Finite Group Databasebeta

← #30595504

Subgroups — #30595504

Order 15616.

676 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 110 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#21461711C211yesmaximal, characteristic
2#21461483C211yesmaximal, characteristic
3#21460640C212yesmaximal, complemented
4#21460639C212yesmaximal, complemented
5#391×D61C211yesmaximal, characteristic
6C22×#11758213—21no—
7C2×#14519834—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C22×Q8×D61C411yescharacteristic
13C2×C4⋊C4×D61C411yescharacteristic
14C2×C42×D61C2211yescharacteristic
15#16839669C2211yescharacteristic
16#16839668C411yescharacteristic
17#16839667C2211yescharacteristic
18#16839666—21no—
19#16839666—21no—
20#95×D61—21no—
21#95×D61—21no—
22C61×#391C411yescharacteristic
23C2×#11758213—24no—
24C2×#11758213—42no—
25C2×#11758213—42no—
26C22×#92420—22no—
27C22×#92420—22no—
28#14519834—42no—
29C22×Dic122—21no—
30C22×Dic122C2×C411yescharacteristic
31C2×Q8×D61—22no—
32C2×Q8×D61—22no—
33C2×Q8×D61—42no—
34C2×Q8×D61—42no—
35C22×C4×D61—21no—
36C22×C4×D61—21no—
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C2×C4⋊Dic61C2×C411yescharacteristic
41C2×#11758196—21no—
42C4⋊C4×D61—42no—
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C2×C4×Dic61C2311yescharacteristic
46C42×D61—42no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53C22×Q8×C61C2×C411yescharacteristic
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C42×C61C2×C411yescharacteristic
56#95×C61—21no—
57#95×C61—21no—
58C23×D61C22⋊C411yescharacteristic
59C2×Q8×C61—21no—
60C2×Q8×C61—41no—
61C2×Q8×C61—41no—
62C2×Q8×C61M1612yescomplemented
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73Q8×D61—48no—
74Q8×D61—48no—
75C2×C4×D61—22no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—22no—
81C2×C4×D61—24no—
82C2×C4×D61—22no—
83C2×C4×D61—22no—
84C2×C4×D61—41no—
85C2×C4×D61—42no—
86C2×C4×D61—42no—
87C2×C4×D61—42no—
88C2×C4×D61—42no—
89C2×C4×D61—42no—
90C2×C4×D61SD1612yes—
91C2×C4×D61Q1612yes—
92C2×Dic122—24no—
93C2×Dic122—21no—
94C2×Dic122—42no—
95C2×Dic122—42no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×Dic122M1612yescomplemented
99C2×#92420—24no—
100C2×#92420—42no—
101C2×#92420—42no—
102C2×#92420—42no—
103C2×#92420—42no—
104C2×#92420—42no—
105#11758213—48no—
106C4⋊C4×C61—41no—
107C42×C61—41no—
108C4⋊Dic61—41no—
109#11758196—42no—
110C4×Dic61—41no—
111C4×Dic61—41no—
112C4×Dic61—41no—
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—21no—
118C22×D61#9912yes—
119C22×D61#10312yes—
120C22×D61#10412yes—
121Q8×C61—42no—
122Q8×C61—42no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61#10312yes—
137C2×C4×C61C2×M1611yescharacteristic
138C2×C4×C61C2×M1611yescharacteristic
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—22no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61#10012yes—
152C2×Dic61C2×SD1611yescharacteristic
153C2×Dic61C2×Q1611yescharacteristic
154C2×Dic61#13712yes—
155C4×D61—24no—
156C4×D61—24no—
157C4×D61—24no—
158C4×D61—24no—
159C4×D61—44no—
160C4×D61—48no—
161C4×D61—84no—
162Dic122—48no—
163Dic122—42no—
164Dic122—42no—
165#92420—44no—
166#92420—82no—
167#7609—611nomaximal, Sylow
168C22×C61#40711yescharacteristic
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#10011yescharacteristic
171C22×C61C2×#10311yescharacteristic
172C22×C61#41911yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C22×C61#42111yescharacteristic
175C2×D61—22no—
176C2×D61—22no—
177C2×D61—22no—
178C2×D61—22no—
179C2×D61—22no—
180C2×D61—22no—
181C2×D61#32814yes—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—22no—
185C4×C61—42no—
186C4×C61—41no—
187C4×C61—81no—
188Dic61—22no—
189Dic61—21no—
190Dic61—21no—
191Dic61—22no—
192Dic61—42no—
193Dic61—81no—
194C2×#391—611no—
195#1817—611no—
196#1728—611no—
197#1272—612no—
198#1272—612no—
199C2×C61#146211yescharacteristic
200C2×C61#148711yescharacteristic
201C2×C61#149111yescharacteristic
202C2×C61C2×#32811yescharacteristic
