Finite Group Databasebeta

← #29576906

Subgroups — #29576906

Order 15616.

612 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455918C211yesmaximal, characteristic
2#21448180C211yesmaximal, characteristic
3#21446763C211yesmaximal, characteristic, complemented
4#21444332C211yesmaximal, characteristic, complemented
5#21443713C211yesmaximal, characteristic, complemented
6#21443674C211yesmaximal, characteristic, complemented
7C61×#1490C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9C2×D4×Dic61—21no—
10#16840097—21no—
11#16840042C412yescomplemented
12#16839967C412yescomplemented
13#16839913—21no—
14#16839862—22no—
15#16839819C2211yescharacteristic
16C4⋊C4×Dic61—21no—
17C4×C4⋊Dic61C2211yescharacteristic
18C22⋊C4×Dic61—22no—
19C4×C22⋊Dic61—21no—
20#16839608C2211yescharacteristic
21C2×#14519684—21no—
22#16839523—22no—
23#16839500—21no—
24#16839500—21no—
25C2×C4×D4×C61C2211yescharacteristic
26C61×#432C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C61×D4⋊C8—22no—
29C23×Dic61—22no—
30C2×#11758200—22no—
31#14519777C2×C411yescharacteristic, complemented
32#14519777C2×C411yescharacteristic, complemented
33#14519769—21no—
34#14519756—21no—
35#14519756—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38D4×Dic61—28no—
39#14519736—21no—
40C2×C4⋊Dic61—21no—
41C2×C22⋊Dic61—22no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61D412yes—
45#14519710—22no—
46C4⋊#92418C2×C412yescomplemented
47#14519700—22no—
48#14519698—22no—
49C22⋊#92418—21no—
50#14519688—21no—
51#14519684—22no—
52#14519684—21no—
53#14519684—21no—
54#14519684D411yescharacteristic
55#14519684D411yescharacteristic
56C22×D4×C61D411yescharacteristic
57C23×C4×C61—21no—
58C2×M16×C61—21no—
59C4×D4×C61—21no—
60C4×D4×C61—21no—
61C4×D4×C61C2×C412yes—
62C4×D4×C61D412yes—
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C22⋊C4×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68#97×C61—21no—
69#97×C61—21no—
70C4×C8×C61—21no—
71C24×C61—21no—
72C2×D4×C61—24no—
73C2×D4×C61—21no—
74C2×D4×C61C22⋊C412yes—
75C22×C4×C61—21no—
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—82no—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×Dic61—21no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C22×Dic61—28no—
86C22×Dic61—82no—
87C22×Dic61—82no—
88C2×Dic122—21no—
89C2×Dic122—21no—
90M16×C61—22no—
91M16×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61C22⋊C412yes—
96C22⋊C4×C61—22no—
97C22⋊C4×C61—22no—
98C42×C61C22×C411yescharacteristic
99C42×C61C2×D411yescharacteristic
100C42×C61#4111yescharacteristic
101C42×C61#4111yescharacteristic
102C22⋊Dic61—28no—
103C22⋊Dic61—41no—
104#11758200—44no—
105C4⋊Dic61—24no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—41no—
111#11758196—41no—
112C4×Dic61—24no—
113C4×Dic61—41no—
114C4×Dic61—42no—
115C4×Dic61—82no—
116C2×#92418—22no—
117C23×C61—24no—
118C23×C61—22no—
119C23×C61—42no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61#12011yescharacteristic
123D4×C61—24no—
124D4×C61—22no—
125D4×C61#10414yes—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—82no—
133C2×C4×C61—82no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12311yescharacteristic
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—24no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—88no—
152Dic122—42no—
153Dic122—81no—
154C8×C61—42no—
155C8×C61—81no—
156#92418—42no—
157#9323—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—24no—
161C22×C61—44no—
162C22×C61—44no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#38611yescharacteristic
166C22×C61#45511yescharacteristic
167C22×C61#45611yescharacteristic
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—42no—
174C4×C61C2×#10412yes—
175Dic61—42no—
176Dic61—44no—
177Dic61—42no—
178Dic61—81no—
179Dic61—81no—
180#3110—611no—
181#2275—611no—
182#1986—611no—
183#1506—611no—
184#1490D6111yescharacteristic
185#1300—611no—
186#1285—611no—
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—44no—
190C2×C61#188211yescharacteristic
