Finite Group Databasebeta

← #30595905

Subgroups — #30595905

Order 15616.

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

normal subgroups only

This group has 444 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
1#21461710C211yesmaximal, characteristic
2#21461709C211yesmaximal, characteristic
3#21460640C212yesmaximal, complemented
4#21460637C212yesmaximal, complemented
5#21454159C211yesmaximal, characteristic
6C2×C22⋊#92420—21no—
7C2×C22⋊#92420—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519756C411yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839667C411yescharacteristic
15#16839667C2211yescharacteristic
16#16839661C411yescharacteristic
17#16839661C2211yescharacteristic
18#16839647—21no—
19#16839647—21no—
20#16839637—21no—
21#16839623—21no—
22C61×#380C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C22⋊#92420—42no—
26C22⋊#92420—42no—
27C22×Dic122C2×C411yescharacteristic
28C23×Dic61D411yescharacteristic
29C23×Dic61D411yescharacteristic
30#14519756—42no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×#11758196—21no—
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61—42no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61C2311yescharacteristic
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47C23×C4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61—21no—
51#95×C61—21no—
52C24×C61C22⋊C411yescharacteristic
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—41no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61—41no—
67C22×Dic61—41no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72C2×Dic122—22no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80C2×#92420—42no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C22⋊Dic61—42no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—41no—
88#11758196—42no—
89C4×Dic61—41no—
90C4×Dic61—41no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61#9912yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—44no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61#13112yes—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—44no—
125C2×Dic61—44no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61#13512yes—
133C2×Dic61#13712yes—
134Dic122—82no—
135#92420—82no—
136#92420—82no—
137#7721—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—41no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—41no—
145C22×C61#40811yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61#41311yescharacteristic
148C22×C61#41611yescharacteristic
149C22×C61#41911yescharacteristic
150C22×C61#42111yescharacteristic
151C22×C61#42111yescharacteristic
152C4×C61—22no—
153C4×C61—41no—
154C4×C61—81no—
155C4×C61—81no—
156Dic61—22no—
157Dic61—22no—
158Dic61—41no—
159Dic61—41no—
160Dic61—81no—
161Dic61—81no—
162#2855—611no—
163#1817—611no—
164#1817—611no—
165#1272—612no—
166#1271—612no—
167C2×C61—42no—
168C2×C61#146811yescharacteristic
169C2×C61#148211yescharacteristic
170C2×C61#149711yescharacteristic
171C2×C61#151111yescharacteristic
172C2×C61#151611yescharacteristic
173C2×C61#153511yescharacteristic
174C2×C61#155411yescharacteristic
175C2×C22⋊Q8—611no—
176C22×C42—611no—
177C2×C22⋊C8—1221no—
178C2×C22⋊C8—1221no—
179#391—611no—
180#391—611no—
181#389—611no—
182#389—611no—
183#387—1221no—
184#387—1221no—
185#385—1221no—
186#381—1221no—
187#380#692311yescharacteristic
188#336—612no—
189#336—612no—
190#336—1221no—
191#336—1221no—
192C61#772111yescharacteristic, complemented, Sylow
193C22×Q8—611no—
194C23×C4—611no—
195C23×C4—611no—
196C23×C4C2×#692311yescharacteristic
197C22×C8—1222no—
198C22×C8—1222no—
199C22⋊Q8—2442no—
200C2×C4⋊C4—611no—
201C2×C4⋊C4—1221no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4C2×#692311yescharacteristic
205C2×C4⋊C4C2×#692311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1221no—
208C2×C42—611no—
209C2×C42—611no—
210C2×C42—611no—
211C2×C42—611no—
212C2×C42—2442no—
213C22⋊C8—2442no—
214C22⋊C8—2442no—
215#95—21no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222C24C22⋊#692311yescharacteristic
223C2×Q8—1222no—
224C2×Q8—2441no—
225C2×Q8—2441no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—611no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—2441no—
242C22×C4—2441no—
243C22×C4C22⋊#692311yescharacteristic
244C22×C4C22×#692311yescharacteristic
245C2×C8—2442no—
246C2×C8—2442no—
247C2×C8—2442no—
248C2×C8—2442no—
249C2×C8—2442no—
250C2×C8—2442no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—2441no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C4⋊C4—2442no—
257C22⋊C4—42no—
258C22⋊C4—2442no—
259C42—2441no—
260C42—2441no—
261C42—2441no—
262C42—2441no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23#1451983612yes—
270C23C2×C22⋊#692311yescharacteristic
271Q8—4882no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—612no—
285C2×C4—612no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1221no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—2444no—
297C2×C4—2444no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4#1451986412yes—
305C8—4882no—
306C8—4882no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22#1684073511yescharacteristic
315C22C2×#1451983611yescharacteristic
316C22#1684074911yescharacteristic
317C22#1684075211yescharacteristic
318C22#1684075511yescharacteristic
319C22#1684075911yescharacteristic
320C22#1684076011yescharacteristic
321C4—22no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—1222no—
326C4—1222no—
327C4—4881no—
328C4—4881no—
329C4—2441no—
330C4—2441no—
331C2—42no—
332C2#2146108611yescharacteristic
333C2#2146113011yescharacteristic
334C2#2146118111yescharacteristic
335C2#2146122711yescharacteristic
336C2#2146124711yescharacteristic
337C2#2146130411yescharacteristic
338C2#2146134511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#380#6923
hypocenterC61#7721
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #21460640C2semidirect
4. #21460637C2semidirect
192. C61#7721semidirect

Maximal subgroups

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

subgroupindex
#214617102
#214617092
#214606402
#214606372
#214541592
#772161