Finite Group Databasebeta

← #29590657

Subgroups — #29590657

Order 15616.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14519694C211yesmaximal, characteristic
2#21444157C211yesmaximal, characteristic
3#21443849C212yesmaximal, complemented
4#21443846C212yesmaximal, complemented
5C61×#2564C211yesmaximal, characteristic
6C2×#14519694—21no—
7#16839726—21no—
8C4×#11758194—21no—
9#16839531C2212yes—
10#16839488C2212yes—
11#16839486—21no—
12C2×C22⋊Q8×C61C411yescharacteristic
13C2×C4×D4×C61C2211yescharacteristic
14C61×#400C2211yescharacteristic
15C61×#399C411yescharacteristic
16C61×#395C2211yescharacteristic
17C61×#393C411yescharacteristic
18C61×#387—21no—
19C61×#386—21no—
20C61×#385—21no—
21C61×#380—21no—
22C2×#95×C61C411yescharacteristic
23C2×#11758194—22no—
24C2×#11758194—41no—
25#14519694—24no—
26C22⋊#92418—22no—
27C22⋊#92418—41no—
28#14519688—41no—
29C4×#92418—41no—
30C22×Q8×C61C2×C411yescharacteristic
31C22×D4×C61D411yescharacteristic
32C23×C4×C61—21no—
33C23×C4×C61C2×C411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C22⋊Q8×C61—41no—
36C22⋊Q8×C61—41no—
37C4×D4×C61—41no—
38C4×D4×C61—41no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61C2311yescharacteristic
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50#95×C61—21no—
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61C2×C411yescharacteristic
56#95×C61C2×C411yescharacteristic
57C24×C61C22⋊C411yescharacteristic
58C24×C61C22⋊C411yescharacteristic
59C2×Q8×C61—41no—
60C2×Q8×C61—81no—
61C2×D4×C61—22no—
62C2×D4×C61—22no—
63C2×D4×C61—41no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61—81no—
75C22×C4×C61—81no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22×C411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C4⋊C4×C61—41no—
83C4⋊C4×C61—42no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—81no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C22⋊C4×C61—81no—
93C42×C61—21no—
94C42×C61—21no—
95#11758194—24no—
96#11758194—42no—
97#11758194—82no—
98C2×#92418—42no—
99C2×#92418—42no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61#10412yes—
110C23×C61C2×C22⋊C411yescharacteristic
111Q8×C61—82no—
112D4×C61—44no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—22no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—42no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—81no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61—84no—
139C2×C4×C61#9911yescharacteristic
140C2×C4×C61#9911yescharacteristic
141C2×C4×C61#13712yes—
142#92418—44no—
143#92418—82no—
144#11793—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—22no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—42no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61—81no—
157C22×C61C2×#9911yescharacteristic
158C22×C61#41911yescharacteristic
159C22×C61C2×#10411yescharacteristic
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—22no—
163C4×C61—42no—
164C4×C61—41no—
165C4×C61—41no—
166C4×C61—41no—
167C4×C61—81no—
168C4×C61—81no—
169C4×C61—81no—
170#2564D6111yescharacteristic
171C4×#100—611no—
172#1465—611no—
173#1354—612no—
174#1353—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—42no—
178C2×C61—81no—
179C2×C61#150811yescharacteristic
180C2×C61#212111yescharacteristic
181C2×C61#213211yescharacteristic
182C2×C22⋊Q8Dic6111yescharacteristic
183C2×C4×D4C2×D6111yescharacteristic
184C2×#100—1221no—
185#407—1221no—
186C4×M16—1221no—
187#400C2×D6111yescharacteristic
188#399Dic6111yescharacteristic
189#395C2×D6111yescharacteristic
190#393Dic6111yescharacteristic
191#387—21no—
192#386—21no—
193#385—21no—
194#380—21no—
195C2×#95Dic6111yescharacteristic
196#343—612no—
197#324—612no—
198#323—1221no—
199C61#1179311yescharacteristic, complemented, Sylow
200C22×Q8C2×Dic6111yescharacteristic
201C22×D4C22⋊D6111yescharacteristic
202C23×C4—21no—
203C23×C4C2×Dic6111yescharacteristic
204C23×C4C22⋊D6111yescharacteristic
205C2×M16—1222no—
206C2×M16—2441no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic6111yescharacteristic
215C2×C4⋊C4C22×D6111yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic6111yescharacteristic
220C2×C22⋊C4C22⋊D6111yescharacteristic
221C2×C42C22⋊D6111yescharacteristic
222#100—1224no—
223C22⋊C8—1222no—
224C22⋊C8—2441no—
225#97—2441no—
226C4×C8—2441no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic6111yescharacteristic
233#95C2×Dic6111yescharacteristic
234C24C22⋊Dic6111yescharacteristic
235C24C22⋊Dic6111yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C22⋊Dic6111yescharacteristic
255C22×C4C22⋊Dic6111yescharacteristic
256C22×C4C22×Dic6111yescharacteristic
257C22×C4C2×C22⋊D6111yescharacteristic
258C22×C4C2×C22⋊D6111yescharacteristic
259M16—1224no—
260M16—4882no—
261M16—2442no—
262C2×C8—2442no—
263C2×C8—2442no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1451970412yes—
287C23C2×C22⋊Dic6111yescharacteristic
288Q8—82no—
289D4—44no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—84no—
316C2×C4#1451969211yescharacteristic
317C2×C4#1451969211yescharacteristic
318C2×C4#1451980812yes—
319C8—4882no—
320C8—2444no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22C2×#1451969211yescharacteristic
334C22#1683977111yescharacteristic
335C22C2×#1451970411yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2144433911yescharacteristic
351C2#2144746511yescharacteristic
352C2#2144750411yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2564C2
socleC22×C61C2×#99
prime radicalC22×D4×C61D4
cosocleC2×C4⋊C4×C61C23
hypercenter#2564D61
hypocenterC61#11793
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #21443849C2semidirect
4. #21443846C2semidirect
199. C61#11793semidirect

Maximal subgroups

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

subgroupindex
C4×#145196942
#214441572
#214438492
#214438462
C61×#25642
#1179361