Finite Group Databasebeta

← #29574739

Subgroups — #29574739

Order 15616.

476 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455890C211yesmaximal, characteristic
2#21446795C211yesmaximal, characteristic, complemented
3C4×#14519704C211yesmaximal, characteristic
4#21444239C211yesmaximal, characteristic, complemented
5#21443708C211yesmaximal, characteristic, complemented
6#21443651C211yesmaximal, characteristic, complemented
7C61×#2991C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840103—21no—
10#16840047C411yescharacteristic, complemented
11#16840042C411yescharacteristic, complemented
12#16839972C411yescharacteristic, complemented
13#16839967C411yescharacteristic, complemented
14#16839924—21no—
15C4×#11758202—21no—
16C4×#11758200—21no—
17#16839815C2211yescharacteristic
18C2×#14519704—21no—
19C4×#11758194C2211yescharacteristic
20C42×Dic61C2211yescharacteristic
21#16839523—22no—
22#16839521—21no—
23#16839492—21no—
24#16839492—21no—
25C4×#41×C61C2211yescharacteristic
26C8×Q8×C61—21no—
27C61×#436—22no—
28C61×#436—21no—
29C61×#431—21no—
30C61×#431C2211yescharacteristic
31C61×#423—21no—
32C61×#423C2211yescharacteristic
33C2×#11758202—21no—
34C2×#11758200—21no—
35C2×#11758194—21no—
36#14519777C2×C411yescharacteristic
37#14519777C2×C411yescharacteristic
38#14519769—21no—
39#14519756—21no—
40#14519756—21no—
41C4×Dic122—21no—
42C4×Dic122—21no—
43#14519736—21no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47#14519710—21no—
48#14519708—21no—
49C4⋊#92418C2×C411yescharacteristic
50C4⋊#92418C2×C411yescharacteristic
51#14519704—24no—
52#14519701—21no—
53#14519700—21no—
54#14519700—21no—
55#14519698—21no—
56C22⋊#92418—21no—
57#14519688—21no—
58C4×#92418—21no—
59C4×#92418—21no—
60C2×#41×C61D411yescharacteristic
61C2×M16×C61—22no—
62C2×M16×C61D412yes—
63C4×Q8×C61C2×C411yescharacteristic
64C4×Q8×C61D411yescharacteristic
65C4×D4×C61—21no—
66C4×D4×C61—21no—
67C4×D4×C61C2×C411yescharacteristic
68C4×D4×C61D411yescharacteristic
69#117×C61—21no—
70#117×C61D411yescharacteristic
71C2×C42×C61—21no—
72C2×C42×C61C2311yescharacteristic
73C4⋊C8×C61—22no—
74C4⋊C8×C61—21no—
75C4⋊C8×C61—21no—
76C4⋊C8×C61D411yescharacteristic
77C4⋊C8×C61D411yescharacteristic
78C22⋊C8×C61—22no—
79C22⋊C8×C61—21no—
80C4×C8×C61—22no—
81C4×C8×C61—21no—
82#41×C61—22no—
83#41×C61—41no—
84C2×Q8×C61C22⋊C411yescharacteristic
85C2×D4×C61—21no—
86C2×D4×C61C22⋊C411yescharacteristic
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61#4111yescharacteristic
93C22×Dic61—41no—
94C2×Dic122—21no—
95C2×Dic122—21no—
96M16×C61—42no—
97M16×C61—82no—
98C2×C8×C61—21no—
99C2×C8×C61—22no—
100C2×C8×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61C22⋊C411yescharacteristic
104C4⋊C4×C61C22⋊C411yescharacteristic
105C22⋊C4×C61—21no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—21no—
108C42×C61—21no—
109C42×C61—21no—
110C42×C61—21no—
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C22⋊Dic61—41no—
116#11758202—42no—
117#11758200—42no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C4×Dic61—22no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—41no—
131#11758194—22no—
132#11758194—41no—
133C2×#92418—21no—
134C2×#92418—21no—
135C2×#92418—41no—
136C23×C61—21no—
137C23×C61#12111yescharacteristic
138Q8×C61—21no—
139Q8×C61#10412yes—
140D4×C61—21no—
141D4×C61—21no—
142D4×C61—42no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61C2×C22⋊C411yescharacteristic
158C2×C4×C61C4×D411yescharacteristic
159C2×C4×C61C4×D411yescharacteristic
160C2×C4×C61#12011yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12311yescharacteristic
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168Dic122—41no—
169Dic122—41no—
170Dic122—41no—
171Dic122—41no—
172C8×C61—42no—
173C8×C61—41no—
174C8×C61—41no—
175#92418—42no—
176#92418—41no—
177#92418—41no—
178#8972—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61#38611yescharacteristic
186C4×C61—21no—
187C4×C61—22no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—41no—
191C4×C61—41no—
192C4×C61C2×#10411yescharacteristic
193C4×C61#42111yescharacteristic
194C4×C61#44311yescharacteristic
195C4×C61#44411yescharacteristic
196C4×C61#45511yescharacteristic
197C4×C61#45611yescharacteristic
198Dic61—44no—
199Dic61—81no—
200Dic61—81no—
201#3107—611no—
202#2991D6111yescharacteristic
203#1994—611no—
204C4×#104—611no—
205#1485—611no—
206#1299—611no—
207#1276—611no—
208C2×C61—41no—
209C2×C61—81no—
