Finite Group Databasebeta

← #30604817

Subgroups — #30604817

Order 15616.

592 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461516C211yesmaximal, characteristic
2D61⋊#323C211yesmaximal, characteristic
3#21460816C212yesmaximal, complemented
4#21460813C212yesmaximal, complemented
5#21451625C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7D61⋊(C22⋊C8)—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×#14519757C411yescharacteristic, complemented
13C2×C22⋊C4×D61C2211yescharacteristic
14#16839697C2211yescharacteristic
15#16839685—21no—
16#16839684—21no—
17#16839675C2211yescharacteristic
18#16839674C411yescharacteristic, complemented
19#16839633—21no—
20#16839621C411yescharacteristic, complemented
21#16839617—21no—
22C61×#399C411yescharacteristic, complemented
23C23×#6923—41no—
24C2×D61⋊C8—41no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—22no—
27C2×C22⋊#6923—41no—
28C2×C22⋊#6923—42no—
29#14519834—41no—
30C22⋊#92420—22no—
31C24×D61D411yescharacteristic
32C22×Dic122C2×C411yescharacteristic, complemented
33C22×C4×D61C2×C411yescharacteristic
34C22×C4×D61D411yescharacteristic
35#14519757—41no—
36#14519757—41no—
37C2×C4⋊Dic61—21no—
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61D411yescharacteristic
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198D411yescharacteristic
47C2×#11758198C2311yescharacteristic
48C22⋊C4×D61—41no—
49C22⋊C4×D61—41no—
50#14519684—21no—
51#14519684—21no—
52#14519684C2×C411yescharacteristic, complemented
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C411yescharacteristic, complemented
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61C2×C411yescharacteristic
57#95×C61C2×C411yescharacteristic, complemented
58C24×C61C22⋊C411yescharacteristic
59C23×D61—21no—
60C23×D61—21no—
61C23×D61—22no—
62C23×D61—81no—
63C23×D61—81no—
64C23×D61C22⋊C411yescharacteristic
65C23×D61SD1612yes—
66C23×D61Q1612yes—
67C23×D61C2×D411yescharacteristic
68C22×#6923—42no—
69C22×#6923—44no—
70C22×#6923—82no—
71C22×#6923—82no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—81no—
85C2×C4×D61—81no—
86C2×Dic122—41no—
87C2×Dic122—81no—
88C22⋊#6923—24no—
89C22⋊#6923—44no—
90C22⋊#6923—42no—
91C2×#92420—42no—
92D61⋊C8—82no—
93C4⋊C4×C61—41no—
94C4⋊C4×C61—81no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—81no—
97C22⋊Dic61—41no—
98C22⋊Dic61—81no—
99#11758198—41no—
100#11758198—41no—
101#11758198—41no—
102#11758198—41no—
103#11758198—42no—
104#11758198—81no—
105#11758198—81no—
106C4⋊Dic61—42no—
107#11758196—81no—
108#11758196—81no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61#10312yes—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—22no—
118C22×D61—22no—
119C22×D61—22no—
120C22×D61—22no—
121C22×D61—41no—
122C22×D61—41no—
123C22×D61—42no—
124C22×D61—42no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—81no—
128C22×D61—81no—
129C22×D61—82no—
130C22×D61—82no—
131C22×D61—84no—
132C22×D61#9911yescharacteristic
133C22×D61#9911yescharacteristic
134C22×D61#10312yes—
135C22×D61#10412yes—
136C22×D61C2×SD1611yescharacteristic
137C22×D61C2×Q1611yescharacteristic
138C2×#6923—44no—
139C2×#6923—88no—
140C2×#6923—82no—
141C2×#6923—82no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C4×D61—42no—
163C4×D61—81no—
164C4×D61—81no—
165C4×D61—81no—
166Dic122—82no—
167#92420—82no—
168#11716—611nomaximal, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—22no—
172C22×C61—42no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—81no—
176C22×C61C2×#9911yescharacteristic
177C22×C61C2×#10311yescharacteristic
178C22×C61C2×#10411yescharacteristic
179C2×D61—21no—
180C2×D61—21no—
181C2×D61—21no—
182C2×D61—21no—
183C2×D61—21no—
184C2×D61—21no—
185C2×D61—42no—
186C2×D61—42no—
187C2×D61—44no—
188C2×D61—44no—
189C2×D61—41no—
190C2×D61—41no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—82no—
194C2×D61—82no—
195C2×D61—81no—
196C2×D61#32812yes—
197C2×D61#32812yes—
198#6923—84no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—81no—
202C4×C61—81no—
203Dic61—41no—
204Dic61—41no—
205Dic61—81no—
206Dic61—81no—
207#2626—611no—
208C2×#342—611no—
209C2×#323—611no—
210#1352—612no—
211#1351—612no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61—42no—
215C2×C61C2×#32811yescharacteristic
216C2×C61#212111yescharacteristic
