Finite Group Databasebeta

← #16807080

Subgroups — #16807080

Order 3328.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131202C214yesmaximal, complemented
2#13131198C212yesmaximal, complemented
3#13131194C212yesmaximal, complemented
4#13131173C212yesmaximal, complemented
5#13131063C211yesmaximal, characteristic
6C2×C22⋊#6092C211yesmaximal, characteristic
7#13131047C212yesmaximal, complemented
8C2×#11718783C211yesmaximal, characteristic
9C22×C13×M16C411yescharacteristic
10C2×#88180C411yescharacteristic
11C23×#1055C2211yescharacteristic
12C2×#88106C2212yes—
13C22×#6092C2212yes—
14#11718783C411yescharacteristic
15#11718783C412yes—
16#11718783C411yescharacteristic
17#11718783C2211yescharacteristic
18#11718783C2211yescharacteristic
19#11718783C2212yes—
20C2×#88055C412yes—
21C2×C8×Dic13C2212yes—
22C22⋊#6092C2211yescharacteristic
23C22⋊#6092C2211yescharacteristic
24C22⋊#6092C2212yes—
25C22⋊#6092C2212yes—
26C22⋊#6092C2212yes—
27#11718260C2214yescomplemented
28#11718260C2214yescomplemented
29#11718258C2214yescomplemented
30#11718257C2214yescomplemented
31C23×C4×C13C2×C411yescharacteristic
32C2×C13×M16—24no—
33C2×C13×M16C2×C411yescharacteristic
34C2×C13×M16C2×C412yes—
35C2×C13×M16C2×C411yescharacteristic
36C22×C8×C13C2×C412yes—
37C2×C22⋊Dic13C812yes—
38#88180C812yes—
39#88180C811yescharacteristic
40#88180C811yescharacteristic
41#88180C2×C412yes—
42#88180C2×C411yescharacteristic
43#88180C2×C411yescharacteristic
44C2×C4⋊Dic13C811yescharacteristic
45C2×C4⋊Dic13C811yescharacteristic
46C2×C4×Dic13C2×C412yes—
47C22×#1055—24no—
48C22×#1055C2×C411yescharacteristic
49C22×#1055C2×C411yescharacteristic
50C22×#1055C2×C411yescharacteristic
51C22×#1055C2311yescharacteristic
52C22×#1055C2311yescharacteristic
53C22×#1055C2311yescharacteristic
54#88106—24no—
55C2×#6092—24no—
56C2×#6092C2312yes—
57C2×#6092C2314yes—
58C2×#6092C2312yes—
59#88055C2×C412yes—
60#88055C2×C412yes—
61#88055C2×C412yes—
62#88055C2×C412yes—
63C8×Dic13C2×C412yes—
64C8×Dic13C2×C412yes—
65C8×Dic13C2312yes—
66C8×Dic13C2312yes—
67#33422—131nomaximal, Sylow
68C24×C13C2×C811yescharacteristic
69C22×C4×C13—22no—
70C22×C4×C13—22no—
71C22×C4×C13C2×C811yescharacteristic
72C22×C4×C13C2×C811yescharacteristic
73C22×C4×C13C2×C811yescharacteristic
74C22×C4×C13C22×C411yescharacteristic
75C22×C4×C13C22×C411yescharacteristic
76C22×C4×C13C22×C411yescharacteristic
77C22×Dic13C2×C812yes—
78C22×Dic13C2×C812yes—
79C13×M16—22no—
80C13×M16—24no—
81C13×M16M1614yes—
82C2×C8×C13—24no—
83C2×C8×C13C22×C412yes—
84C2×C8×C13C22×C412yes—
85C22⋊Dic13C2×C812yes—
86C22⋊Dic13C2×C812yes—
87C22⋊Dic13C2×C812yes—
88C22⋊Dic13C2×C812yes—
89C4⋊Dic13C2×C812yes—
90C4⋊Dic13C2×C812yes—
91C4⋊Dic13C2×C811yescharacteristic
92C4⋊Dic13C2×C811yescharacteristic
93C4⋊Dic13C2×C811yescharacteristic
