Finite Group Databasebeta

← #30598343

Subgroups — #30598343

Order 15616.

712 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C4×#14519838C211yesmaximal, characteristic
2#21461281C212yesmaximal
3#21461161C212yesmaximal
4#21461081C211yesmaximal, characteristic
5C2×#16840488C211yesmaximal, characteristic
6#16840783C2212yes—
7C2×#14519838—21no—
8C4×#11758217C2212yes—
9#16840642—21no—
10#16840619—24no—
11Q8⋊#92420—24no—
12#16840608—21no—
13#16840607—21no—
14#16840488—24no—
15#16840488—24no—
16C2×#14519776—21no—
17C2×#14519776C411yescharacteristic
18C2×#14519769—21no—
19C2×#14519769C411yescharacteristic
20C2×#14519768—21no—
21C2×#14519768C2211yescharacteristic
22C2×Q8×Dic61C411yescharacteristic
23C2×Q8×Dic61C2211yescharacteristic
24C2×C4×Dic122—21no—
25C2×C4×Dic122C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic
27C2×#11758217—22no—
28C4⋊#92420—22no—
29#14519838—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33C4×#92420—22no—
34C22×Dic122—21no—
35C22×Dic122D411yescharacteristic
36#14519776—28no—
37#14519776—22no—
38#14519776—22no—
39#14519769—28no—
40#14519769—22no—
41#14519769—22no—
42#14519768—28no—
43#14519768—22no—
44#14519768D414yes—
45Q8×Dic61—22no—
46Q8×Dic61—22no—
47Q8×Dic61—22no—
48Q8×Dic61D414yes—
49C4×Dic122—28no—
50C4×Dic122—22no—
51C4×Dic122—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×C4⋊Dic61D411yescharacteristic
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758196D411yescharacteristic
63C2×#11758196D411yescharacteristic
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C2×C4×Dic61C2311yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61—22no—
70C4×Q8×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C42×C61—21no—
74C2×C42×C61C2×C411yescharacteristic
75C2×Q8×C61—24no—
76C2×Q8×C61—21no—
77C2×Q8×C61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C2×Dic122—24no—
90C2×Dic122—22no—
91C2×Dic122—22no—
92C2×Dic122—41no—
93C2×Dic122—42no—
94C2×Dic122—42no—
95#11758217—24no—
96#11758217—42no—
97C2×#92420—42no—
98C2×#92420—42no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61C22⋊C412yes—
102C42×C61—24no—
103C42×C61—21no—
104C42×C61—21no—
105C4⋊Dic61—24no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—24no—
109C4⋊Dic61—24no—
110C4⋊Dic61—22no—
111C4⋊Dic61C22⋊C412yes—
112C4⋊Dic61C22⋊C412yes—
113#11758196—24no—
114#11758196—24no—
115#11758196—24no—
116#11758196—24no—
117#11758196—22no—
118#11758196—22no—
119C4×Dic61—24no—
120C4×Dic61—21no—
121C4×Dic61—21no—
122C4×Dic61—24no—
123C4×Dic61C2×D412yes—
124C4×Dic61C2×D412yes—
125C23×C61C2×C22⋊C411yescharacteristic
126Q8×C61—24no—
127Q8×C61—22no—
128Q8×C61#10114yescomplemented
129C2×C4×C61—24no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—22no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—21no—
147C2×Dic61—22no—
148C2×Dic61—22no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61#10412yes—
155C2×Dic61#12012yes—
156C2×Dic61#12012yes—
157Dic122—44no—
158Dic122—48no—
159#92420—44no—
160#92420—82no—
161#8374—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61#37911yescharacteristic
165C22×C61C2×#10411yescharacteristic
166C22×C61#42111yescharacteristic
167C4×C61—24no—
168C4×C61—21no—
169C4×C61—41no—
170C4×C61—41no—
171C4×C61—42no—
172C4×C61C2×#10112yes—
173Dic61—22no—
174Dic61—22no—
175Dic61—22no—
176Dic61—41no—
177Dic61—42no—
178Dic61—42no—
179C2×#557—611no—
180C4×#101—611no—
181#1528—612no—
182#1491—612no—
183#1467—611no—
184C2×C61—21no—
