Finite Group Databasebeta

← #30670330

Subgroups — #30670330

Order 16768.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862245C211yesmaximal, characteristic, complemented
2C2×#16862224C211yesmaximal, characteristic
3#22553254C211yesmaximal, characteristic, complemented
4#22553225C211yesmaximal, characteristic
5#22553225C211yesmaximal, characteristic
6#22553217C211yesmaximal, characteristic, complemented
7#22553216C211yesmaximal, characteristic, complemented
8#22553216C211yesmaximal, characteristic, complemented
9#22553216C211yesmaximal, characteristic, complemented
10#22553204C211yesmaximal, characteristic
11#22553202C211yesmaximal, characteristic
12#22553171C211yesmaximal, characteristic
13#22553163C211yesmaximal, characteristic, complemented
14#22553159C211yesmaximal, characteristic
15#400×C131C211yesmaximal, characteristic, complemented
16C22×Dic262C2211yescharacteristic
17C22×Dic262C2211yescharacteristic
18C22×Dic262C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20#16862245—24no—
21#16862224—24no—
22C2×C4⋊Dic131C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2211yescharacteristic
25C2×Dic131⋊C4C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×C4×Dic131C2211yescharacteristic
34C2×C4×Dic131C2211yescharacteristic
35C2×C4×Dic131C2211yescharacteristic
36C2×C4×Dic131C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C4⋊C4×C131C2211yescharacteristic
48#95×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C23×D131C2311yescharacteristic
54C2×Dic262—22no—
55C2×Dic262—22no—
56C2×Dic262—22no—
57C2×Dic262—22no—
58C2×Dic262—22no—
59C2×Dic262—22no—
60C2×Dic262—22no—
61C2×Dic262—41no—
62C2×Dic262—41no—
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C22×Dic131C2311yescharacteristic
68C22×Dic131C2311yescharacteristic
69C22×Dic131C2311yescharacteristic
70C2×C4×D131—22no—
71C2×C4×D131—22no—
72C2×C4×D131—22no—
73C4⋊Dic131—22no—
74Dic131⋊C4—22no—
75Dic131⋊C4—22no—
76Dic131⋊C4—22no—
77Dic131⋊C4D414yes—
78(C2×D131)⋊C4—22no—
79(C2×D131)⋊C4—22no—
80(C2×D131)⋊C4—22no—
81(C2×D131)⋊C4—41no—
82(C2×D131)⋊C4—41no—
83(C2×D131)⋊C4Q814yescomplemented
84C4×Dic131—22no—
85C4×Dic131—22no—
86C4×Dic131—41no—
87C4×Dic131—41no—
88C22×C4×C131C2311yescharacteristic
89C22×C4×C131C2311yescharacteristic
90C22×C4×C131C2311yescharacteristic
91C22×C4×C131C2311yescharacteristic
92C22×C4×C131C2311yescharacteristic
93C22×C4×C131C2311yescharacteristic
94C22×C4×C131C2311yescharacteristic
95C4⋊C4×C131—22no—
96C4⋊C4×C131—22no—
97C22×D131—21no—
98C22×D131—21no—
99C22×D131—21no—
100C22×D131—21no—
101C22×D131—21no—
102C22×D131—21no—
103C22×D131C2×Q812yes—
104Dic262—44no—
105Dic262—44no—
106Dic262—44no—
107C2×Dic131—21no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—21no—
112C2×Dic131—21no—
113C2×Dic131—21no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131—21no—
117C2×Dic131—21no—
118C2×Dic131—21no—
119C2×Dic131—21no—
120C2×Dic131—21no—
121C2×Dic131—21no—
122C2×Dic131C2×D412yes—
123C2×Dic131C2×D412yes—
124C2×Dic131C2×Q812yes—
125C2×Dic131#4112yes—
126C2×Dic131#4112yes—
127C2×Dic131#4112yes—
128C4×D131—44no—
129C23×C131C2411yescharacteristic
130C2×C4×C131—21no—
131C2×C4×C131—21no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—21no—
135C2×C4×C131—21no—
136C2×C4×C131—21no—
137C2×C4×C131—21no—
138C2×C4×C131—21no—
139C2×C4×C131—21no—
140C2×C4×C131—21no—
141C2×C4×C131—21no—
142C2×C4×C131—21no—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131—21no—
148C2×C4×C131—21no—
149C2×C4×C131C2×D412yes—
150C2×C4×C131C2×Q812yes—
151C22×C131C22×D411yescharacteristic
152C22×C131C22×Q811yescharacteristic
153C22×C131C2×#4111yescharacteristic
154C22×C131C2×#4111yescharacteristic
155C22×C131C2×#4111yescharacteristic
156C22×C131H(5,𝔽2)11yescharacteristic
157C22×C1312−1+411yescharacteristic
158C2×D131—22no—
159C2×D131—22no—
160C2×D131—22no—
161C2×D131—41no—
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—41no—
165C4×C131—22no—
166C4×C131—22no—
167C4×C131—41no—
168C4×C131—41no—
169C4×C131—41no—
170C4×C131—41no—
171C4×C131—41no—
172Dic131—22no—
173Dic131—22no—
174Dic131—22no—
175Dic131—41no—
176Dic131—41no—
177Dic131—41no—
178Dic131#12414yes—
179C2×C131C2×#12411yescharacteristic
180C2×C131#53311yescharacteristic
