Finite Group Databasebeta

← #30600556

Subgroups — #30600556

Order 15616.

516 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462296C211yesmaximal, characteristic
2#21461258C212yesmaximal
3D61⋊#324C211yesmaximal, characteristic
4#21460681C212yesmaximal
5#21451633C211yesmaximal, characteristic
6D61⋊(C22⋊C8)C2212yes—
7#16840711—24no—
8#16840705—21no—
9#16840704C2212yes—
10C4×C22⋊#6923—21no—
11#16840615—24no—
12#16840608—21no—
13#16840607—21no—
14C2×#14519757C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839700C411yescharacteristic, complemented
17#16839696—21no—
18#16839694—21no—
19#16839683C411yescharacteristic, complemented
20#16839682C411yescharacteristic, complemented
21#16839636—21no—
22#16839624—21no—
23#16839621C2211yescharacteristic
24C61×#400C411yescharacteristic, complemented
25C2×D61⋊C8—22no—
26C2×C4⋊#6923—22no—
27C2×C22⋊#6923—22no—
28C2×C4×#6923—22no—
29#14519834—22no—
30C22⋊#92420—22no—
31(C2×#6923)⋊C4—24no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4D414yes—
34C22×Dic122C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38#14519757—22no—
39#14519757—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196C2×C411yescharacteristic
46C2×#11758196D411yescharacteristic
47C4⋊C4×D61—22no—
48C4⋊C4×D61—22no—
49C2×#11758198—21no—
50C2×#11758198—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684C2×C411yescharacteristic
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C4⋊C4×C61C2×C411yescharacteristic
58#95×C61—21no—
59#95×C61—21no—
60#95×C61C2×C411yescharacteristic
61C23×D61C2×D411yescharacteristic
62C22×#6923—24no—
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×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×C4×D61—21no—
74C2×C4×D61—21no—
75C2×C4×D61—21no—
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—21no—
79C2×C4×D61—81no—
80C2×C4×D61—81no—
81C2×C4×D61C2×D412yes—
82C2×C4×D61#4112yes—
83C2×Dic122—22no—
84C2×Dic122—81no—
85C22⋊#6923—42no—
86C22⋊#6923—42no—
87C2×#92420—42no—
88C4⋊#6923—82no—
89C4×#6923—82no—
90D61⋊C8—82no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93#11758198—41no—
94#11758198—41no—
95#11758198—42no—
96C4⋊Dic61—22no—
97C4⋊Dic61—42no—
98C4⋊Dic61—81no—
99#11758196—22no—
100#11758196—22no—
101#11758196—81no—
102#11758196—81no—
103C23×C61C2×C22⋊C411yescharacteristic
104C22×D61—41no—
105C22×D61—41no—
106C22×D61#12011yescharacteristic
107C22×D61#12311yescharacteristic
108C22×D61#13512yes—
109C22×D61#13712yes—
110C2×#6923—44no—
111C2×#6923—44no—
112C2×#6923—44no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—42no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C4×D412yes—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61#12112yes—
138C4×D61—42no—
139C4×D61—42no—
140C4×D61—42no—
141C4×D61—42no—
142C4×D61—42no—
143C4×D61—81no—
144Dic122—82no—
145#92420—82no—
146#9202—611nomaximal, Sylow
147C22×C61—21no—
148C22×C61—21no—
149C22×C61#38611yescharacteristic
150C22×C61#41611yescharacteristic
151C22×C61#41911yescharacteristic
152C2×D61—21no—
153C2×D61—21no—
154C2×D61—21no—
155C2×D61—21no—
156C2×D61—42no—
157C2×D61—42no—
158C2×D61#45212yes—
159C2×D61#48512yes—
160#6923—84no—
161C4×C61—22no—
162C4×C61—41no—
163C4×C61—41no—
164C4×C61—81no—
165C4×C61—81no—
166Dic61—22no—
167Dic61—41no—
168Dic61—41no—
169Dic61—81no—
170Dic61—81no—
171#2627—611no—
172#2313—611no—
173#1520—612no—
174C2×#324—611no—
175#1289—612no—
176C2×C61—21no—
177C2×C61—21no—
