Finite Group Databasebeta

← #29840074

Subgroups — #29840074

Order 15616.

2806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448696C218yesmaximal, complemented
2#21448688C218yesmaximal, complemented
3#21448437C212yesmaximal, complemented
4#21448422C214yesmaximal, complemented
5#21448202C212yesmaximal
6#21448202C214yesmaximal
7#21448176C212yesmaximal, complemented
8C23×C4⋊C4×C61C211yesmaximal, characteristic
9C2×#14519736C2214yes—
10C2×#14519736C2218yes—
11#16839661C22116yescomplemented
12#16839648C22132yescomplemented
13#16839625C2214yes—
14#16839625C2218yes—
15#16839622C2218yes—
16#16839622C2214yes—
17#16839622C2214yes—
18#16839622C2218yes—
19#16839618C2214yes—
20C4⋊C4×Dic61C22116yescomplemented
21C4×C22⋊Dic61C2218yes—
22C4×C22⋊Dic61C2214yes—
23C4×C22⋊Dic61C2214yes—
24C4×C22⋊Dic61C2218yes—
25C24×C4×C61C2211yescharacteristic
26C24×C4×C61C2212yes—
27C22×C4⋊C4×C61C4116yescomplemented
28C22×C4⋊C4×C61C2212yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2214yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2214yes—
33#14519736—216no—
34#14519736—416no—
35C2×C4⋊Dic61C2318yes—
36C2×C4⋊Dic61C23116yes—
37C2×C22⋊Dic61C2318yes—
38C2×C22⋊Dic61C2318yes—
39C2×C22⋊Dic61C2318yes—
40C2×C4×Dic61C2318yes—
41C2×C4×Dic61C23116yes—
42#14519684C23116yes—
43#14519684C2318yes—
44#14519684C2318yes—
45#14519684C23116yes—
46C25×C61C2311yescharacteristic
47C23×C4×C61—28no—
48C23×C4×C61—48no—
49C23×C4×C61C2×C418yes—
50C23×C4×C61C2×C4116yes—
51C23×C4×C61C2314yes—
52C23×C4×C61C2312yes—
53C23×C4×C61C2312yes—
54C23×C4×C61C2311yescharacteristic
55C23×C4×C61C2312yes—
56C23×C4×C61C2314yes—
57C23×C4×C61C2312yes—
58C23×C4×C61C2311yescharacteristic
59C2×C4⋊C4×C61C2×C4132yes—
60C2×C4⋊C4×C61C2×C4116yes—
61C2×C4⋊C4×C61C2×C4132yes—
62C2×C4⋊C4×C61C2×C418yes—
63C2×C4⋊C4×C61C2×C418yes—
64C2×C4⋊C4×C61C2314yes—
65C2×C4⋊C4×C61C2318yes—
66C2×C4⋊C4×C61C2314yes—
67C24×C61—24no—
68C24×C61—24no—
69C24×C61—24no—
70C24×C61C22×C414yes—
71C24×C61C2411yescharacteristic
72C24×C61C2411yescharacteristic
73C24×C61C2411yescharacteristic
74C22×C4×C61—24no—
75C22×C4×C61—216no—
76C22×C4×C61—232no—
77C22×C4×C61—28no—
78C22×C4×C61—28no—
79C22×C4×C61—432no—
80C22×C4×C61—44no—
81C22×C4×C61—48no—
82C22×C4×C61—44no—
83C22×C4×C61C22×C414yes—
84C22×C4×C61C22×C4116yes—
85C22×C4×C61C22×C418yes—
86C22×C4×C61C22×C418yes—
87C22×C4×C61C22×C418yes—
88C22×C4×C61C22×C418yes—
89C22×C4×C61C22×C4116yes—
90C22×C4×C61C22×C414yes—
91C22×C4×C61#4118yes—
92C22×C4×C61C2412yes—
93C22×C4×C61C2414yes—
94C22×C4×C61C2414yes—
95C22×C4×C61C2412yes—
96C22×Dic61C24116yes—
97C4⋊C4×C61C22×C4132yes—
98C4⋊C4×C61C22×C4116yes—
99C4⋊C4×C61C22×C4116yes—
100C22⋊Dic61—216no—
101C22⋊Dic61—216no—
102C22⋊Dic61—216no—
103C4⋊Dic61—216no—
104C4⋊Dic61—416no—
105C4×Dic61—216no—
106C4×Dic61—416no—