217C2×C61#212711yescharacteristic
218D61—22no—
219D61—22no—
220D61—41no—
221D61—41no—
222C2×C22⋊Q8—611no—
223C22×C22⋊C4—611no—
224C22×C22⋊C4—1221no—
225C2×C22⋊C8—1221no—
226#399—611no—
227#399#692311yescharacteristic, complemented
228#396—1221no—
229#396—1221no—
230#393—611no—
231#393—611no—
232#385—1221no—
233#380—611no—
234#379—1221no—
235#342—612no—
236#342—1221no—
237#323—612no—
238#323—1221no—
239C61#1171611yescharacteristic, complemented, Sylow
240C25—611no—
241C22×Q8—611no—
242C23×C4—611no—
243C23×C4—611no—
244C23×C4—2441no—
245C22×C8—2441no—
246C22⋊Q8—2441no—
247C22⋊Q8—2441no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—1221no—
250C2×C4⋊C4—1221no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4C2×#692311yescharacteristic, complemented
253C2×C22⋊C4—21no—
254C2×C22⋊C4—611no—
255C2×C22⋊C4—611no—
256C2×C22⋊C4—611no—
257C2×C22⋊C4—1222no—
258C2×C22⋊C4—1224no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C22⋊C4—1221no—
262C2×C22⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—2441no—
265C2×C22⋊C4—2441no—
266C2×C22⋊C4—2442no—
267C2×C22⋊C4—2441no—
268C2×C22⋊C4C2×#692311yescharacteristic
269C22⋊C8—1222no—
270C22⋊C8—2441no—
271#95—611no—
272#95—1221no—
273#95—1221no—
274#95C2×#692311yescharacteristic, complemented
275C24—611no—
276C24—612no—
277C24—612no—
278C24—611no—
279C24—1222no—
280C24—1221no—
281C24—1221no—
282C24—4881no—
283C24—4881no—
284C24C22⋊#692311yescharacteristic
285C2×Q8—4881no—
286C2×Q8—2441no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—4881no—
298C22×C4—4881no—
299C22×C4—4882no—
300C22×C4—4882no—
301C22×C4—2444no—
302C22×C4—2442no—
303C22×C4C22⋊#692311yescharacteristic
304C22×C4C22×#692311yescharacteristic
305C2×C8—4882no—
306C2×C8—2442no—
307C4⋊C4—41no—
308C4⋊C4—81no—
309C4⋊C4—4881no—
310C4⋊C4—4881no—
311C4⋊C4—2442no—
312C22⋊C4—41no—
313C22⋊C4—81no—
314C22⋊C4—1224no—
315C22⋊C4—4881no—
316C22⋊C4—4881no—
317C22⋊C4—4881no—
318C22⋊C4—2441no—
319C22⋊C4—2441no—
320C22⋊C4—2442no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4—2441no—
324C22⋊C4—2442no—
325C22⋊C4—2444no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—611no—
331C23—611no—
332C23—612no—
333C23—612no—
334C23—611no—
335C23—611no—
336C23—1222no—
337C23—1222no—
338C23—1222no—
339C23—1222no—
340C23—1221no—
341C23—1221no—
342C23—4881no—
343C23—4881no—
344C23—4882no—
345C23—4882no—
346C23—4884no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23—2442no—
352C23—2442no—
353C23#1451984012yes—
354C23C2×C22⋊#692311yescharacteristic
355Q8—4882no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—1221no—
367C2×C4—1221no—
368C2×C4—4881no—
369C2×C4—4881no—
370C2×C4—4881no—
371C2×C4—4888no—
372C2×C4—4882no—
373C2×C4—4882no—
374C2×C4—2442no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4—2444no—
380C2×C4—2441no—
381C2×C4—2441no—
382C2×C4—2441no—
383C2×C4—2441no—
384C8—4882no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—81no—
392C22—612no—
393C22—612no—
394C22—1221no—
395C22—1221no—
396C22—1221no—
397C22—1221no—
398C22—1221no—
399C22—1221no—
400C22—4881no—
401C22—4882no—
402C22—4882no—
403C22—2442no—
404C22—2442no—
405C22—2441no—
406C22—2441no—
407C22—2444no—
408C22—2444no—
409C22—2441no—
410C22—2441no—
411C22D61⋊#9911yescharacteristic
412C22C2×#1451984011yescharacteristic
413C22D61⋊#10411yescharacteristic
414C4—41no—
415C4—41no—
416C4—81no—
417C4—81no—
418C4—4881no—
419C4—4884no—
420C4—4881no—
421C4—2441no—
422C4—2441no—
423C2—21no—
424C2—21no—
425C2—42no—
426C2—1222no—
427C2—1222no—
428C2—2441no—
429C2—2441no—
430C2D61⋊#32811yescharacteristic
431C2#2146207711yescharacteristic
432C2#2146209511yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61C2×#99
prime radicalC24×D61D4
cosocleC2×#11758198C23
hypercenter#399#6923
hypocenterC61#11716
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
4. #21460813C2semidirect
12. C2×#14519757C4semidirect
18. #16839674C4semidirect
20. #16839621C4semidirect
22. C61×#399C4semidirect
32. C22×Dic122C2×C4semidirect
52. #14519684C2×C4semidirect
54. C2×C4⋊C4×C61C2×C4semidirect
57. #95×C61C2×C4semidirect
227. #399#6923semidirect
239. C61#11716semidirect
252. C2×C4⋊C4C2×#6923semidirect
274. #95C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214615162
D61⋊#3232
#214608162
#214608132
#214516252
#1171661