Finite Group Databasebeta

← #30601444

Subgroups — #30601444

Order 15616.

382 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 98 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#21462000C211yesmaximal, characteristic
2#21461997C211yesmaximal, characteristic
3#21461930C211yesmaximal, characteristic
4#21461612C211yesmaximal, characteristic
5#21460698C211yesmaximal, characteristic
6#21460648C211yesmaximal, characteristic
7#333×D61C211yesmaximal, characteristic
8C2×#14519846—21no—
9D61⋊(C4⋊C8)C2211yescharacteristic
10C4×D61⋊C8C2211yescharacteristic
11C2×C8×#6923—21no—
12#16840708C2211yescharacteristic
13C4×C4⋊#6923C2211yescharacteristic
14#16840636—21no—
15#16840635—21no—
16#16840635—21no—
17#16840635—21no—
18#16840635—21no—
19(C2×#6923)⋊C8—21no—
20C4⋊Q8×D61C2211yescharacteristic
21C4⋊C8×D61C2211yescharacteristic
22C4⋊C8×D61C2211yescharacteristic
23#16839511C411yescharacteristic, complemented
24#16839510C411yescharacteristic, complemented
25#16839510C411yescharacteristic, complemented
26C61×#333C411yescharacteristic, complemented
27C2×D61⋊C8—21no—
28C2×D61⋊C8D411yescharacteristic
29C2×D61⋊C8D411yescharacteristic
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—21no—
32C2×C4⋊#6923—21no—
33C2×C4⋊#6923D411yescharacteristic
34C2×C4⋊#6923D411yescharacteristic
35C2×C4×#6923—21no—
36#14519846—42no—
37#14519844—21no—
38C4⋊#92420—21no—
39C4×#92420—21no—
40C4×#92420—21no—
41C8×#6923—42no—
42(C2×#6923)⋊C4—21no—
43C2×Q8×D61—21no—
44C2×C8×D61—21no—
45C2×C8×D61—21no—
46#14519777C2×C411yescharacteristic
47#14519776—21no—
48#14519776—21no—
49#14519775—21no—
50C4⋊C4×D61—21no—
51C4⋊C4×D61—21no—
52C42×D61C2311yescharacteristic
53C4⋊#92418C2×C411yescharacteristic
54C4⋊#92418C2×C411yescharacteristic
55#14519706—21no—
56#14519706—21no—
57C4⋊Q8×C61C2×C411yescharacteristic
58C4⋊C8×C61C2×C411yescharacteristic
59C4⋊C8×C61C2×C411yescharacteristic
60C22×#6923—21no—
61C22×#6923—21no—
62C22×#6923—41no—
63C2×Q8×C61—21no—
64Q8×D61—81no—
65Q8×D61—81no—
66C2×C4×D61—41no—
67C2×C4×D61C2×D411yescharacteristic
68C2×C4×D61C2×D411yescharacteristic
69C2×C4×D61#4111yescharacteristic
70C2×Dic122—21no—
71C2×Dic122—41no—
72C2×#92420—21no—
73C2×#92420—21no—
74C2×#92420—41no—
75C4⋊#6923—21no—
76C4⋊#6923—21no—
77C4⋊#6923—41no—
78C4⋊#6923—41no—
79C4⋊#6923—41no—
80C4⋊#6923—41no—
81C4⋊#6923—81no—
82C4⋊#6923SD1612yes—
83C4⋊#6923Q1612yes—
84C4×#6923—41no—
85C4×#6923—41no—
86D61⋊C8—21no—
87D61⋊C8—21no—
88D61⋊C8—21no—
89D61⋊C8—41no—
90D61⋊C8—41no—
91D61⋊C8Q1612yes—
92C2×C8×C61—21no—
93C2×C8×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C42×C61C22×C411yescharacteristic
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99#11758196—41no—
100#11758196—41no—
101C4×Dic61#4111yescharacteristic
102C4×Dic61#4111yescharacteristic
103C4×Dic61#4111yescharacteristic
104C2×#92418—21no—
105C2×#92418—21no—
106C8×D61—41no—
107C8×D61—41no—
108C8×D61—41no—
109C8×D61—41no—
110C22×D61#12111yescharacteristic
111C2×#6923—22no—
112C2×#6923—22no—
113C2×#6923—41no—
114C2×#6923—41no—
115C2×#6923—41no—
116C2×#6923—41no—
117C2×#6923—81no—
118C2×#6923—81no—
119C2×#6923—81no—
120Q8×C61—81no—
121C2×C4×C61—41no—
122C2×C4×C61#11711yescharacteristic
123C2×C4×C61C4×D411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×Dic61—41no—
126C2×Dic61#12311yescharacteristic
127C2×Dic61#12411yescharacteristic
128C2×Dic61#12611yescharacteristic
129C4×D61—21no—
130C4×D61—21no—
131C4×D61—81no—
132C4×D61—81no—
133C4×D61C2×SD1611yescharacteristic
134C4×D61C2×Q1611yescharacteristic
135C4×D61C2×Q1611yescharacteristic
136C4×D61#13511yescharacteristic
137C4×D61#13511yescharacteristic
138C4×D61#13511yescharacteristic
139C4×D61AGL(1,ℤ/8)11yescharacteristic
140C4×D61#13711yescharacteristic
141Dic122—82no—
142Dic122—81no—
143#92420—22no—
144#92420—41no—
145#92420—81no—
146C8×C61—41no—
147C8×C61—41no—
148#92418—41no—
149#92418—41no—
150#9568—611nomaximal, Sylow
151C22×C61#38811yescharacteristic
152C2×D61#46211yescharacteristic
153C2×D61#46311yescharacteristic
154C2×D61#46511yescharacteristic
155C2×D61#46711yescharacteristic
156C2×D61#47111yescharacteristic
157C2×D61#47211yescharacteristic
158#6923—42no—
159#6923—42no—
160#6923—161no—
161C4×C61—21no—
162C4×C61—81no—