191C2×C61#189611yescharacteristic
192C2×C61#190411yescharacteristic
193#533—611no—
194C2×C4×D4—1221no—
195C2×C4×D4C2×D6111yescharacteristic
196#484—1221no—
197#474—612no—
198#461—612no—
199#450—1221no—
200#440—1222no—
201#432—611no—
202#432C2×D6111yescharacteristic
203C4×M16C2×D6111yescharacteristic
204C4×C4⋊C4—611no—
205C4×C4⋊C4—1221no—
206C4×C22⋊C4—1221no—
207C4×C22⋊C4—1222no—
208#376—611no—
209C2×#95—1221no—
210#339—1222no—
211#328—1221no—
212#328—1221no—
213D4⋊C8—22no—
214C61#932311yescharacteristic, complemented, Sylow
215C22×D4D24411yescharacteristic
216C23×C4—21no—
217C23×C4—1222no—
218C2×SD16—1222no—
219C2×M16—21no—
220C4⋊Q8—611no—
221C4⋊Q8—611no—
222#125—1221no—
223C22⋊Q8—1221no—
224C22⋊Q8—1221no—
225C4×Q8—1221no—
226C4×Q8—1221no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—1228no—
230C4×D4C4×D6112yes—
231C4×D4C22⋊D6112yes—
232#117—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4D24411yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—1222no—
237C2×C42—612no—
238C2×C42—1221no—
239C2×C42—1222no—
240C2×C42C22×D6111yescharacteristic
241#107—1222no—
242C4⋊C8—21no—
243C4⋊C8—612no—
244#103—1222no—
245#102—1222no—
246C22⋊C8—21no—
247C22⋊C8—1221no—
248#97—21no—
249#97—21no—
250#97—1221no—
251C4×C8—21no—
252#95—611no—
253#95—611no—
254#95—1222no—
255#95—1221no—
256#95—1221no—
257C24—21no—
258C2×Q8—1221no—
259C2×Q8—1221no—
260C2×D4—24no—
261C2×D4—21no—
262C2×D4#1175819812yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—82no—
267C22×C4—1228no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—1222no—
271C22×C4—4882no—
272C22×C4—4882no—
273C22×C4C2×D24411yescharacteristic
274C22×C4D4×D6111yescharacteristic
275C22×C4D4×D6111yescharacteristic
276SD16—2444no—
277M16—22no—
278M16—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—1222no—
282C4⋊C4—21no—
283C4⋊C4—1224no—
284C4⋊C4—1221no—
285C4⋊C4—1221no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C4⋊C4#1175819812yes—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—1228no—
294C22⋊C4—2441no—
295C42—1224no—
296C42—4882no—
297C42—2442no—
298C42—2441no—
299C42C2×C4×D6111yescharacteristic
300C42#1175822411yescharacteristic
301C42#1175822411yescharacteristic
302C42C2×C22⋊D6111yescharacteristic
303C23—24no—
304C23—22no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23#1451974711yescharacteristic
309Q8—4881no—
310Q8—2442no—
311D4—24no—
312D4—22no—
313D4#1451970314yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—1222no—
323C2×C4—1222no—
324C2×C4—1224no—
325C2×C4—4888no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2442no—
330C2×C4—2441no—
331C2×C4—2442no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4C2×#1175819811yescharacteristic
335C2×C4#1451973711yescharacteristic
336C2×C4#1451973711yescharacteristic
337C2×C4#1451974911yescharacteristic
338C2×C4#1451974911yescharacteristic
339C2×C4#1451976111yescharacteristic
340C8—42no—
341C8—81no—
342C8—2442no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22#1683963911yescharacteristic
351C22#1683994111yescharacteristic
352C22#1683994611yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—4881no—
360C4—4881no—
361C4—2442no—
362C4—2444no—
363C4—2442no—
364C4C2×#1451970312yes—
365C2—21no—
366C2—21no—
367C2—44no—
368C2#2144618611yescharacteristic
369C2#2144626011yescharacteristic
370C2#2144630411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1490C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1490D61
hypocenterC61#9323
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
3. #21446763C2semidirect
4. #21444332C2semidirect
5. #21443713C2semidirect
6. #21443674C2semidirect
11. #16840042C4semidirect
12. #16839967C4semidirect
31. #14519777C2×C4semidirect
32. #14519777C2×C4semidirect
46. C4⋊#92418C2×C4semidirect
214. C61#9323semidirect

Maximal subgroups

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

subgroupindex
#214559182
#214481802
#214467632
#214443322
#214437132
#214436742
C61×#14902
#932361