94C4⋊Dic13C2×C811yescharacteristic
95C4×Dic13C2×C812yes—
96C4×Dic13C2×C812yes—
97C4×Dic13C22×C412yes—
98C4×Dic13C22×C412yes—
99C2×#1055—22no—
100C2×#1055—22no—
101C2×#1055—22no—
102C2×#1055—22no—
103C2×#1055—22no—
104C2×#1055C22×C411yescharacteristic
105C2×#1055C22×C411yescharacteristic
106C2×#1055C22×C411yescharacteristic
107C2×#1055#4114yes—
108C2×#1055C2411yescharacteristic
109#6092—22no—
110#6092—22no—
111#6092—24no—
112C2×#405—131no—
113#2164—134no—
114#2163—132no—
115#2162—132no—
116#2151—132no—
117#2112—131no—
118C2×C22⋊C16—131no—
119#2108—132no—
120C23×C13—22no—
121C23×C13—22no—
122C23×C13—41no—
123C23×C13C22×C811yescharacteristic
124C23×C13C22×C811yescharacteristic
125C23×C13C22×C811yescharacteristic
126C2×C4×C13—22no—
127C2×C4×C13—21no—
128C2×C4×C13—21no—
129C2×C4×C13—21no—
130C2×C4×C13—21no—
131C2×C4×C13—22no—
132C2×C4×C13—22no—
133C2×C4×C13#11712yes—
134C2×C4×C13C22×C811yescharacteristic
135C2×C4×C13C22×C811yescharacteristic
136C2×C4×C13C22×C811yescharacteristic
137C2×C4×C13C2×M1612yes—
138C2×C4×C13C23×C411yescharacteristic
139C2×Dic13—24no—
140C2×Dic13—42no—
141C2×Dic13C22×C812yes—
142C2×Dic13C22×C812yes—
143C2×Dic13C22×C812yes—
144C2×Dic13C22×C812yes—
145C8×C13—22no—
146C8×C13C2×M1614yes—
147#1055—21no—
148#1055—21no—
149#1055#11712yes—
150#1055C2×#4112yes—
151C22×M16#26111yescharacteristic
152C23×C8—131no—
153C2×#117—131no—
154C2×M32—132no—
155C22×C16—132no—
156#405—131no—
157#405—131no—
158#405—132no—
159#405—131no—
160#405—132no—
161#405—131no—
162C2×#97—132no—
163C2×C4×C8—132no—
164C4⋊C16—134no—
165C4⋊C16—134no—
166C22⋊C16—131no—
167C22⋊C16—131no—
168C22⋊C16—132no—
169C22⋊C16—132no—
170C22⋊C16—132no—
171#346—134no—
172C4×C16—134no—
173C22×C13—21no—
174C22×C13—21no—
175C22×C13—21no—
176C22×C13—21no—
177C22×C13—21no—
178C22×C13—21no—
179C22×C13—21no—
180C22×C13—44no—
181C22×C13#43112yes—
182C22×C13#50412yes—
183C22×C13C23×C811yescharacteristic
184C4×C13—21no—
185C4×C13—21no—
186C4×C13#43112yes—
187C4×C13C2×#11711yescharacteristic
188C4×C13C22×M1611yescharacteristic
189Dic13—22no—
190Dic13—22no—
191Dic13—22no—
192Dic13—22no—
193C23×C4C2×#26111yescharacteristic
194C2×M16—24no—
195C2×M16C2×#26111yescharacteristic
196C2×M16C2×#26112yes—
197C2×M16C2×#26111yescharacteristic
198C22×C8—131no—
199C22×C8—131no—
200C22×C8—131no—
201C22×C8—131no—
202C22×C8—131no—
203C22×C8—131no—
204C22×C8—264no—
205C22×C8C2×#26112yes—
206#117—131no—
207#117—131no—
208#117—132no—
209#117—132no—
210#117—131no—
211#117—131no—
212C2×C4⋊C4—131no—
213C2×C4⋊C4—131no—
214C2×C22⋊C4—132no—
215C2×C42—132no—
216M32—264no—
217C2×C16—132no—
218C2×C16—134no—
219C2×C16—132no—
220C2×C16—264no—
221#97—132no—
222#97—132no—
223#97—132no—
224#97—132no—
225C4×C8—132no—
226C4×C8—132no—
227C4×C8—132no—
228C4×C8—132no—