185C2×C61—21no—
186C2×C61#178411yescharacteristic
187C2×C61#178911yescharacteristic
188C2×C61#179611yescharacteristic
189#557—1224no—
190#557—1224no—
191C2×C4⋊Q8—611no—
192C2×C4⋊Q8—1221no—
193C2×#125—611no—
194C2×#125—611no—
195C2×#125—1221no—
196C2×#125—1221no—
197C2×C4×Q8—611no—
198C2×C4×Q8—611no—
199C2×C4×Q8—611no—
200C2×C4×Q8—1221no—
201C2×C4×Q8#692311yescharacteristic
202#432—612no—
203C2×#101—1221no—
204C4×M16—612no—
205#343—1221no—
206#330—1224no—
207Q8⋊C8—1224no—
208#324—1221no—
209#324—1221no—
210C61#837411yescharacteristic, complemented, Sylow
211C22×Q8—611no—
212C22×Q8—1221no—
213C22×Q8C2×#692311yescharacteristic
214C2×M16—1222no—
215C4⋊Q8—1222no—
216C4⋊Q8—1228no—
217C4⋊Q8—1222no—
218#125—614no—
219#125—1222no—
220#125—1222no—
221#125—1228no—
222#125—1222no—
223#125—1228no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—614no—
227C4×Q8—1222no—
228C4×Q8—1222no—
229C4×Q8—1228no—
230C4×Q8—1222no—
231C4×Q8—1222no—
232C4×Q8—1222no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—611no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—1221no—
245C2×C4⋊C4—1221no—
246C2×C4⋊C4C2×#692311yescharacteristic
247C2×C42—21no—
248C2×C42—611no—
249C2×C42—611no—
250C2×C42—1221no—
251C2×C42—1221no—
252C2×C42C2×#692311yescharacteristic
253C4⋊C8—1222no—
254#101—1224no—
255C22⋊C8—1222no—
256#97—1222no—
257#97—1222no—
258C4×C8—1222no—
259C2×Q8—24no—
260C2×Q8—21no—
261C2×Q8—1222no—
262C2×Q8—1222no—
263C2×Q8—1224no—
264C2×Q8—2441no—
265C2×Q8—2442no—
266C2×Q8—2442no—
267C2×Q8C22⋊#692312yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4C22⋊#692311yescharacteristic
277C22×C4C22⋊#692311yescharacteristic
278C22×C4C22×#692311yescharacteristic
279M16—1224no—
280M16—2442no—
281C2×C8—2442no—
282C2×C8—2442no—
283C4⋊C4—24no—
284C4⋊C4—21no—
285C4⋊C4—612no—
286C4⋊C4—612no—
287C4⋊C4—1222no—
288C4⋊C4—1222no—
289C4⋊C4—1222no—
290C4⋊C4—1224no—
291C4⋊C4—1224no—
292C4⋊C4—1224no—
293C4⋊C4—1224no—
294C4⋊C4—1224no—
295C4⋊C4—1224no—
296C4⋊C4—1224no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C4⋊C4C22⋊#692312yes—
300C42—24no—
301C42—21no—
302C42—21no—
303C42—612no—
304C42—612no—
305C42—1224no—
306C42—1224no—
307C42—1221no—
308C42—1221no—
309C23C2×C22⋊#692311yescharacteristic
310Q8—24no—
311Q8—22no—
312Q8—2444no—
313Q8—2448no—
314Q8#1451983814yescomplemented
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—612no—
327C2×C4—612no—
328C2×C4—612no—
329C2×C4—1221no—
330C2×C4—1221no—
331C2×C4—1221no—
332C2×C4—1221no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—1221no—
336C2×C4—2441no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4C2×C22⋊#692311yescharacteristic
342C2×C4C2×C22⋊#692311yescharacteristic
343C8—4882no—
344C8—2444no—
345C22—21no—
346C22—21no—
347C22#1684070311yescharacteristic
348C22D61⋊#10411yescharacteristic
349C22#1684075911yescharacteristic
350C4—24no—
351C4—21no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—1222no—
356C4—1222no—
357C4—1222no—
358C4—2442no—
359C4—2441no—
360C4—2442no—
361C4C2×#1451983812yes—
362C2—21no—
363C2—21no—
364C2#2146165311yescharacteristic
365C2#2146166111yescharacteristic
366C2#2146167111yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8374
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
128. Q8×C61#101semidirect
210. C61#8374semidirect
314. Q8#14519838semidirect

Maximal subgroups

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

subgroupindex
C4×#145198382
#214612812
#214611612
#214610812
C2×#168404882
#837461