Finite Group Databasebeta

← #30606299

Subgroups — #30606299

Order 15616.

854 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×#16840743C212yesmaximal
2#21460914C212yesmaximal, complemented
3#21460913C212yesmaximal, complemented
4#21448308C211yesmaximal, characteristic
5C2×#14519856—22no—
6#16840743—24no—
7#16840743C418yescomplemented
8C2×#14519836—22no—
9C2×#14519836C2214yes—
10C22×#11758226C2211yescharacteristic
11C2×#14519735C411yescharacteristic, complemented
12C2×C42×D61C411yescharacteristic, complemented
13#16839669C411yescharacteristic, complemented
14#16839664C2211yescharacteristic
15#16839663C2211yescharacteristic
16#16839640—22no—
17#16839609—22no—
18C61×#391C411yescharacteristic, complemented
19#14519856—28no—
20#14519856—44no—
21C2×C4⋊#6923—42no—
22C2×C22⋊#6923—24no—
23C2×C4×#6923—42no—
24#14519836—28no—
25#14519836—24no—
26#14519836C2×C418yescomplemented
27C2×#11758226—42no—
28C2×#11758226—41no—
29C2×#11758226—41no—
30C2×#11758226C2×C414yes—
31C2×#11758226D412yes—
32C2×#11758226Q812yes—
33C22×D244D412yes—
34C22×D244C2311yescharacteristic
35C22×C4×D61—21no—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61D412yes—
38#14519735—41no—
39#14519735—41no—
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×C4×Dic61—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61C2×C411yescharacteristic
47C42×D61—42no—
48#14519684—22no—
49C22×Q8×C61C2×C411yescharacteristic
50C2×C4⋊C4×C61C2×C411yescharacteristic
51C2×C42×C61C2×C411yescharacteristic
52#95×C61—22no—
53C23×D61C22⋊C411yescharacteristic
54C23×D61C2×D412yes—
55C22×#6923—44no—
56C2×Q8×C61—41no—
57C2×Q8×C61—41no—
58C2×Q8×C61C4212yes—
59C2×Q8×C61C4⋊C412yes—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61C22⋊C411yescharacteristic
67#11758226—44no—
68#11758226—44no—
69#11758226—42no—
70#11758226—42no—
71#11758226—42no—
72#11758226—42no—
73C2×D244—42no—
74C2×D244—42no—
75C2×D244—41no—
76C2×D244—41no—
77C2×D244C22⋊C414yes—
78C2×D244C4⋊C414yes—
79C2×D244C22×C412yes—
80C2×D244C2×D411yescharacteristic
81C2×D244C2×Q811yescharacteristic
82C2×C4×D61—22no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—22no—
86C2×C4×D61—22no—
87C2×C4×D61—24no—
88C2×C4×D61—42no—
89C2×C4×D61—42no—
90C2×C4×D61—42no—
91C2×C4×D61C4212yes—
92C2×C4×D61C22⋊C414yes—
93C2×C4×D61C4⋊C414yes—
94C2×C4×D61C4⋊C412yes—
95C22⋊#6923—28no—
96C22⋊#6923—44no—
97C4⋊#6923—48no—
98C4×#6923—48no—
99C4⋊C4×C61—41no—
100C42×C61—41no—
101#11758198—44no—
102#11758198—44no—
103#11758198—42no—
104C4⋊Dic61—41no—
105C4×Dic61—42no—
106C4×Dic61—41no—
107C4×Dic61—41no—
108C4×Dic61—41no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—22no—
113C22×D61—22no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61#9512yes—
119C22×D61C2×C22⋊C412yes—
120C22×D61C2×C4⋊C412yes—
121C2×#6923—48no—
122C2×#6923—84no—
123C2×#6923—84no—
124Q8×C61—42no—
125Q8×C61—42no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61#9514yes—
136C2×C4×C61C2×C4211yescharacteristic
137C2×C4×C61C2×C4⋊C411yescharacteristic
138C2×Dic61—22no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61#9512yes—
144C2×Dic61#9912yes—
145C2×Dic61#9911yescharacteristic
146C2×Dic61#9911yescharacteristic
147D244—44no—
148D244—44no—
149D244—42no—
150D244—41no—
151D244—41no—
152C4×D61—24no—
153C4×D61—24no—
154C4×D61—22no—
155C4×D61—22no—
156C4×D61—24no—
157C4×D61—24no—
158C4×D61—22no—
159C4×D61—22no—
160C4×D61—48no—
161#12581—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61C2×#9511yescharacteristic
165C22×C61C2×#9911yescharacteristic
166C22×C61C2×#9911yescharacteristic
167C2×D61—22no—
168C2×D61—22no—
169C2×D61—22no—
170C2×D61—42no—
171C2×D61—42no—
172C2×D61—42no—
173C2×D61—41no—
174C2×D61—41no—
175C2×D61—42no—
176C2×D61—42no—
177C2×D61—42no—
178C2×D61—82no—
179#6923—88no—
180C4×C61—24no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61—81no—
185Dic61—42no—
186Dic61—81no—
187Dic61#34212yes—
188Dic61#34212yes—
189C2×#410—612no—
190#2297—611no—
191#1393—612no—
192#1393—612no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61C2×#34211yescharacteristic
196C2×C61#212611yescharacteristic
