Finite Group Databasebeta

← #30604732

Subgroups — #30604732

Order 15616.

552 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 88 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#21461518C211yesmaximal, characteristic, complemented
2#21461052C211yesmaximal, characteristic, complemented
3#21460799C212yesmaximal, complemented
4#21460797C212yesmaximal
5#21449057C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840778—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840608—21no—
11#16840606C2212yes—
12C2×C4×C22⋊D61—21no—
13C2×#14519734C2211yescharacteristic
14#16839693C411yescharacteristic, complemented
15#16839688C2211yescharacteristic
16#16839685—21no—
17#16839678—21no—
18#16839676C411yescharacteristic, complemented
19#16839670C2211yescharacteristic
20C4×C4⋊Dic61C411yescharacteristic, complemented
21C4×C22⋊Dic61—21no—
22C2×C4×D4×C61C411yescharacteristic, complemented
23#14519856—24no—
24#14519856—42no—
25C2×C4⋊#6923—41no—
26C2×C22⋊#6923—22no—
27C2×C4×#6923—41no—
28C4⋊#92420—41no—
29C22⋊#92420—22no—
30C4×#92420—41no—
31C22×C22⋊D61—21no—
32C22×C4×D61D411yescharacteristic
33C4×C22⋊D61—42no—
34C4×C22⋊D61—42no—
35#14519734—41no—
36#14519734—41no—
37C2×C4⋊Dic61—21no—
38C2×#11758196—21no—
39C2×#11758196C2×C411yescharacteristic
40C2×#11758196D411yescharacteristic
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×#11758198—21no—
45C2×#11758198D411yescharacteristic
46C2×#11758198C2311yescharacteristic
47C2×C4×Dic61—21no—
48C2×C4×Dic61D411yescharacteristic
49#14519684—21no—
50#14519684—21no—
51#14519684C2×C411yescharacteristic
52#14519684C2×C411yescharacteristic
53C22×D4×C61C2×C411yescharacteristic
54C23×C4×C61—21no—
55C4×D4×C61—41no—
56C4×D4×C61—41no—
57C2×C4⋊C4×C61C2×C411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C42×C61C2×C411yescharacteristic
60C24×C61—21no—
61C23×D61C2×D411yescharacteristic
62C22×#6923—42no—
63C2×D4×C61—41no—
64C2×D4×C61—42no—
65C22×C4×C61—21no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—22no—
85C2×C4×D61D812yes—
86C2×C4×D61SD1612yes—
87C22⋊#6923—24no—
88C22⋊#6923—42no—
89C2×#92420—42no—
90C4⋊#6923—44no—
91C4×#6923—44no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C42×C61—21no—
95C42×C61—21no—
96C22⋊Dic61—42no—
97C22⋊Dic61—42no—
98C22⋊Dic61—42no—
99#11758198—41no—
100#11758198—41no—
101#11758198—41no—
102#11758198—41no—
103C4⋊Dic61—42no—
104#11758196—41no—
105#11758196—41no—
106#11758196—42no—
107C4×Dic61—21no—
108C4×Dic61—21no—
109C4×Dic61—42no—
110C23×C61—22no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61C2×C22⋊C411yescharacteristic
118C22×D61—21no—
119C22×D61—21no—
120C22×D61—41no—
121C22×D61—41no—
122C22×D61C2×D811yescharacteristic
123C22×D61C2×SD1611yescharacteristic
124C2×#6923—44no—
125C2×#6923—82no—
126C2×#6923—82no—
127D4×C61—42no—
128D4×C61—42no—
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—82no—
137C2×C4×C61—82no—
138C2×C4×C61#9911yescharacteristic
139C2×C4×C61#9911yescharacteristic
140C2×C4×C61#10212yes—
141C2×C4×C61#10412yes—
142C22⋊D61—44no—
143C22⋊D61—44no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—21no—
148C2×Dic61—21no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61#10212yes—
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—22no—
160C4×D61—22no—
161#92420—82no—
162#11691—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61C2×#9911yescharacteristic
173C22×C61C2×#10211yescharacteristic
174C22×C61C2×#10411yescharacteristic
175C2×D61—21no—
176C2×D61—21no—
177C2×D61—41no—
178C2×D61—41no—
179C2×D61—41no—
180C2×D61—41no—
181C2×D61—81no—
182#6923—84no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—81no—
186C4×C61#32712yes—
187C4×C61#32712yes—
188Dic61—22no—
189Dic61—42no—
190Dic61—41no—
191Dic61—41no—
192Dic61—81no—
193#2392—611no—
194#1740—611no—
195#1458—611no—
196#1344—612no—
197#1343—612no—
198C2×C61—21no—
199C2×C61—21no—
200C2×C61—82no—
201C2×C61C2×#32711yescharacteristic
202C2×C61#212011yescharacteristic