107C23×C61—232no—
108C23×C61—28no—
109C23×C61—28no—
110C23×C61—28no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—22no—
114C23×C61—22no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61C23×C412yes—
118C23×C61C23×C412yes—
119C23×C61C23×C412yes—
120C23×C61C2×#4114yes—
121C23×C61H(5,𝔽2)14yes—
122C23×C612−1+414yes—
123C23×C61C2511yescharacteristic
124C2×C4×C61—28no—
125C2×C4×C61—232no—
126C2×C4×C61—24no—
127C2×C4×C61—28no—
128C2×C4×C61—28no—
129C2×C4×C61—216no—
130C2×C4×C61—216no—
131C2×C4×C61—48no—
132C2×C4×C61—416no—
133C2×C4×C61—48no—
134C2×C4×C61#117116yescomplemented
135C2×C4×C61C23×C414yes—
136C2×C4×C61C23×C418yes—
137C2×C4×C61C23×C418yes—
138C2×C4×C61C23×C414yes—
139C2×C4×C61C2×#4118yes—
140C2×Dic61—216no—
141C2×Dic61—216no—
142C2×Dic61—216no—
143#34178—611nomaximal, Sylow
144C22×C61—232no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—22no—
148C22×C61—22no—
149C22×C61—22no—
150C22×C61—22no—
151C22×C61—28no—
152C22×C61—28no—
153C22×C61—28no—
154C22×C61C2×#11714yes—
155C22×C61#51814yes—
156C22×C61#51914yes—
157C22×C61C24×C411yescharacteristic
158C22×C61C22×#4112yes—
159C22×C61C2×H(5,𝔽2)12yes—
160C22×C61C2×2−1+412yes—
161C4×C61—24no—
162C4×C61—28no—
163C4×C61—28no—
164C4×C61C2×#11718yes—
165Dic61—416no—
166C23×C4⋊C4D6111yescharacteristic
167#2345—618no—
168#2344—618no—
169#2316—612no—
170#2313—614no—
171#2280—612no—
172#2280—614no—
173#2274—612no—
174C2×C61—24no—
175C2×C61—24no—
176C2×C61—24no—
177C2×C61C22×#11711yescharacteristic
178C2×C61C2×#51811yescharacteristic
179C2×C61C2×#51911yescharacteristic
180C2×C61#349214yes—
181C24×C4C2×D6111yescharacteristic
182C24×C4C2×D6112yes—
183C2×#117—614no—
184C2×#117—618no—
185C22×C4⋊C4Dic61116yescomplemented
186C22×C4⋊C4C2×D6112yes—
187C22×C4⋊C4C2×D6111yescharacteristic
188C22×C4⋊C4C2×D6114yes—
189C22×C4⋊C4C2×D6114yes—
190C22×C4⋊C4C2×D6111yescharacteristic
191#389—6116no—
192#387—6132no—
193#381—614no—
194#381—618no—
195#380—618no—
196#380—614no—
197#380—618no—
198#380—614no—
199#379—614no—
200C4×C4⋊C4—6116no—
201C4×C22⋊C4—618no—
202C4×C22⋊C4—614no—
203C4×C22⋊C4—614no—
204C4×C22⋊C4—618no—
205C61#3417811yescharacteristic, complemented, Sylow
206C25C22×D6111yescharacteristic
207C23×C4—28no—
208C23×C4—48no—
209C23×C4C2×Dic6118yes—
210C23×C4C2×Dic61116yes—
211C23×C4C22×D6114yes—
212C23×C4C22×D6112yes—
213C23×C4C22×D6112yes—
214C23×C4C22×D6111yescharacteristic
215C23×C4C22×D6112yes—
216C23×C4C22×D6114yes—
217C23×C4C22×D6112yes—
218C23×C4C22×D6111yescharacteristic
219#117—12216no—
220#117—24416no—
221C2×C4⋊C4—618no—
222C2×C4⋊C4—6116no—
223C2×C4⋊C4C2×Dic61132yes—
224C2×C4⋊C4C2×Dic61116yes—
225C2×C4⋊C4C2×Dic61132yes—
226C2×C4⋊C4C2×Dic6118yes—
227C2×C4⋊C4C2×Dic6118yes—
228C2×C4⋊C4C22×D6114yes—
229C2×C4⋊C4C22×D6118yes—
230C2×C4⋊C4C22×D6114yes—
231C2×C22⋊C4—618no—
