Finite Group Databasebeta

← #30668712

Subgroups — #30668712

Order 16768.

484 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862224C211yesmaximal, characteristic
2#22553218C211yesmaximal, characteristic, complemented
3#22553216C211yesmaximal, characteristic, complemented
4#22553204C211yesmaximal, characteristic
5#22553203C211yesmaximal, characteristic
6#22553203C211yesmaximal, characteristic
7#22553202C211yesmaximal, characteristic
8#22553198C211yesmaximal, characteristic, complemented
9#22553196C211yesmaximal, characteristic, complemented
10#22553194C211yesmaximal, characteristic, complemented
11#22553193C211yesmaximal, characteristic, complemented
12#22553174C211yesmaximal, characteristic
13C4×Dic131⋊C4C211yesmaximal, characteristic, complemented
14C4×(C2×D131)⋊C4C211yesmaximal, characteristic
15C4×C4⋊C4×C131C211yesmaximal, characteristic, complemented
16C22×Dic262C2211yescharacteristic
17C22×C4×D131C2211yescharacteristic
18#16862224C418yescomplemented
19C2×C4⋊Dic131C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2211yescharacteristic
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×C4×Dic131C2211yescharacteristic
31C2×C4×Dic131C2211yescharacteristic
32C2×C4×Dic131C2211yescharacteristic
33C131⋊#95C2211yescharacteristic
34C131⋊#95C2211yescharacteristic
35C131⋊#95C2211yescharacteristic
36C131⋊#95C2211yescharacteristic
37C131⋊#95C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C2×C4⋊C4×C131C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C42×C131C2211yescharacteristic
48C2×C42×C131C2211yescharacteristic
49C2×C42×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52C23×D131C2311yescharacteristic
53C2×Dic262—41no—
54C2×Dic262—41no—
55C2×Dic262C2×C414yes—
56C22×Dic131C2311yescharacteristic
57C22×Dic131C2311yescharacteristic
58C22×Dic131C2311yescharacteristic
59C22×Dic131C2311yescharacteristic
60C22×Dic131C2311yescharacteristic
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C2×C4×D131—41no—
64C2×C4×D131—41no—
65C2×C4×D131C2×C414yes—
66C4⋊Dic131—41no—
67Dic131⋊C4—22no—
68Dic131⋊C4—41no—
69Dic131⋊C4C2×C414yes—
70Dic131⋊C4C2×C414yes—
71(C2×D131)⋊C4—22no—
72(C2×D131)⋊C4—22no—
73(C2×D131)⋊C4—41no—
74(C2×D131)⋊C4—41no—
75(C2×D131)⋊C4C2×C414yes—
76(C2×D131)⋊C4C2×C414yes—
77C4×Dic131—22no—
78C4×Dic131—41no—
79C4×Dic131—41no—
80C22×C4×C131C2311yescharacteristic
81C22×C4×C131C2311yescharacteristic
82C22×C4×C131C2311yescharacteristic
83C22×C4×C131C2311yescharacteristic
84C22×C4×C131C2311yescharacteristic
85C22×C4×C131C2311yescharacteristic
86C22×C4×C131C2311yescharacteristic
87C4⋊C4×C131—22no—
88C4⋊C4×C131C2×C414yes—
89C42×C131—22no—
90C42×C131—41no—
91C42×C131—41no—
92C22×D131—21no—
93C22×D131—21no—
94C22×D131—21no—
95C22×D131—21no—
96C22×D131—21no—
97C22×D131—21no—
98C22×D131C22×C412yes—
99Dic262—44no—
100C2×Dic131—21no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131—21no—
104C2×Dic131—21no—
105C2×Dic131—21no—
106C2×Dic131—21no—
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×Dic131C22×C412yes—
119C2×Dic131C22×C412yes—
120C2×Dic131C22×C412yes—
121C4×D131—44no—
122C23×C131C2411yescharacteristic
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—21no—
126C2×C4×C131—21no—
127C2×C4×C131—21no—
128C2×C4×C131—21no—
129C2×C4×C131—21no—
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×C131C22×C412yes—
140C2×C4×C131C22×C412yes—
141C2×C4×C131C22×C412yes—
142C2×C4×C131#4112yes—
143C2×C4×C131#4112yes—
144C22×C131C23×C411yescharacteristic
145C22×C131C2×#4111yescharacteristic
146C22×C131C2×#4111yescharacteristic
147C22×C131H(5,𝔽2)11yescharacteristic
148C22×C131H(5,𝔽2)11yescharacteristic
149C22×C1312−1+411yescharacteristic
150C22×C1312−1+411yescharacteristic
151C2×D131—22no—
152C2×D131—22no—
153C2×D131—22no—
154C2×D131—41no—
155C2×D131—41no—
156C2×D131—41no—
157C2×D131—41no—
158C4×C131—22no—
159C4×C131—22no—
160C4×C131—22no—
161C4×C131—22no—
162C4×C131—22no—
163C4×C131—41no—
164C4×C131—41no—
165Dic131—22no—
166Dic131—22no—
167Dic131—22no—
168Dic131—41no—
169Dic131—41no—
170Dic131—41no—
171Dic131—41no—
172C2×C131C4×#4111yescharacteristic
173C2×C131#51811yescharacteristic
174C2×C131#51911yescharacteristic
175C2×C131#53911yescharacteristic
176C2×C131#53911yescharacteristic
177C2×C131#54211yescharacteristic