178C2×C61#187311yescharacteristic
179C2×C61#191211yescharacteristic
180C2×C61#191311yescharacteristic
181D61—22no—
182D61—22no—
183C2×C22⋊Q8—611no—
184C22×C4⋊C4—611no—
185C2×C22⋊C8—612no—
186#400—611no—
187#400#692311yescharacteristic, complemented
188#398—1221no—
189#398—1221no—
190#395—611no—
191#395—611no—
192#387—1224no—
193#385—1221no—
194#381—1221no—
195#381—1221no—
196#380—612no—
197#380—611no—
198C4×C22⋊C4—1221no—
199#328—1224no—
200#324—1221no—
201#324—1221no—
202C61#920211yescharacteristic, complemented, Sylow
203C22×Q8—611no—
204C23×C4—611no—
205C23×C4—611no—
206C23×C4—611no—
207C22×C8—1222no—
208C22⋊Q8—1222no—
209C22⋊Q8—1222no—
210C2×C4⋊C4—611no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1222no—
214C2×C4⋊C4—1222no—
215C2×C4⋊C4—1221no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1222no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4C2×#692311yescharacteristic
221C2×C4⋊C4C2×#692311yescharacteristic
222C2×C22⋊C4—1221no—
223C2×C22⋊C4—1221no—
224C2×C22⋊C4—1222no—
225C2×C42—1222no—
226C22⋊C8—1222no—
227C22⋊C8—1222no—
228#95—21no—
229#95—21no—
230#95—614no—
231#95—611no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95—1222no—
236#95—1224no—
237#95—1221no—
238#95C2×#692311yescharacteristic
239C24—611no—
240C2×Q8—1222no—
241C2×Q8—4881no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—612no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1222no—
254C22×C4—1222no—
255C22×C4—1224no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—4881no—
259C22×C4—4881no—
260C22×C4C22⋊#692311yescharacteristic
261C22×C4C22⋊#692311yescharacteristic
262C22×C4C22×#692311yescharacteristic
263C2×C8—4882no—
264C2×C8—2442no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—1222no—
268C4⋊C4—1222no—
269C4⋊C4—1222no—
270C4⋊C4—4881no—
271C4⋊C4—4881no—
272C4⋊C4—4881no—
273C4⋊C4—4882no—
274C4⋊C4—2442no—
275C22⋊C4—2442no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2442no—
279C22⋊C4—2442no—
280C42—4882no—
281C23—611no—
282C23—611no—
283C23—612no—
284C23—612no—
285C23—2441no—
286C23—2441no—
287C23C2×C22⋊#692311yescharacteristic
288Q8—4882no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—612no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—4881no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2444no—
317C2×C4—2444no—
318C2×C4—2444no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4D4×#692312yes—
323C8—4882no—
324C22—21no—
325C22—21no—
326C22—612no—
327C22—612no—
328C22—1221no—
329C22—1221no—
330C22—1221no—
331C22—1221no—
332C22—2442no—
333C22—2442no—
334C22#1684071011yescharacteristic
335C22#1684075211yescharacteristic
336C22#1684075511yescharacteristic
337C4—22no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—1222no—
343C4—4881no—
344C4—4884no—
345C4—4881no—
346C4—2441no—
347C4—2441no—
348C2—21no—
349C2—21no—
350C2—1222no—
351C2—1222no—
352C2#2146180511yescharacteristic
353C2#2146186711yescharacteristic
354C2#2146186911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#400C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#400#6923
hypocenterC61#9202
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
16. #16839700C4semidirect
19. #16839683C4semidirect
20. #16839682C4semidirect
24. C61×#400C4semidirect
187. #400#6923semidirect
202. C61#9202semidirect

Maximal subgroups

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

subgroupindex
#214622962
#214612582
D61⋊#3242
#214606812
#214516332
#920261