Finite Group Databasebeta

← #30680571

Subgroups — #30680571

Order 16768.

912 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
1C131⋊#564C212yesmaximal, complemented
2#22554066C212yesmaximal, complemented
3#22554038C212yesmaximal, complemented
4#22554020C212yesmaximal, complemented
5#22554004C212yesmaximal, complemented
6#22554003C212yesmaximal, complemented
7#22553995C214yesmaximal, complemented
8#22553941C212yesmaximal, complemented
9#22553939C212yesmaximal, complemented
10#22553923C212yesmaximal, complemented
11#22553920C214yesmaximal, complemented
12#22553911C212yesmaximal
13#22553846C211yesmaximal, characteristic
14C2×#16862238C211yesmaximal, characteristic, complemented
15#533×C131C211yesmaximal, characteristic, complemented
16#16862245C2212yes—
17#16862245C2214yes—
18#16862244C2214yes—
19#16862241C2214yes—
20#16862241C2214yes—
21#16862241C2214yes—
22#16862240C2212yes—
23#16862240C2214yes—
24#16862238C2212yes—
25#16862238C2214yes—
26#16862238C2212yes—
27#16862238C2212yes—
28#16862238C2212yes—
29#16862238C2211yescharacteristic
30#16862237C2214yes—
31#16862237C2214yes—
32#16862237C2212yes—
33#16862236C2214yes—
34#16862236C2214yes—
35#16862236C2214yes—
36#16862236C2214yes—
37#16862236C2214yes—
38#16862234C2211yescharacteristic
39#16862231C2214yes—
40#16862230C2214yes—
41#16862229C2212yes—
42#16862226C2214yes—
43#16862225C2211yescharacteristic
44#16862224C2214yes—
45#16862223C2214yes—
46#16862222C2214yes—
47#16862222C2212yes—
48#16862222C2212yes—
49#16862220C2211yescharacteristic
50Q8×Dic131C2212yes—
51Q8×Dic131C2212yes—
52C4×Dic262C2214yes—
53C4×Dic262C2211yescharacteristic
54C4×D524C2211yescharacteristic
55C4×C22⋊D131C2212yes—
56(C22⋊D131)⋊C4C2214yes—
57#16862204C2212yes—
58#16862203C2214yes—
59#16862203C2212yes—
60#16862202C2214yes—
61C2×C4⋊Dic131C2211yescharacteristic
62C2×C4⋊Dic131C2211yescharacteristic
63C2×Dic131⋊C4C2212yes—
64C2×Dic131⋊C4C2212yes—
65C4⋊C4×D131C2212yes—
66C4⋊Q8×C131C2211yescharacteristic
67C4⋊Q8×C131C2211yescharacteristic
68#125×C131C2212yes—
69C22⋊Q8×C131C2212yes—
70C22⋊Q8×C131C2212yes—
71C4×Q8×C131C2212yes—
72C4×Q8×C131C2212yes—
73#117×C131C2212yes—
74C2×C42×C131C2211yescharacteristic
75C2×Dic262C2314yes—
76C2×Dic262C2311yescharacteristic
77C2×D524C2311yescharacteristic
78C2×C22⋊D131C2312yes—
79C22×Dic131C2312yes—
80C22×Dic131C2312yes—
81C2×C4×D131C2312yes—
82C2×C4×D131C2314yes—
83C4⋊Dic131C2311yescharacteristic
84C4⋊Dic131C2311yescharacteristic
85C4⋊Dic131C2314yes—
86C4⋊Dic131C2312yes—
87C4⋊Dic131C2312yes—
88C4⋊Dic131C2314yes—
89C4⋊Dic131C2314yes—
90C4⋊Dic131C2312yes—
91C4⋊Dic131C2312yes—
92Dic131⋊C4C2312yes—
93Dic131⋊C4C2314yes—
94Dic131⋊C4C2314yes—
95Dic131⋊C4C2314yes—
96Dic131⋊C4C2314yes—
97Dic131⋊C4C2314yes—
98Dic131⋊C4C2314yes—
99Dic131⋊C4C2314yes—
100Dic131⋊C4C2314yes—
101Dic131⋊C4C2312yes—
102Dic131⋊C4C2312yes—
103Dic131⋊C4C2314yes—
104Dic131⋊C4C2312yes—
105C22⋊Dic131C2312yes—
106(C2×D131)⋊C4C2312yes—
107(C2×D131)⋊C4C2312yes—
108(C2×D131)⋊C4C2314yes—
109(C2×D131)⋊C4C2314yes—
110(C2×D131)⋊C4C2314yes—
111(C2×D131)⋊C4C2312yes—
112(C2×D131)⋊C4C2312yes—
113C4×Dic131C2312yes—
114C4×Dic131C2314yes—