229C2×C13—22no—
230C2×C13—22no—
231C2×C13—41no—
232C2×C13C2×#43111yescharacteristic
233C2×C13C2×#50411yescharacteristic
234C2×C13#340711yescharacteristic
235C24D13⋊C811yescharacteristic
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—132no—
239C22×C4—132no—
240C22×C4D13⋊C811yescharacteristic
241C22×C4D13⋊C811yescharacteristic
242C22×C4D13⋊C811yescharacteristic
243C22×C4C22×#26111yescharacteristic
244C22×C4C22×#26111yescharacteristic
245C22×C4C22×#26111yescharacteristic
246M16—22no—
247M16—24no—
248M16#612214yes—
249C2×C8—24no—
250C2×C8—134no—
251C2×C8—131no—
252C2×C8—131no—
253C2×C8—131no—
254C2×C8—131no—
255C2×C8—262no—
256C2×C8—262no—
257C2×C8—262no—
258C2×C8—262no—
259C2×C8—262no—
260C2×C8C22×#26112yes—
261C2×C8C22×#26112yes—
262C4⋊C4—132no—
263C4⋊C4—132no—
264C4⋊C4—131no—
265C4⋊C4—131no—
266C4⋊C4—131no—
267C4⋊C4—131no—
268C22⋊C4—132no—
269C22⋊C4—132no—
270C22⋊C4—132no—
271C22⋊C4—132no—
272C42—132no—
273C42—132no—
274C42—132no—
275C42—132no—
276C16—262no—
277C16—262no—
278C16—264no—
279C13#3342211yescharacteristic, complemented, Sylow
280C23—22no—
281C23—22no—
282C23—41no—
283C23C2×D13⋊C811yescharacteristic
284C23C2×D13⋊C811yescharacteristic
285C23C2×D13⋊C811yescharacteristic
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—132no—
294C2×C4—132no—
295C2×C4—132no—
296C2×C4—132no—
297C2×C4—264no—
298C2×C4—522no—
299C2×C4C2×D13⋊C811yescharacteristic
300C2×C4C2×D13⋊C811yescharacteristic
301C2×C4C2×D13⋊C811yescharacteristic
302C2×C4#8823712yes—
303C2×C4C2×#612212yes—
304C2×C4C23×#26111yescharacteristic
305C8—22no—
306C8—132no—
307C8—132no—
308C8—261no—
309C8—261no—
310C8C2×#612214yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22#1171909612yes—
320C22#1171912512yes—
321C22C22×D13⋊C811yescharacteristic
322C4—21no—
323C4—21no—
324C4—262no—
325C4—262no—
326C4—262no—
327C4—262no—
328C4#1171912512yes—
329C4C2×#8823711yescharacteristic
330C4C22×#612211yescharacteristic
331C2—22no—
332C2—22no—
333C2—41no—
334C2C2×#1171912511yescharacteristic
335C2C2×#1171909611yescharacteristic
336C2#1313180511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#261
commutator, socleC22×C13C23×C8
FittingC22×C13×M16C4
prime radicalC24×C13C2×C8
cosocleC2×#1055C24
hypercenterC22×M16#261
hypocenterC13#33422
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #13131202C2semidirect
2. #13131198C2semidirect
3. #13131194C2semidirect
4. #13131173C2semidirect
7. #13131047C2semidirect
27. #11718260C22semidirect
28. #11718260C22semidirect
29. #11718258C22semidirect
30. #11718257C22semidirect
279. C13#33422semidirect

Maximal subgroups

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

subgroupindex
#131312022
#131311982
#131311942
#131311732
#131310632
C2×C22⋊#60922
#131310472
C2×#117187832
#3342213