197C2×C61#212611yescharacteristic
198D61—42no—
199D61—42no—
200D61—42no—
201C22×#41—611no—
202C2×#117—611no—
203C2×#117—1222no—
204C22×C42—611no—
205#410—618no—
206#410—1224no—
207C2×#99—614no—
208C2×#99—1222no—
209#391—611no—
210#391#692311yescharacteristic, complemented
211#390—611no—
212#390—611no—
213#386—1222no—
214C4×C22⋊C4—1222no—
215C61#1258111yescharacteristic, complemented, Sylow
216C2×#41—614no—
217C2×#41—612no—
218C2×#41—612no—
219C2×#41—2441no—
220C2×#41—2442no—
221C2×#41—2441no—
222C22×Q8C2×#692311yescharacteristic
223C22×D4—612no—
224C22×D4—611no—
225C23×C4—612no—
226C23×C4—611no—
227C23×C4—1221no—
228#117—1228no—
229#117—2444no—
230#117—2441no—
231#117—2441no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—2442no—
234C2×C4⋊C4C2×#692311yescharacteristic
235C2×C22⋊C4—1224no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C42—611no—
240C2×C42—1221no—
241C2×C42—1221no—
242C2×C42—2442no—
243C2×C42—2442no—
244C2×C42C2×#692311yescharacteristic
245#99—618no—
246#99—1224no—
247#99—1228no—
248#95—22no—
249#95—1222no—
250C24—612no—
251C24—611no—
252#41—2442no—
253#41—2444no—
254#41—2442no—
255#41—2444no—
256#41—2442no—
257#41—2442no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8C4×#692312yes—
261C2×Q8C4⋊#692312yes—
262C2×D4—614no—
263C2×D4—614no—
264C2×D4—612no—
265C2×D4—611no—
266C2×D4—611no—
267C2×D4—2441no—
268C2×D4—2442no—
269C2×D4—2441no—
270C2×D4—2442no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—611no—
274C22×C4—614no—
275C22×C4—614no—
276C22×C4—612no—
277C22×C4—612no—
278C22×C4—1222no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1224no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1221no—
286C22×C4—2442no—
287C22×C4—2442no—
288C22×C4—2442no—
289C22×C4—2444no—
290C22×C4C22⋊#692312yes—
291C22×C4C22×#692311yescharacteristic
292C4⋊C4—41no—
293C4⋊C4—2441no—
294C4⋊C4—2448no—
295C22⋊C4—1228no—
296C22⋊C4—2444no—
297C22⋊C4—2444no—
298C22⋊C4—2442no—
299C22⋊C4—2444no—
300C42—41no—
301C42—2441no—
302C42—2442no—
303C42—2441no—
304C42—2441no—
305C42—2448no—
306C23—612no—
307C23—612no—
308C23—612no—
309C23—1222no—
310C23—1221no—
311C23—1221no—
312C23—1222no—
313C23—2442no—
314C23—2442no—
315C23—2441no—
316C23—2441no—
317C23C2×C22⋊#692311yescharacteristic
318Q8—42no—
319Q8—42no—
320D4—2444no—
321D4—2444no—
322D4—2442no—
323D4—2441no—
324D4—2441no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—612no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—612no—
338C2×C4—1224no—
339C2×C4—1224no—
340C2×C4—1224no—
341C2×C4—1224no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1222no—
347C2×C4—4884no—
348C2×C4—4884no—
349C2×C4—2448no—
350C2×C4—2442no—
351C2×C4—2448no—
352C2×C4—2441no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4(C2×#6923)⋊C414yes—
356C2×C4C2×C4×#692311yescharacteristic
357C2×C4C2×C4⋊#692311yescharacteristic
358C22—21no—
359C22—21no—
360C22—1222no—
361C22—1222no—
362C22—1222no—
363C22—4882no—
364C22—2442no—
365C22—2442no—
366C22—2442no—
367C22—2442no—
368C22—2442no—
369C22—2441no—
370C22—2442no—
371C22—2441no—
372C22C2×(C2×#6923)⋊C411yescharacteristic
373C22D61⋊#9911yescharacteristic
374C22D61⋊#9911yescharacteristic
375C4—24no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—81no—
380C4—612no—
381C4—612no—
382C4—4888no—
383C4—4881no—
384C4—2442no—
385C2—21no—
386C2—21no—
387C2—2442no—
388C2—2442no—
389C2—2442no—
390C2#2146151611yescharacteristic
391C2#2146209111yescharacteristic
392C2#2146209311yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC22×C61C2×#99
prime radicalC22×#11758226C22
cosocleC22×D244C23
hypercenter#391#6923
hypocenterC61#12581
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #21460914C2semidirect
3. #21460913C2semidirect
7. #16840743C4semidirect
11. C2×#14519735C4semidirect
12. C2×C42×D61C4semidirect
13. #16839669C4semidirect
18. C61×#391C4semidirect
26. #14519836C2×C4semidirect
210. #391#6923semidirect
215. C61#12581semidirect

Maximal subgroups

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

subgroupindex
C2×#168407432
#214609142
#214609132
#214483082
#1258161