232C2×C22⋊C4—618no—
233C2×C22⋊C4—618no—
234C2×C42—618no—
235C2×C42—6116no—
236#95—6116no—
237#95—618no—
238#95—618no—
239#95—6116no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24C22×Dic6114yes—
244C24C23×D6111yescharacteristic
245C24C23×D6111yescharacteristic
246C24C23×D6111yescharacteristic
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—232no—
252C22×C4—432no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—6116no—
257C22×C4#1175822218yes—
258C22×C4C22×Dic6114yes—
259C22×C4C22×Dic61116yes—
260C22×C4C22×Dic6118yes—
261C22×C4C22×Dic6118yes—
262C22×C4C22×Dic6118yes—
263C22×C4C22×Dic6118yes—
264C22×C4C22×Dic61116yes—
265C22×C4C22×Dic6114yes—
266C22×C4C23×D6112yes—
267C22×C4C23×D6114yes—
268C22×C4C23×D6114yes—
269C22×C4C23×D6112yes—
270C4⋊C4—12216no—
271C4⋊C4—24416no—
272C4⋊C4C22×Dic61132yes—
273C4⋊C4C22×Dic61116yes—
274C4⋊C4C22×Dic61116yes—
275C22⋊C4—12216no—
276C22⋊C4—12216no—
277C22⋊C4—12216no—
278C42—12216no—
279C42—24416no—
280C23—232no—
281C23—28no—
282C23—28no—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23C23×Dic6112yes—
291C23C23×Dic6112yes—
292C23C23×Dic6112yes—
293C23C2×#1175822214yes—
294C23#1451982214yes—
295C23#1451982514yes—
296C23C24×D6111yescharacteristic
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—12216no—
308C2×C4—12216no—
309C2×C4—12216no—
310C2×C4#14519736116yescomplemented
311C2×C4C23×Dic6114yes—
312C2×C4C23×Dic6118yes—
313C2×C4C23×Dic6118yes—
314C2×C4C23×Dic6114yes—
315C2×C4C2×#1175822218yes—
316C22—232no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—28no—
324C22—28no—
325C22—28no—
326C22C2×#1451973614yes—
327C22#1684023714yes—
328C22#1684024014yes—
329C22C24×Dic6111yescharacteristic
330C22C22×#1175822212yes—
331C22C2×#1451982212yes—
332C22C2×#1451982512yes—
333C4—24no—
334C4—28no—
335C4—28no—
336C4—24416no—
337C4C2×#1451973618yes—
338C2—24no—
339C2—24no—
340C2—24no—
341C2C22×#1451973611yescharacteristic
342C2C2×#1684023711yescharacteristic
343C2C2×#1684024011yescharacteristic
344C2#2145952814yes—
subgroupG/H
center, FrattiniC23C24×D61
commutatorC22×C61C24×C4
FittingC23×C4⋊C4×C61C2
socle, cosocleC23×C61C25
prime radicalC25×C61C23
hypercenterC23×C4⋊C4D61
hypocenterC61#34178
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #21448696C2semidirect
2. #21448688C2semidirect
3. #21448437C2semidirect
4. #21448422C2semidirect
7. #21448176C2semidirect
11. #16839661C22semidirect
12. #16839648C22semidirect
20. C4⋊C4×Dic61C22semidirect
27. C22×C4⋊C4×C61C4semidirect
134. C2×C4×C61#117semidirect
185. C22×C4⋊C4Dic61semidirect
205. C61#34178semidirect
310. C2×C4#14519736semidirect

Maximal subgroups

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

subgroupindex
#214486962
#214486882
#214484372
#214484222
#214482022
#214481762
C23×C4⋊C4×C612
#3417861