115C4×Dic131C2314yes—
116C4×Dic131C2314yes—
117C2×Q8×C131C2312yes—
118C2×Q8×C131C2312yes—
119C22×C4×C131C2311yescharacteristic
120C22×C4×C131C2311yescharacteristic
121C22×C4×C131C2311yescharacteristic
122C4⋊C4×C131C2312yes—
123C4⋊C4×C131C2314yes—
124C4⋊C4×C131C2314yes—
125C4⋊C4×C131C2312yes—
126C4⋊C4×C131C2312yes—
127C4⋊C4×C131C2312yes—
128C22⋊C4×C131C2314yes—
129C42×C131C2311yescharacteristic
130C42×C131C2314yes—
131C42×C131C2311yescharacteristic
132C42×C131C2311yescharacteristic
133C42×C131C2311yescharacteristic
134C22×D131C2412yes—
135Dic262—44no—
136Dic262—41no—
137D524—41no—
138C22⋊D131—24no—
139C2×Dic131—24no—
140C2×Dic131—42no—
141C2×Dic131C2412yes—
142C2×Dic131C2412yes—
143C2×Dic131C2414yes—
144C2×Dic131C2412yes—
145C2×Dic131C2414yes—
146C4×D131—24no—
147C4×D131—44no—
148C23×C131C2411yescharacteristic
149Q8×C131—24no—
150Q8×C131—42no—
151C2×C4×C131—22no—
152C2×C4×C131—22no—
153C2×C4×C131—41no—
154C2×C4×C131C2411yescharacteristic
155C2×C4×C131C2411yescharacteristic
156C2×C4×C131C2411yescharacteristic
157C2×C4×C131C2411yescharacteristic
158C2×C4×C131C2411yescharacteristic
159C2×C4×C131C2414yes—
160C2×C4×C131C2414yes—
161C2×C4×C131C2411yescharacteristic
162C22×C131—21no—
163C22×C131—21no—
164C22×C1312−1+412yes—
165C22×C131C2511yescharacteristic
166C2×D131—22no—
167C2×D131—22no—
168C2×D131—22no—
169C4×C131—21no—
170C4×C131—21no—
171C4×C131—21no—
172C4×C131—24no—
173C4×C131—21no—
174C4×C131—21no—
175C4×C131—24no—
176C4×C1312−1+412yes—
177Dic131—22no—
178Dic131—24no—
179Dic131—22no—
180Dic131—24no—
181Dic131—22no—
182C2×C131—22no—
183C2×C131C2×2−1+411yescharacteristic
184C2×C131C2×2−1+411yescharacteristic
185C2×C131#58511yescharacteristic
186D131—42no—
187C131#356311yescharacteristic, complemented, Sylow
188#3563—1311nomaximal, Sylow
189#564—1312no—
190#563—1312no—
191#557—1312no—
192#554—1312no—
193#552—1312no—
194#552—1312no—
195#552—1314no—
196#544—1312no—
197#544—1312no—
198#541—1312no—
199#541—1314no—
200#539—1312no—
201#533D13111yescharacteristic, complemented
202#529—1311no—
203C2×#125—1311no—
204C4⋊Q8—1314no—
205C4⋊Q8—1312no—
206C4⋊Q8—1314no—
207C4⋊Q8C2×D13111yescharacteristic
208C4⋊Q8C2×D13111yescharacteristic
209#126—1314no—
210#126—1314no—
211#126—1314no—
212#126—1312no—
213#126—1314no—
214#125—1314no—
215#125—1314no—
216#125—1314no—
217#125—1314no—
218#125—1314no—
219#125—1314no—
220#125—1314no—
221#125—1312no—
222#125—1312no—
223#125—1312no—
224#125—1312no—
225#125—1312no—
226#125—1311no—
227#125—1314no—
228#125C2×D13112yes—
229#124—1311no—
230#123—1312no—
231#123—1314no—
232#123—1314no—
233C22⋊Q8—1314no—
234C22⋊Q8—1312no—
235C22⋊Q8—1312no—
236C22⋊Q8—1311no—
237C22⋊Q8—1314no—
238C22⋊Q8—1314no—
239C22⋊Q8—1314no—
240C22⋊Q8C2×D13112yes—
241C22⋊Q8C2×D13112yes—
242#121—1311no—
243C4×Q8—1312no—
244C4×Q8—1312no—
245C4×Q8—1311no—
246C4×Q8—1314no—
247C4×Q8C2×D13112yes—
248C4×Q8C2×D13112yes—
249C4×D4—1312no—
250C4×D4—1314no—
251C4×D4—1311no—
252#117—1312no—
253#117—1314no—
254#117—1312no—
255#117—1314no—
256#117C2×D13112yes—
257C2×C4⋊C4—1312no—
258C2×C4⋊C4—1312no—
259C2×C4⋊C4—1311no—
260C2×C4⋊C4—1311no—
261C2×C4⋊C4—1312no—
