Finite Group Databasebeta

← #30604838

Subgroups — #30604838

Order 15616.

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

normal subgroups only

This group has 452 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
1C2×#16840642C211yesmaximal, characteristic
2C2×#16840608C211yesmaximal, characteristic
3#21460820C212yesmaximal, complemented
4#21460818C212yesmaximal, complemented
5#21451634C211yesmaximal, characteristic
6C22×#11758217—21no—
7C2×C22⋊#92420—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C22×#11758196C2211yescharacteristic
13#16839700C411yescharacteristic
14#16839700C2211yescharacteristic
15#16839695—21no—
16#16839695—21no—
17#16839681C411yescharacteristic
18#16839681C2211yescharacteristic
19#16839634—21no—
20#16839623—21no—
21#16839622C411yescharacteristic
22C2×C22⋊Q8×C61C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C22×#92420—41no—
28C22×#92420—41no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C2×C4⋊Dic61—21no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—41no—
37C2×#11758196—41no—
38C2×#11758196C2×C411yescharacteristic
39C2×#11758196D411yescharacteristic
40C2×#11758196D411yescharacteristic
41C2×#11758196C2311yescharacteristic
42C2×C22⋊Dic61—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684C2×C411yescharacteristic
50#14519684C2×C411yescharacteristic
51C22×Q8×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C22⋊Q8×C61—41no—
54C22⋊Q8×C61—41no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C24×C61C22⋊C411yescharacteristic
59C2×Q8×C61—41no—
60C2×Q8×C61—81no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—81no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—81no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61SD1612yes—
79C22×Dic61Q1612yes—
80C22×Dic61C2×D411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82#11758217—24no—
83#11758217—44no—
84#11758217—42no—
85C2×#92420—44no—
86C2×#92420—42no—
87C2×#92420—42no—
88C2×#92420—82no—
89C4⋊C4×C61—42no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊Dic61—42no—
94C4⋊Dic61—81no—
95#11758196—41no—
96#11758196—41no—
97#11758196—41no—
98#11758196—41no—
99#11758196—81no—
100#11758196—81no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61#9911yescharacteristic
104C23×C61#9911yescharacteristic
105C23×C61#10312yes—
106C23×C61#10412yes—
107C23×C61C2×C22⋊C411yescharacteristic
108Q8×C61—82no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—81no—
119C2×C4×C61—81no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—22no—
123C2×Dic61—22no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—42no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—81no—
140C2×Dic61#10312yes—
141C2×Dic61C2×SD1611yescharacteristic
142C2×Dic61C2×Q1611yescharacteristic
143#92420—44no—
144#92420—82no—
145#11722—611nomaximal, Sylow
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—42no—
153C22×C61—42no—
154C22×C61#32812yes—
155C22×C61#32812yes—
156C22×C61C2×#9911yescharacteristic
157C22×C61C2×#10311yescharacteristic
158C22×C61C2×#10411yescharacteristic
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—81no—
162C4×C61—81no—
163Dic61—21no—
164Dic61—22no—
165Dic61—21no—
166Dic61—41no—
167Dic61—41no—
168Dic61—81no—
169Dic61—81no—
170#2627—611no—
171C2×#343—611no—
172C2×#324—611no—
173#1354—612no—
174#1353—612no—
175C2×C61—21no—
176C2×C61—22no—
177C2×C61—22no—
178C2×C61—21no—
179C2×C61C2×#32811yescharacteristic
180C2×C61#212111yescharacteristic
181C2×C61#213311yescharacteristic
182C22×M16—1221no—
183C2×C22⋊Q8#692311yescharacteristic
184C22×C4⋊C4—611no—
185C2×C22⋊C8—1221no—
186#400—611no—
187#400—611no—
188#398—1221no—
189#398—1221no—
190#395—611no—
191#395—611no—
192#385—1221no—
193#381—1221no—
194#380—611no—
195#343—612no—
196#343—1221no—
197#324—612no—
198#324—1221no—
199C61#1172211yescharacteristic, complemented, Sylow
200C22×Q8C2×#692311yescharacteristic
201C23×C4—611no—
202C23×C4—611no—
203C23×C4C2×#692311yescharacteristic
204C2×M16—1222no—
205C2×M16—1224no—
206C2×M16—2442no—
207C2×M16—2441no—
208C22×C8—2441no—
209C22×C8—2441no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—611no—
214C2×C4⋊C4—611no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—1221no—
220C2×C4⋊C4—2441no—
221C2×C4⋊C4—2441no—
222C2×C4⋊C4C2×#692311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—1221no—
225C22⋊C8—1222no—
226C22⋊C8—2441no—
227#95—611no—
228#95—611no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235C24C22⋊#692311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—612no—
244C22×C4—612no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1222no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—4881no—
257C22×C4C22⋊#692311yescharacteristic
258C22×C4C22×#692311yescharacteristic
259M16—1224no—
260M16—2442no—
261M16—2444no—
262C2×C8—4882no—
263C2×C8—2442no—
264C2×C8—2442no—
265C2×C8—2444no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—4881no—
269C4⋊C4—4881no—
270C4⋊C4—4881no—
271C4⋊C4—2441no—
272C4⋊C4—2441no—
273C4⋊C4—2441no—
274C4⋊C4—2441no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—2442no—
278C23—41no—
279C23—41no—
280C23#1451983511yescharacteristic
281C23#1451983511yescharacteristic
282C23#1451984012yes—
283C23#1451984212yes—
284C23C2×C22⋊#692311yescharacteristic
285Q8—82no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—612no—
298C2×C4—611no—
299C2×C4—611no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1222no—
305C2×C4—1222no—
306C2×C4—1222no—
307C2×C4—1222no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—4881no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C8—4882no—
321C8—2444no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1684061612yes—
331C22#1684061612yes—
332C22C2×#1451983511yescharacteristic
333C22C2×#1451984011yescharacteristic
334C22C2×#1451984211yescharacteristic
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—1221no—
340C4—1222no—
341C4—1221no—
342C4—4881no—
343C4—4881no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1684061611yescharacteristic
351C2#2146207411yescharacteristic
352C2#2146211411yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC2×#11758196C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11722
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460820C2semidirect
4. #21460818C2semidirect
199. C61#11722semidirect

Maximal subgroups

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

subgroupindex
C2×#168406422
C2×#168406082
#214608202
#214608182
#214516342
#1172261