181C2×C131#54211yescharacteristic
182C2×C131#54611yescharacteristic
183C2×C131D4×Q811yescharacteristic
184C2×C131#55411yescharacteristic
185C2×C131#55611yescharacteristic
186D131—42no—
187C131#249211yescharacteristic, complemented, Sylow
188#2492—1311nomaximal, Sylow
189C2×C4⋊Q8—1311no—
190C2×C22⋊Q8—1311no—
191#400—1311no—
192#400D13111yescharacteristic, complemented
193#393—1311no—
194#393—1311no—
195#391—1311no—
196#391—1311no—
197#391—1311no—
198#391—1311no—
199#388—1311no—
200#388—1311no—
201#380—1311no—
202C4×C4⋊C4—1311no—
203C4×C22⋊C4—1311no—
204C22×Q8—1311no—
205C22×Q8—1311no—
206C22×Q8—1311no—
207C23×C4—1311no—
208C4⋊Q8—2624no—
209C22⋊Q8—2624no—
210C2×C4⋊C4—1311no—
211C2×C4⋊C4—1311no—
212C2×C4⋊C4—1311no—
213C2×C4⋊C4—1311no—
214C2×C4⋊C4—1311no—
215C2×C4⋊C4C2×D13111yescharacteristic
216C2×C4⋊C4C2×D13111yescharacteristic
217C2×C22⋊C4—1311no—
218C2×C22⋊C4—1311no—
219C2×C22⋊C4—1311no—
220C2×C22⋊C4—1311no—
221C2×C22⋊C4—1311no—
222C2×C22⋊C4—1311no—
223C2×C42—1311no—
224C2×C42—1311no—
225C2×C42—1311no—
226C2×C42—1311no—
227#95—1311no—
228#95—1311no—
229#95—1311no—
230#95—1311no—
231#95—1311no—
232#95—1311no—
233#95—1311no—
234#95—1311no—
235#95—1311no—
236#95C2×D13111yescharacteristic
237#95C2×D13111yescharacteristic
238#95C2×D13111yescharacteristic
239#95C2×D13111yescharacteristic
240#95C2×D13111yescharacteristic
241C24—1311no—
242C2×Q8—2622no—
243C2×Q8—2622no—
244C2×Q8—2622no—
245C2×Q8—2622no—
246C2×Q8—2622no—
247C2×Q8—2622no—
248C2×Q8—2622no—
249C2×Q8—5241no—
250C2×Q8—5241no—
251C22×C4—2622no—
252C22×C4—2622no—
253C22×C4—2622no—
254C22×C4—1311no—
255C22×C4—1311no—
256C22×C4—1311no—
257C22×C4—1311no—
258C22×C4—1311no—
259C22×C4—1311no—
260C22×C4—1311no—
261C22×C4C22×D13111yescharacteristic
262C22×C4C22×D13111yescharacteristic
263C22×C4C22×D13111yescharacteristic
264C22×C4C22×D13111yescharacteristic
265C22×C4C22×D13111yescharacteristic
266C22×C4C22×D13111yescharacteristic
267C22×C4C22×D13111yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—2622no—
271C4⋊C4—2622no—
272C4⋊C4—2622no—
273C4⋊C4—2622no—
274C4⋊C4—1314no—
275C22⋊C4—2622no—
276C22⋊C4—2622no—
277C22⋊C4—2622no—
278C22⋊C4—5241no—
279C22⋊C4—5241no—
280C22⋊C4—1314no—
281C42—2622no—
282C42—2622no—
283C42—5241no—
284C42—5241no—
285C23—2621no—
286C23—2621no—
287C23—2621no—
288C23—2621no—
289C23—2621no—
290C23—2621no—
291C23—1312no—
292C23C23×D13111yescharacteristic
293Q8—5244no—
294Q8—5244no—
295Q8—5244no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—2621no—
316C2×C4—2621no—
317C2×C4—2621no—
318C2×C4—2621no—
319C2×C4—2621no—
320C2×C4—2621no—
321C2×C4—2621no—
322C2×C4—2621no—
323C2×C4—2621no—
324C2×C4—2621no—
325C2×C4—2621no—
326C2×C4—2621no—
327C2×C4—2621no—
328C2×C4—2621no—
329C2×C4—2621no—
330C2×C4—5244no—
331C2×C4—1312no—
332C2×C4—1312no—
333C2×C4—1312no—
334C2×C4—1312no—
335C2×C4—1312no—
336C2×C4—1312no—
337C2×C4D4×D13112yes—
338C2×C4Q8×D13112yes—
339C22—2622no—
340C22—2622no—
341C22—2622no—
342C22—5241no—
343C22—5241no—
344C22—5241no—
345C22—5241no—
346C22C2×D4×D13111yescharacteristic
347C22C2×Q8×D13111yescharacteristic
348C22#41×D13111yescharacteristic
349C22#41×D13111yescharacteristic
350C22#41×D13111yescharacteristic
351C22#1686229211yescharacteristic
352C22#1686229311yescharacteristic
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2622no—
361C4—2622no—
362C4—2622no—
363C4—5241no—
364C4—5241no—
365C4—5241no—
366C4—1314no—
367C2—5242no—
368C2#124×D13111yescharacteristic
369C2#2255386811yescharacteristic
370C2#2255392811yescharacteristic
371C2#2255395411yescharacteristic
372C2#2255397511yescharacteristic
373C2#2255401811yescharacteristic
374C2#2255402911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#400×C131C2
prime radicalC23×D131C23
hypercenter#400D131
hypocenterC131#2492
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
3. #22553254C2semidirect
6. #22553217C2semidirect
7. #22553216C2semidirect
8. #22553216C2semidirect
9. #22553216C2semidirect
13. #22553163C2semidirect
15. #400×C131C2semidirect
83. (C2×D131)⋊C4Q8semidirect
187. C131#2492semidirect
192. #400D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622242
#225532542
#225532252
#225532172
#225532162
#225532042
#225532022
#225531712
#225531632
#225531592
#400×C1312
#2492131