178C2×C131#54211yescharacteristic
179D131—42no—
180C131#233211yescharacteristic, complemented, Sylow
181#2332—1311nomaximal, Sylow
182C2×C22⋊Q8—1311no—
183#391—1311no—
184#391—1311no—
185#388—1311no—
186#388—1311no—
187#388—1311no—
188#388—1311no—
189#387—1311no—
190#387—1311no—
191#387—1311no—
192#387—1311no—
193#381—1311no—
194C4×C4⋊C4—1311no—
195C4×C4⋊C4D13111yescharacteristic, complemented
196C4×C22⋊C4—1311no—
197C22×Q8—1311no—
198C23×C4—1311no—
199C22⋊Q8—1318no—
200C2×C4⋊C4—1311no—
201C2×C4⋊C4—1311no—
202C2×C4⋊C4—1311no—
203C2×C4⋊C4—1311no—
204C2×C4⋊C4—1311no—
205C2×C4⋊C4C2×D13111yescharacteristic
206C2×C4⋊C4C2×D13111yescharacteristic
207C2×C22⋊C4—1311no—
208C2×C22⋊C4—1311no—
209C2×C22⋊C4—1311no—
210C2×C22⋊C4—1311no—
211C2×C22⋊C4—1311no—
212C2×C22⋊C4—1311no—
213C2×C42—1311no—
214C2×C42—1311no—
215C2×C42—1311no—
216C2×C42C2×D13111yescharacteristic
217C2×C42C2×D13111yescharacteristic
218C2×C42C2×D13111yescharacteristic
219#95—1311no—
220#95—1311no—
221#95—1311no—
222#95—1311no—
223#95—1311no—
224#95—1311no—
225#95—1311no—
226#95—1311no—
227#95—1311no—
228#95—1311no—
229#95—1311no—
230#95—1311no—
231#95C2×D13111yescharacteristic
232#95C2×D13111yescharacteristic
233C24—1311no—
234C2×Q8—5241no—
235C2×Q8—5241no—
236C2×Q8—1314no—
237C22×C4—5241no—
238C22×C4—5241no—
239C22×C4—1311no—
240C22×C4—1311no—
241C22×C4—1311no—
242C22×C4—1311no—
243C22×C4—1311no—
244C22×C4—1311no—
245C22×C4—1311no—
246C22×C4—1314no—
247C22×C4C22×D13111yescharacteristic
248C22×C4C22×D13111yescharacteristic
249C22×C4C22×D13111yescharacteristic
250C22×C4C22×D13111yescharacteristic
251C22×C4C22×D13111yescharacteristic
252C22×C4C22×D13111yescharacteristic
253C22×C4C22×D13111yescharacteristic
254C4⋊C4—22no—
255C4⋊C4—2622no—
256C4⋊C4—5241no—
257C4⋊C4—5241no—
258C4⋊C4—1314no—
259C4⋊C4—1314no—
260C4⋊C4C4×D13114yes—
261C22⋊C4—2622no—
262C22⋊C4—2622no—
263C22⋊C4—5241no—
264C22⋊C4—5241no—
265C22⋊C4—1314no—
266C22⋊C4—1314no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—2622no—
271C42—5241no—
272C42—5241no—
273C23—2621no—
274C23—2621no—
275C23—2621no—
276C23—2621no—
277C23—2621no—
278C23—2621no—
279C23—1312no—
280C23C23×D13111yescharacteristic
281Q8—5244no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—2621no—
299C2×C4—2621no—
300C2×C4—2621no—
301C2×C4—2621no—
302C2×C4—2621no—
303C2×C4—2621no—
304C2×C4—2621no—
305C2×C4—2621no—
306C2×C4—2621no—
307C2×C4—2621no—
308C2×C4—2621no—
309C2×C4—2621no—
310C2×C4—2621no—
311C2×C4—2621no—
312C2×C4—2621no—
313C2×C4—2621no—
314C2×C4—2621no—
315C2×C4—2621no—
316C2×C4—5244no—
317C2×C4—1312no—
318C2×C4—1312no—
319C2×C4—1312no—
320C2×C4C2×C4×D13112yes—
321C2×C4C2×C4×D13112yes—
322C2×C4C2×C4×D13112yes—
323C2×C4#1452351312yes—
324C2×C4#1452351312yes—
325C22—2622no—
326C22—2622no—
327C22—2622no—
328C22—5241no—
329C22—5241no—
330C22—5241no—
331C22—5241no—
332C22C22×C4×D13111yescharacteristic
333C22C2×#1452351311yescharacteristic
334C22C2×#1452351311yescharacteristic
335C22#1686229111yescharacteristic
336C22#1686229211yescharacteristic
337C22#1686229311yescharacteristic
338C22#1686229411yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—2622no—
347C4—2622no—
348C4—2622no—
349C4—5241no—
350C4—5241no—
351C4—5241no—
352C4—5241no—
353C2—5242no—
354C2C4×#1452351311yescharacteristic
355C2#2255378411yescharacteristic
356C2#2255378811yescharacteristic
357C2#2255390411yescharacteristic
358C2#2255390911yescharacteristic
359C2#2255393311yescharacteristic
360C2#2255393411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutatorC22×C131C23×C4
FittingC4×C4⋊C4×C131C2
socle, cosocleC23×C131C24
prime radicalC23×D131C23
hypercenterC4×C4⋊C4D131
hypocenterC131#2332
LCSG ▷ C22×C131 ▷ C131
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22553218C2semidirect
3. #22553216C2semidirect
8. #22553198C2semidirect
9. #22553196C2semidirect
10. #22553194C2semidirect
11. #22553193C2semidirect
13. C4×Dic131⋊C4C2semidirect
15. C4×C4⋊C4×C131C2semidirect
18. #16862224C4semidirect
180. C131#2332semidirect
195. C4×C4⋊C4D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622242
#225532182
#225532162
#225532042
#225532032
#225532022
#225531982
#225531962
#225531942
#225531932
#225531742
C4×Dic131⋊C42
C4×(C2×D131)⋊C42
C4×C4⋊C4×C1312
#2332131