203C2×C61#151111yescharacteristic
204C2×C61#152711yescharacteristic
205C2×C61#154811yescharacteristic
206D61—24no—
207C23×Q8—611no—
208C22×M16—1221no—
209C22×C4⋊C4—611no—
210C22×C42—611no—
211C2×C22⋊C8—1221no—
212#391—611no—
213#391—611no—
214#391—611no—
215#391—1221no—
216#391—1221no—
217#391#692311yescharacteristic
218C2×#95—1221no—
219C2×#95—1221no—
220#336—612no—
221#336—612no—
222#336—1221no—
223#336—1221no—
224C61#760911yescharacteristic, complemented, Sylow
225C22×Q8—611no—
226C22×Q8—1221no—
227C22×Q8—1222no—
228C22×Q8—1222no—
229C22×Q8—2442no—
230C22×Q8—2442no—
231C22×Q8C2×#692311yescharacteristic
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—611no—
235C23×C4—1221no—
236C23×C4—1221no—
237C2×M16—1224no—
238C2×M16—2442no—
239C2×M16—2442no—
240C22×C8—1222no—
241C22×C8—1222no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4—2442no—
245C2×C4⋊C4C2×#692311yescharacteristic
246C2×C42—611no—
247C2×C42—611no—
248C2×C42—611no—
249C2×C42—2442no—
250C2×C42C2×#692311yescharacteristic
251C22⋊C8—2442no—
252#95—21no—
253#95—21no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260C24—611no—
261C2×Q8—21no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—612no—
265C2×Q8—1224no—
266C2×Q8—1221no—
267C2×Q8—2441no—
268C2×Q8—2441no—
269C2×Q8—2442no—
270C2×Q8—2442no—
271C2×Q8—2448no—
272C2×Q8—2448no—
273C2×Q8#1175821312yescomplemented
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—612no—
280C22×C4—612no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—1222no—
286C22×C4—1222no—
287C22×C4—1224no—
288C22×C4—1221no—
289C22×C4—1221no—
290C22×C4—1221no—
291C22×C4—1221no—
292C22×C4—2441no—
293C22×C4—2442no—
294C22×C4—2442no—
295C22×C4—2442no—
296C22×C4—2442no—
297C22×C4—2442no—
298C22×C4C22⋊#692311yescharacteristic
299C22×C4C22⋊#692311yescharacteristic
300C22×C4C22×#692311yescharacteristic
301M16—2448no—
302C2×C8—1224no—
303C2×C8—2442no—
304C2×C8—2442no—
305C2×C8—2442no—
306C2×C8—2442no—
307C2×C8—2442no—
308C4⋊C4—41no—
309C4⋊C4—2441no—
310C4⋊C4—2442no—
311C42—41no—
312C42—2441no—
313C42—2441no—
314C42—2441no—
315C23—612no—
316C23—612no—
317C23—612no—
318C23—1221no—
319C23—1221no—
320C23—1221no—
321C23—1221no—
322C23C2×C22⋊#692311yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—2442no—
326Q8—2442no—
327Q8—2448no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—612no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—612no—
345C2×C4—1224no—
346C2×C4—1224no—
347C2×C4—1224no—
348C2×C4—1224no—
349C2×C4—1221no—
350C2×C4—1221no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—1221no—
354C2×C4—1221no—
355C2×C4—1222no—
356C2×C4—4884no—
357C2×C4—2444no—
358C2×C4—2448no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4#1451984012yes—
365C2×C4D61⋊M1611yescharacteristic
366C2×C4C2×#1175821311yescharacteristic
367C8—4882no—
368C8—2444no—
369C22—614no—
370C22—1222no—
371C22—1222no—
372C22—1222no—
373C22—1222no—
374C22—1222no—
375C22—1222no—
376C22#1684073311yescharacteristic
377C22D61⋊#9911yescharacteristic
378C22D61⋊#10011yescharacteristic
379C22C2×#1451984011yescharacteristic
380C22#1684075511yescharacteristic
381C22D61⋊#10411yescharacteristic
382C22#1684075911yescharacteristic
383C4—21no—
384C4—21no—
385C4—22no—
386C4—42no—
387C4—41no—
388C4—81no—
389C4—1222no—
390C4—1221no—
391C4—1221no—
392C4—1222no—
393C4—4881no—
394C4—2442no—
395C2—1224no—
396C2#2146106511yescharacteristic
397C2#2146114811yescharacteristic
398C2#2146116211yescharacteristic
399C2D61⋊#32811yescharacteristic
400C2#2146122811yescharacteristic
401C2#2146127911yescharacteristic
402C2#2146133311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#7609
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #21460640C2semidirect
4. #21460639C2semidirect
62. C2×Q8×C61M16semidirect
98. C2×Dic122M16semidirect
224. C61#7609semidirect
273. C2×Q8#11758213semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214614832
#214606402
#214606392
#391×D612
#760961