262C2×C42C2×D13111yescharacteristic
263C2×Q8—1314no—
264C2×Q8—1311no—
265C2×Q8C22×D13112yes—
266C2×Q8C22×D13112yes—
267C2×D4—1312no—
268C2×D4—1311no—
269C22×C4—1312no—
270C22×C4—1312no—
271C22×C4—1312no—
272C22×C4—1314no—
273C22×C4C22×D13111yescharacteristic
274C22×C4C22×D13111yescharacteristic
275C22×C4C22×D13111yescharacteristic
276C4⋊C4—1314no—
277C4⋊C4—1314no—
278C4⋊C4—1314no—
279C4⋊C4—1314no—
280C4⋊C4—1314no—
281C4⋊C4—1314no—
282C4⋊C4—1314no—
283C4⋊C4—1314no—
284C4⋊C4—1312no—
285C4⋊C4—1312no—
286C4⋊C4—1314no—
287C4⋊C4—1312no—
288C4⋊C4—1312no—
289C4⋊C4—1312no—
290C4⋊C4—1312no—
291C4⋊C4—1312no—
292C4⋊C4—1312no—
293C4⋊C4—1311no—
294C4⋊C4—1311no—
295C4⋊C4—1314no—
296C4⋊C4—1314no—
297C4⋊C4—1314no—
298C4⋊C4C22×D13112yes—
299C4⋊C4C22×D13112yes—
300C4⋊C4C22×D13112yes—
301C4⋊C4C22×D13114yes—
302C4⋊C4C22×D13114yes—
303C4⋊C4C22×D13112yes—
304C22⋊C4—1312no—
305C22⋊C4—1312no—
306C22⋊C4—1312no—
307C22⋊C4—1314no—
308C22⋊C4—1312no—
309C22⋊C4—1314no—
310C22⋊C4—1314no—
311C22⋊C4—1312no—
312C22⋊C4C22×D13114yes—
313C42—1314no—
314C42—1312no—
315C42—1314no—
316C42—1314no—
317C42C22×D13111yescharacteristic
318C42C22×D13111yescharacteristic
319C42C22×D13111yescharacteristic
320C42C22×D13111yescharacteristic
321C42C22×D13114yes—
322C23—1312no—
323C23C23×D13111yescharacteristic
324Q8—24no—
325Q8—42no—
326Q8—5241no—
327Q8—5244no—
328D4—2624no—
329D4—5241no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—2624no—
334C2×C4—2624no—
335C2×C4—5244no—
336C2×C4—5242no—
337C2×C4—1312no—
338C2×C4—1312no—
339C2×C4—1312no—
340C2×C4—1314no—
341C2×C4—1314no—
342C2×C4C23×D13114yes—
343C2×C4C23×D13114yes—
344C2×C4C23×D13111yescharacteristic
345C2×C4C23×D13111yescharacteristic
346C2×C4C23×D13111yescharacteristic
347C2×C4C23×D13111yescharacteristic
348C2×C4C23×D13111yescharacteristic
349C2×C4C23×D13111yescharacteristic
350C22—21no—
351C22—21no—
352C22—2622no—
353C22—2622no—
354C22—2622no—
355C22#1686229412yes—
356C22C24×D13111yescharacteristic
357C4—21no—
358C4—24no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—24no—
364C4—2622no—
365C4—2624no—
366C4—2622no—
367C4—2624no—
368C4—2622no—
369C4#1686229312yes—
370C2—22no—
371C2—5242no—
372C2C2×#1686229311yescharacteristic
373C2C2×#1686229411yescharacteristic
374C2#2255414811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D131
commutator, socle, cosocleC22×C131C25
Fitting#533×C131C2
prime radical#16862220C22
hypercenter#533D131
hypocenterC131#3563
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C131⋊#564C2semidirect
2. #22554066C2semidirect
3. #22554038C2semidirect
4. #22554020C2semidirect
5. #22554004C2semidirect
6. #22554003C2semidirect
7. #22553995C2semidirect
8. #22553941C2semidirect
9. #22553939C2semidirect
10. #22553923C2semidirect
11. #22553920C2semidirect
14. C2×#16862238C2semidirect
15. #533×C131C2semidirect
187. C131#3563semidirect
201. #533D131semidirect

Maximal subgroups

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

subgroupindex
C131⋊#5642
#225540662
#225540382
#225540202
#225540042
#225540032
#225539952
#225539412
#225539392
#225539232
#225539202
#225539112
#225538462
C2×#168622382
#533×C1312
#3563131