163C4×C61C4×SD1611yescharacteristic
164C4×C61C4×Q1611yescharacteristic
165C4×C61C4×Q1611yescharacteristic
166C4×C61#44011yescharacteristic
167Dic61—21no—
168Dic61—81no—
169Dic61#48411yescharacteristic
170Dic61#48511yescharacteristic
171Dic61#48711yescharacteristic
172Dic61#48811yescharacteristic
173#1990—611no—
174#1987—611no—
175#1947—611no—
176C8×C4⋊C4—611no—
177C2×#333—611no—
178#1296—611no—
179#1277—611no—
180C2×C61#192811yescharacteristic
181C2×C61#193111yescharacteristic
182C2×C61#193311yescharacteristic
183D61#167211yescharacteristic
184D61#167611yescharacteristic
185D61#167711yescharacteristic
186D61#168311yescharacteristic
187C2×C4⋊Q8—611no—
188C2×#107—1221no—
189C2×C4⋊C8—611no—
190C2×C4⋊C8—611no—
191C2×C4⋊C8—611no—
192C2×C4×C8—611no—
193C2×C4×C8—1221no—
194#384—611no—
195C4×C4⋊C4—611no—
196#340—1221no—
197#340—1221no—
198#340—1221no—
199#340—1221no—
200#340—1221no—
201#336—1221no—
202#333—611no—
203#333—611no—
204#333—611no—
205#333#692311yescharacteristic, complemented
206C61#956811yescharacteristic, complemented, Sylow
207C22×Q8—1221no—
208C22×C8—611no—
209C22×C8—611no—
210C22×C8—1221no—
211C22×C8—1221no—
212C22×C8—1221no—
213C4⋊Q8—611no—
214C4⋊Q8—1221no—
215C4⋊Q8—1221no—
216C4⋊Q8—1221no—
217C4⋊Q8C2×#692311yescharacteristic
218C2×C4⋊C4—611no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C42—611no—
226C2×C42—1221no—
227#107—2442no—
228C4⋊C8—611no—
229C4⋊C8—611no—
230C4⋊C8—1221no—
231C4⋊C8—1221no—
232C4⋊C8—1221no—
233C4⋊C8—1221no—
234C4⋊C8C2×#692311yescharacteristic
235C4⋊C8C2×#692311yescharacteristic
236C4×C8—1221no—
237C4×C8—1221no—
238C4×C8—2442no—
239#95—1221no—
240C2×Q8—21no—
241C2×Q8—1221no—
242C2×Q8—4881no—
243C2×Q8—4881no—
244C2×Q8—2441no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—1221no—
249C22×C4—1221no—
250C22×C4—2441no—
251C22×C4—2441no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—612no—
255C2×C8—1221no—
256C2×C8—1221no—
257C2×C8—1221no—
258C2×C8—1221no—
259C2×C8—1221no—
260C2×C8—1221no—
261C2×C8—1221no—
262C2×C8—2441no—
263C2×C8—2441no—
264C2×C8—2441no—
265C2×C8—2441no—
266C2×C8—2441no—
267C2×C8—2441no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—612no—
272C4⋊C4—612no—
273C4⋊C4—1221no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—4881no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C42—611no—
285C42—611no—
286C42—611no—
287C42—2441no—
288C42—2441no—
289C42C22×#692311yescharacteristic
290C23—611no—
291Q8—81no—
292Q8—4881no—
293Q8—4882no—
294C2×C4—41no—
295C2×C4—611no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—611no—
299C2×C4—611no—
300C2×C4—611no—
301C2×C4—611no—
302C2×C4—611no—
303C2×C4—611no—
304C2×C4—611no—
305C2×C4—611no—
306C2×C4—1221no—
307C2×C4—1221no—
308C2×C4—1222no—
309C2×C4—1222no—
310C2×C4—4881no—
311C2×C4—4881no—
312C2×C4—4881no—
313C2×C4—4881no—
314C2×C4—4881no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4#1451985611yescharacteristic
321C2×C4D4×#692311yescharacteristic
322C2×C4D4×#692311yescharacteristic
323C8—41no—
324C8—41no—
325C8—1222no—
326C8—4881no—
327C8—2441no—
328C8—2441no—
329C8—2441no—
330C22—611no—
331C22—611no—
332C22—611no—
333C22—611no—
334C22—611no—
335C22—611no—
336C22#1684071211yescharacteristic
337C4—21no—
338C4—81no—
339C4—611no—
340C4—611no—
341C4—611no—
342C4—611no—
343C4—1221no—
344C4—9761no—
345C4—4881no—
346C4—2442no—
347C4—2442no—
348C4SD16×#692311yescharacteristic
349C4Q16×#692311yescharacteristic
350C4Q16×#692311yescharacteristic
351C4#1684080111yescharacteristic
352C2—611no—
353C2—611no—
354C2—611no—
355C2—611no—
356C2#2146189911yescharacteristic
357C2#2146190511yescharacteristic
358C2#2146190811yescharacteristic
subgroupG/H
centerC22#16840712
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#333C4
socleC22×C61#388
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#333#6923
hypocenterC61#9568
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
23. #16839511C4semidirect
24. #16839510C4semidirect
25. #16839510C4semidirect
26. C61×#333C4semidirect
205. #333#6923semidirect
206. C61#9568semidirect

Maximal subgroups

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

subgroupindex
#214620002
#214619972
#214619302
#214616122
#214606982
#214606482
#333×D612
#956861