Finite Group Databasebeta

← #30605011

Subgroups — #30605011

Order 15616.

462 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 72 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
2#21461069C211yesmaximal, characteristic
3#21460820C212yesmaximal, complemented
4#21460803C212yesmaximal, complemented
5#21451200C211yesmaximal, characteristic
6C22×#11758217—21no—
7#16840732—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608—21no—
11#16840608C2212yes—
12C22×#11758196C2211yescharacteristic
13#16839693C411yescharacteristic
14#16839693C2211yescharacteristic
15#16839681C411yescharacteristic
16#16839681C2211yescharacteristic
17#16839657—21no—
18#16839657—21no—
19#16839634—21no—
20#16839634—21no—
21C4×C22⋊Dic61C411yescharacteristic
22C2×C22⋊Q8×C61C411yescharacteristic
23C2×#11758217—24no—
24C2×#11758217—22no—
25C2×#11758217—41no—
26C2×#11758217—42no—
27C2×#11758217—41no—
28C22×#92420—41no—
29C22⋊#92420—22no—
30C22⋊#92420—41no—
31C23×Dic61D411yescharacteristic
32C23×Dic61D411yescharacteristic
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196—41no—
38C2×#11758196—41no—
39C2×#11758196C2×C411yescharacteristic
40C2×#11758196D411yescharacteristic
41C2×#11758196D411yescharacteristic
42C2×#11758196C2311yescharacteristic
43C2×C22⋊Dic61—21no—
44C2×C4×Dic61—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—21no—
76C22×Dic61—21no—
77C22×Dic61—81no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82#11758217—24no—
83#11758217—44no—
84#11758217—42no—
85#11758217—82no—
86C2×#92420—44no—
87C2×#92420—42no—
88C2×#92420—42no—
89C4⋊C4×C61—42no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊Dic61—24no—
94#11758196—41no—
95#11758196—41no—
96#11758196—41no—
97#11758196—41no—
98#11758196—42no—
99#11758196—41no—
100#11758196—41no—
101#11758196—81no—
102#11758196—81no—
103C4×Dic61—81no—
104C23×C61—22no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—41no—
108C23×C61#9911yescharacteristic
109C23×C61#9911yescharacteristic
110C23×C61C2×C22⋊C411yescharacteristic
111Q8×C61—82no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—81no—
123C2×C4×C61—81no—
124C2×Dic61—22no—
125C2×Dic61—22no—
126C2×Dic61—22no—
127C2×Dic61—22no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—22no—
133C2×Dic61—22no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—81no—
144C2×Dic61#10412yes—
145C2×Dic61AGL(1,ℤ/8)11yescharacteristic
146C2×Dic61#13711yescharacteristic
147#92420—44no—
148#92420—82no—
149#11768—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—42no—
158C22×C61—42no—
159C22×C61#35612yes—
160C22×C61C2×#9911yescharacteristic
161C22×C61#41711yescharacteristic
162C22×C61C2×#10411yescharacteristic
163C4×C61—41no—
164C4×C61—41no—
165C4×C61—81no—
166C4×C61—81no—
167Dic61—21no—
168Dic61—22no—
169Dic61—21no—
170Dic61—42no—
171Dic61—42no—
172Dic61—41no—
173Dic61—41no—
174#2583—611no—
175C2×#343—611no—
176#1463—611no—
177#1354—612no—
178#1346—612no—
179C2×C61—21no—
180C2×C61—21no—
181C2×C61—22no—
182C2×C61—41no—
183C2×C61#150411yescharacteristic
184C2×C61#212111yescharacteristic
185C2×C61C2×#35611yescharacteristic
186C22×M16—1221no—
187C2×C22⋊Q8#692311yescharacteristic
188C22×C4⋊C4—611no—
189#407—1221no—
190#398—611no—
191#398—611no—
192#395—611no—
193#395—611no—
194#389—1221no—
195#389—1221no—
196#385—1221no—
197#385—1221no—
198C4×C22⋊C4—611no—
199#343—612no—
200#343—1221no—
201#324—612no—
202#324—1221no—
203C61#1176811yescharacteristic, complemented, Sylow
204C22×Q8C2×#692311yescharacteristic
205C23×C4—611no—
206C23×C4—611no—
207C23×C4C2×#692311yescharacteristic
208C2×M16—1222no—
209C2×M16—1224no—
210C2×M16—2441no—
211C2×M16—2442no—
212C2×M16—2441no—
213C22×C8—2441no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—611no—
218C2×C4⋊C4—611no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—611no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—2441no—
226C2×C4⋊C4—2441no—
227C2×C4⋊C4C2×#692311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1221no—
230C2×C42—1221no—
231C22⋊C8—1222no—
232C22⋊C8—2441no—
233#95—611no—
234#95—611no—
235#95—1221no—
236#95—1221no—
237#95—1221no—
238#95—1221no—
239C24C22⋊#692311yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—81no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—1222no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—1221no—
255C22×C4—1221no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—4881no—
261C22×C4C22⋊#692311yescharacteristic
262C22×C4C22×#692311yescharacteristic
263M16—1224no—
264M16—4882no—
265M16—2442no—
266M16—2444no—
267C2×C8—2442no—
268C2×C8—2444no—
269C2×C8—2442no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—4881no—
273C4⋊C4—4881no—
274C4⋊C4—2441no—
275C4⋊C4—2441no—
276C4⋊C4—2441no—
277C4⋊C4—2441no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C4⋊C4—2442no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—1224no—
284C42—4881no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1451983511yescharacteristic
290C23#1451983511yescharacteristic
291C23C2×C22⋊#692311yescharacteristic
292Q8—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—611no—
306C2×C4—611no—
307C2×C4—612no—
308C2×C4—1222no—
309C2×C4—1222no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1222no—
317C2×C4—1222no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—4881no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2442no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C8—4882no—
329C8—2444no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1684067212yes—
340C22C2×#1451983511yescharacteristic
341C22#1684075311yescharacteristic
342C22D61⋊#10411yescharacteristic
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—1221no—
348C4—1222no—
349C4—1221no—
350C4—2442no—
351C4—2442no—
352C4—2441no—
353C4—2441no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2146120511yescharacteristic
359C2#2146207411yescharacteristic
360C2C2×#1684067211yescharacteristic
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#11768
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. #21460803C2semidirect
203. C61#11768semidirect

Maximal subgroups

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

subgroupindex
C2×#168406422
#214610692
#214608202
#214608032
#214512002
#1176861