Finite Group Databasebeta

← #30602179

Subgroups — #30602179

Order 15616.

664 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 122 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#21461675C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3#21460657C214yesmaximal
4C2×#16839667C211yesmaximal, characteristic
5C2×#14519844—21no—
6C2×C22⋊#92420C2212yes—
7C2×C22⋊#92420C2212yes—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628—21no—
11C23×Dic122C411yescharacteristic
12C22×C4×Dic61C2211yescharacteristic
13#16839667—21no—
14#16839667—21no—
15#16839667—21no—
16#16839667—21no—
17C2×#14519684C412yes—
18C2×#14519684C2212yes—
19C22×C4⋊C4×C61C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519844—42no—
23C22⋊#92420—22no—
24C22⋊#92420—24no—
25C22⋊#92420Q814yescomplemented
26#14519832—42no—
27C22×Dic122—21no—
28C22×Dic122—21no—
29C22×Dic122—24no—
30C22×Dic122—42no—
31C22×Dic122—42no—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C412yes—
34C23×Dic61C2311yescharacteristic
35C2×C4×Dic61—42no—
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40#14519684—22no—
41#14519684—22no—
42#14519684—22no—
43#14519684—22no—
44C23×C4×C61C2×C412yes—
45C23×C4×C61C2×C411yescharacteristic
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—42no—
49C24×C61C22×C411yescharacteristic
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—24no—
55C22×C4×C61—42no—
56C22×C4×C61—42no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—41no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61—42no—
67C22×Dic61—42no—
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61M1614yes—
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×Q812yes—
74C22×Dic61#4112yes—
75C2×Dic122—24no—
76C2×Dic122—24no—
77C2×Dic122—416no—
78C2×Dic122—42no—
79C2×Dic122—42no—
80C2×Dic122—42no—
81C2×Dic122—42no—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C2×#92420—42no—
87C2×#92420—42no—
88C4⋊C4×C61—42no—
89C4⋊C4×C61—42no—
90C4×Dic61—22no—
91C4×Dic61—22no—
92C4×Dic61—41no—
93C4×Dic61—41no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61C2×C22⋊C411yescharacteristic
99C23×C61C4×Q812yes—
100C23×C61C2×M1612yes—
101C23×C61#13112yes—
102C2×C4×C61—28no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—42no—
110C2×C4×C61—84no—
111C2×C4×C61#9912yes—
112C2×C4×C61#10012yes—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—24no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—44no—
126C2×Dic61—44no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—44no—
130C2×Dic61#10112yes—
131C2×Dic61C22⋊Q812yes—
132C2×Dic61#12411yescharacteristic
133C2×Dic61C4⋊Q811yescharacteristic
134Dic122—48no—
135Dic122—48no—
136#92420—44no—
137#92420—82no—
138#9882—611nomaximal, Sylow
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61#39111yescharacteristic
146C22×C61#40711yescharacteristic
147C22×C61#40811yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C22×C61#41011yescharacteristic
150C22×C61C2×#10011yescharacteristic
151C22×C61C2×#10111yescharacteristic
152C22×C61#44614yes—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—82no—
156Dic61—22no—
157Dic61—22no—
158Dic61—21no—
159Dic61—21no—
160Dic61—44no—
161Dic61—41no—
162C2×#391—611no—
163#1797—611no—
164#1749—611no—
165#1280—614no—
166C2×C61—24no—
167C2×C61#146211yescharacteristic
168C2×C61#146711yescharacteristic
169C2×C61#146911yescharacteristic
170C2×C61#147211yescharacteristic
171C2×C61#198511yescharacteristic
172C2×C61#199611yescharacteristic
173C2×C61#199711yescharacteristic
174C23×Q8—611no—
175C22×C4⋊C4#692311yescharacteristic
176C22×C42—611no—
177C2×C4⋊C8—1221no—
178C2×C22⋊C8—612no—
179C2×C22⋊C8—612no—
180C2×#97—1221no—
181#391—1221no—
182#391—1221no—
183#391—1221no—
184#391—1221no—
185C2×#95—612no—
186C2×#95—612no—
187#336—1221no—
188#336—1221no—
189C61#988211yescharacteristic, complemented, Sylow
190C22×Q8—1224no—
191C22×Q8—1221no—
192C22×Q8—1221no—
193C22×Q8—2442no—
194C22×Q8—2442no—
195C23×C4—611no—
196C23×C4—611no—
197C23×C4—612no—
198C23×C4C2×#692312yes—
199C23×C4C2×#692311yescharacteristic
200C22×C8—1222no—
201C22×C8—1222no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C42—611no—
206C2×C42—611no—
207C2×C42—611no—
208C2×C42—611no—
209C2×C42—2442no—
210C4⋊C8—2442no—
211C22⋊C8—614no—
212C22⋊C8—1222no—
213C22⋊C8—1224no—
214#97—2442no—
215#95—1222no—
216#95—1222no—
217#95—1222no—
218#95—1222no—
219C24C22×#692311yescharacteristic
220C2×Q8—1224no—
221C2×Q8—1224no—
222C2×Q8—24416no—
223C2×Q8—2442no—
224C2×Q8—2442no—
225C2×Q8—2442no—
226C2×Q8—2442no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—611no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—611no—
240C22×C4—614no—
241C22×C4—1222no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2442no—
249C22×C4—2442no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22⋊#692311yescharacteristic
252C2×C8—1224no—
253C2×C8—2442no—
254C2×C8—2442no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42—1222no—
261C42—1222no—
262C42—2441no—
263C42—2441no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C2×C22⋊#692311yescharacteristic
269C23Q8×#692312yes—
270C23D61⋊M1612yes—
271C23#1451986412yes—
272Q8—2448no—
273Q8—2448no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—612no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—612no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—1221no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1224no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2444no—
304C2×C4#1451983612yes—
305C2×C4#1451983712yes—
306C8—4882no—
307C8—2444no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22(C2×#6923)⋊Q811yescharacteristic
315C22#1684073411yescharacteristic
316C22#1684073611yescharacteristic
317C22C2×#1451983611yescharacteristic
318C22#1684074211yescharacteristic
319C22C2×#1451983711yescharacteristic
320C22D61⋊#10111yescharacteristic
321C22#1684081414yes—
322C4—22no—
323C4—22no—
324C4—82no—
325C4—1222no—
326C4—1221no—
327C4—1222no—
328C4—1221no—
329C4—2444no—
330C4—2441no—
331C2—24no—
332C2#2146106611yescharacteristic
333C2#2146108311yescharacteristic
334C2#2146108911yescharacteristic
335C2#2146109711yescharacteristic
336C2#2146199111yescharacteristic
337C2#2146200911yescharacteristic
338C2#2146201111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutator, prime radicalC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#9882
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
25. C22⋊#92420Q8semidirect
189. C61#9882semidirect

Maximal subgroups

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

subgroupindex
#214616752
#214615452
#214606572
C2×#168396672
#988261