210C2×C61#187111yescharacteristic
211C2×C61#187611yescharacteristic
212C2×C61#188211yescharacteristic
213#533—611no—
214C4×#41C2×D6111yescharacteristic
215#485—1221no—
216#475—611no—
217#474—611no—
218#462—611no—
219#461—611no—
220#452—1221no—
221C8×Q8—21no—
222C4×Q16—1221no—
223C4×SD16—1221no—
224#436—22no—
225#436—21no—
226#431—21no—
227#431—611no—
228#431C2×D6111yescharacteristic
229#423—21no—
230#423C2×D6111yescharacteristic
231C2×#104—1221no—
232C4×M16—611no—
233C43—611no—
234#339—1222no—
235#339—1221no—
236Q8⋊C8—1221no—
237Q8⋊C8—1221no—
238C61#897211yescharacteristic, complemented, Sylow
239C2×#41D24411yescharacteristic
240C2×Q16—1221no—
241C2×SD16—1221no—
242C2×M16—22no—
243C2×M16—1221no—
244C2×M16C22⋊D6112yes—
245C4⋊Q8—611no—
246C4⋊Q8—611no—
247#125—1221no—
248C22⋊Q8—1221no—
249C22⋊Q8—1221no—
250C4×Q8—1221no—
251C4×Q8—1221no—
252C4×Q8C4×D6111yescharacteristic
253C4×Q8C22⋊D6111yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C4×D6111yescharacteristic
257C4×D4C22⋊D6111yescharacteristic
258#117—21no—
259#117—1221no—
260#117D24411yescharacteristic
261C2×C42—21no—
262C2×C42—1221no—
263C2×C42—1221no—
264C2×C42—1221no—
265C2×C42C22×D6111yescharacteristic
266#107—1221no—
267#106—1221no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—611no—
272C4⋊C8—611no—
273C4⋊C8C22⋊D6111yescharacteristic
274C4⋊C8C22⋊D6111yescharacteristic
275#104—1224no—
276#103—1221no—
277#103—1221no—
278#103—1221no—
279#102—1221no—
280C22⋊C8—22no—
281C22⋊C8—21no—
282C22⋊C8—1221no—
283#97—1221no—
284C4×C8—22no—
285C4×C8—21no—
286C4×C8—1221no—
287C4×C8—1221no—
288#41—22no—
289#41—41no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×Q8#1175819811yescharacteristic
293C2×D4—21no—
294C2×D4#1175819811yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—2441no—
299C22×C4C2×D24411yescharacteristic
300C22×C4#1175822211yescharacteristic
301C22×C4C2×C22⋊D6111yescharacteristic
302Q16—2442no—
303SD16—2442no—
304M16—42no—
305M16—82no—
306M16—1222no—
307M16—2441no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—1221no—
312C2×C8—1221no—
313C2×C8—2441no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—1221no—
317C4⋊C4—1221no—
318C4⋊C4—1221no—
319C4⋊C4—1221no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4#1175819811yescharacteristic
323C4⋊C4#1175819811yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2441no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—1222no—
332C42—1222no—
333C42—1222no—
334C42—2441no—
335C42—2441no—
336C42—2441no—
337C42—2441no—
338C42C2×C4×D6111yescharacteristic
339C42#1175822211yescharacteristic
340C42C2×C22⋊D6111yescharacteristic
341C42C2×C22⋊D6111yescharacteristic
342C23—21no—
343C23#1451975111yescharacteristic
344Q8—21no—
345Q8—2441no—
346Q8—2441no—
347Q8—2441no—
348Q8—2441no—
349Q8#1451970312yes—
350D4—21no—
351D4—21no—
352D4—42no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4—2442no—
370C2×C4—2442no—
371C2×C4—2442no—
372C2×C4C2×#1175819811yescharacteristic
373C2×C4C4×C22⋊D6111yescharacteristic
374C2×C4C4×C22⋊D6111yescharacteristic
375C2×C4#1451974611yescharacteristic
376C2×C4#1451975111yescharacteristic
377C2×C4#1451976011yescharacteristic
378C8—42no—
379C8—41no—
380C8—41no—
381C8—2442no—
382C8—2441no—
383C8—2441no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22#1683964111yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—4881no—
398C4—4881no—
399C4—2444no—
400C4C2×#1451970311yescharacteristic
401C4#1683977811yescharacteristic
402C4#1683987711yescharacteristic
403C4#1683988211yescharacteristic
404C4#1683994011yescharacteristic
405C4#1683994511yescharacteristic
406C2—41no—
407C2—81no—
408C2#2144612111yescharacteristic
409C2#2144614611yescharacteristic
410C2#2144618811yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2991C2
socleC22×C61#386
prime radicalC2×#41×C61D4
cosocleC2×C42×C61C23
hypercenter#2991D61
hypocenterC61#8972
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2991

Complements

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

NH ≅ G/Ntype
2. #21446795C2semidirect
4. #21444239C2semidirect
5. #21443708C2semidirect
6. #21443651C2semidirect
10. #16840047C4semidirect
11. #16840042C4semidirect
12. #16839972C4semidirect
13. #16839967C4semidirect
238. C61#8972semidirect

Maximal subgroups

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

subgroupindex
#214558902
#214467952
C4×#145197042
#214442392
#214437082
#214436512
C61×#29912
#897261