8C22⋊D6111yescharacteristic
238C2×Q16—21no—
239C2×Q16—21no—
240C2×M16—611no—
241C2×M16—611no—
242C2×M16—1221no—
243C22×C8—1221no—
244C4⋊Q8C2×Dic6111yescharacteristic
245C4⋊Q8C2×Dic6111yescharacteristic
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C2×Dic6111yescharacteristic
254C4×Q8C2×Dic6111yescharacteristic
255C4×Q8C22⋊D6111yescharacteristic
256C4×Q8C22⋊D6111yescharacteristic
257#117—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4C22⋊D6111yescharacteristic
261C2×C42—21no—
262C2×C42—1221no—
263C2×C42C22×D6111yescharacteristic
264#107—21no—
265#107—21no—
266C4⋊C8—612no—
267C4⋊C8—1222no—
268C4⋊C8—1222no—
269C4⋊C8—1221no—
270C4⋊C8C2×Dic6111yescharacteristic
271C4⋊C8C2×Dic6111yescharacteristic
272#103—21no—
273#103—21no—
274#103—21no—
275#103—21no—
276#103—1224no—
277C22⋊C8—21no—
278#97—1222no—
279#97—1221no—
280C4×C8—21no—
281C4×C8—1222no—
282C4×C8—1221no—
283C4×C8—1221no—
284C4×C8—1221no—
285#95—1221no—
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—41no—
291C2×Q8C22⋊Dic6111yescharacteristic
292C2×Q8C22⋊Dic6111yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—2441no—
296C22×C4D4×D6111yescharacteristic
297C22×C4#1175822411yescharacteristic
298C22×C4C2×C22⋊D6111yescharacteristic
299Q16—42no—
300Q16—42no—
301M16—612no—
302M16—1221no—
303M16—1221no—
304M16—1221no—
305M16—2441no—
306M16—2441no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—1221no—
310C2×C8—1222no—
311C2×C8—1222no—
312C2×C8—1221no—
313C2×C8—2441no—
314C2×C8—2441no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—1222no—
327C4⋊C4—2441no—
328C4⋊C4C22⋊Dic6111yescharacteristic
329C4⋊C4C22⋊Dic6111yescharacteristic
330C22⋊C4—41no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—41no—
336C42—2441no—
337C42—2441no—
338C42D4×D6111yescharacteristic
339C42#1175822411yescharacteristic
340C42C22×Dic6111yescharacteristic
341C42C2×C22⋊D6111yescharacteristic
342C23#1451974911yescharacteristic
343Q8—21no—
344Q8—21no—
345Q8—21no—
346Q8—42no—
347Q8—42no—
348Q8—41no—
349Q8—41no—
350Q8#1451970412yes—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—4881no—
363C2×C4—4881no—
364C2×C4—2442no—
365C2×C4C2×C22⋊Dic6111yescharacteristic
366C2×C4D4×Dic6111yescharacteristic
367C2×C4D4×Dic6111yescharacteristic
368C2×C4#1451974511yescharacteristic
369C2×C4#1451974911yescharacteristic
370C2×C4#1451975811yescharacteristic
371C2×C4Q16×D6111yescharacteristic
372C2×C4#1451980011yescharacteristic
373C2×C4#1451980911yescharacteristic
374C2×C4#1451980911yescharacteristic
375C8—41no—
376C8—41no—
377C8—1222no—
378C8—2442no—
379C8—2441no—
380C8—2442no—
381C22—21no—
382C22—21no—
383C22#1683964211yescharacteristic
384C22#1683997011yescharacteristic
385C22#1683998611yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—4882no—
396C4C2×#1451970411yescharacteristic
397C4#1683977911yescharacteristic
398C4Q16×Dic6111yescharacteristic
399C4#1683986911yescharacteristic
400C4#1683991611yescharacteristic
401C4#1683992311yescharacteristic
402C2—21no—
403C2—21no—
404C2#2144612211yescharacteristic
405C2#2144616211yescharacteristic
406C2#2144618311yescharacteristic
subgroupG/H
centerC2×C4#14519745
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3104C2
socleC22×C61#386
prime radicalC23×C61#121
cosocleC2×C42×C61C23
hypercenter#3104D61
hypocenterC61#8990
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
25. C61×#477C4semidirect
26. C61×#477C4semidirect
27. C61×#462C4semidirect
28. C61×#462C4semidirect
131. #11758194Q16semidirect
214. #477Dic61semidirect
215. #477Dic61semidirect
216. #462Dic61semidirect
217. #462Dic61semidirect
236. C61#8990semidirect

Maximal subgroups

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

subgroupindex
Q8×#117581942
#214467642
C4×#145197012
#214442742
#214437212
#214436552
C61×#31042
#899061