203C2×C61#212611yescharacteristic
204D61—41no—
205D61—41no—
206C2×C4×D4—1221no—
207C2×C4×D4#692311yescharacteristic, complemented
208C2×#117—611no—
209C2×#117—1221no—
210#431—1221no—
211#398—611no—
212#397—611no—
213#396—1221no—
214#394—611no—
215#394—1221no—
216#392—611no—
217C4×C4⋊C4—611no—
218C4×C22⋊C4—1221no—
219#342—612no—
220#342—1221no—
221#324—1221no—
222#323—612no—
223C61#1169111yescharacteristic, complemented, Sylow
224C22×D4—1221no—
225C22×D4C2×#692311yescharacteristic
226C23×C4—21no—
227C23×C4—611no—
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—2442no—
231C4×D4—2442no—
232#117—1224no—
233#117—2441no—
234#117—2441no—
235#117—2442no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—611no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—2441no—
241C2×C4⋊C4C2×#692311yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—611no—
244C2×C22⋊C4—611no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C42—611no—
251C2×C42—1221no—
252C2×C42—2441no—
253C2×C42C2×#692311yescharacteristic
254C4⋊C8—2441no—
255C22⋊C8—1222no—
256C4×C8—2441no—
257#95—611no—
258#95—611no—
259#95—1221no—
260#95—1221no—
261C24—21no—
262C24—611no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—2442no—
266C2×D4—2442no—
267C2×D4—2441no—
268C2×D4—2441no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—611no—
275C22×C4—612no—
276C22×C4—612no—
277C22×C4—611no—
278C22×C4—611no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—1222no—
282C22×C4—1221no—
283C22×C4—1221no—
284C22×C4—2442no—
285C22×C4C22⋊#692311yescharacteristic
286C22×C4C22⋊#692311yescharacteristic
287C22×C4C22×#692311yescharacteristic
288C2×C8—2442no—
289C4⋊C4—41no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4—2442no—
293C4⋊C4—2442no—
294C4⋊C4—2444no—
295C22⋊C4—42no—
296C22⋊C4—1224no—
297C22⋊C4—2441no—
298C22⋊C4—2441no—
299C22⋊C4—2441no—
300C22⋊C4—2441no—
301C22⋊C4—2442no—
302C22⋊C4—2442no—
303C22⋊C4—2442no—
304C22⋊C4—2442no—
305C42—21no—
306C42—21no—
307C42—1221no—
308C42—1221no—
309C42—2444no—
310C42—2442no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—611no—
319C23—611no—
320C23—1221no—
321C23—1221no—
322C23—2441no—
323C23—2441no—
324C23C2×C22⋊#692311yescharacteristic
325D4—42no—
326D4—42no—
327D4—2444no—
328D4—2444no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—612no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—1222no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1221no—
347C2×C4—1221no—
348C2×C4—4882no—
349C2×C4—4882no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2444no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4#1451983511yescharacteristic
359C2×C4#1451983511yescharacteristic
360C2×C4#1451983912yes—
361C2×C4#1451984112yes—
362C8—4882no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—1221no—
373C22—1221no—
374C22—4881no—
375C22—2441no—
376C22—2441no—
377C22—2441no—
378C22—2441no—
379C22C2×#1451983511yescharacteristic
380C22C2×#1451983911yescharacteristic
381C22C2×#1451984111yescharacteristic
382C4—41no—
383C4—41no—
384C4—81no—
385C4—1222no—
386C4—4881no—
387C4—4884no—
388C4—2442no—
389C4—2441no—
390C4—2441no—
391C4#1684061312yes—
392C4#1684061312yes—
393C2—21no—
394C2—21no—
395C2—82no—
396C2—2441no—
397C2—2441no—
398C2C2×#1684061311yescharacteristic
399C2#2146207011yescharacteristic
400C2#2146209011yescharacteristic
subgroupG/H
centerC2×C4#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C2×#99
prime radical#16839670C22
cosocleC2×#11758198C23
hypercenterC2×C4×D4#6923
hypocenterC61#11691
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461518C2semidirect
2. #21461052C2semidirect
3. #21460799C2semidirect
14. #16839693C4semidirect
18. #16839676C4semidirect
20. C4×C4⋊Dic61C4semidirect
22. C2×C4×D4×C61C4semidirect
207. C2×C4×D4#6923semidirect
223. C61#11691semidirect

Maximal subgroups

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

subgroupindex
#214615182
#214610522
#214607992
#214607972
#214490572
#1169161