Finite Group Databasebeta

← #30671140

Subgroups — #30671140

Order 16768.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16862224C211yesmaximal, characteristic, complemented
2C2×C4×D524C211yesmaximal, characteristic
3#22553254C211yesmaximal, characteristic, complemented
4#22553247C211yesmaximal, characteristic, complemented
5#22553233C211yesmaximal, characteristic, complemented
6#22553225C211yesmaximal, characteristic, complemented
7#22553196C211yesmaximal, characteristic, complemented
8#22553190C211yesmaximal, characteristic
9#22553190C211yesmaximal, characteristic
10#22553186C211yesmaximal, characteristic, complemented
11#22553183C211yesmaximal, characteristic, complemented
12#22553171C211yesmaximal, characteristic, complemented
13#22553169C211yesmaximal, characteristic, complemented
14#95×D131C211yesmaximal, characteristic, complemented
15#387×C131C211yesmaximal, characteristic, complemented
16C22×Dic262C2211yescharacteristic
17C22×D524C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20C22×C4×D131C2211yescharacteristic
21C22×C4×D131C2211yescharacteristic
22#16862224—24no—
23C4×D524—24no—
24C2×C4⋊Dic131C2211yescharacteristic
25C2×C4⋊Dic131C2211yescharacteristic
26C2×Dic131⋊C4C2211yescharacteristic
27C2×Dic131⋊C4C2211yescharacteristic
28C2×Dic131⋊C4C2211yescharacteristic
29C2×Dic131⋊C4C2211yescharacteristic
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×(C2×D131)⋊C4C2211yescharacteristic
32C2×(C2×D131)⋊C4C2211yescharacteristic
33C2×(C2×D131)⋊C4C2211yescharacteristic
34C2×(C2×D131)⋊C4C2211yescharacteristic
35C2×(C2×D131)⋊C4C2211yescharacteristic
36C2×(C2×D131)⋊C4C2211yescharacteristic
37C2×(C2×D131)⋊C4C2211yescharacteristic
38C131⋊#95C2211yescharacteristic
39C131⋊#95C2211yescharacteristic
40C131⋊#95C2211yescharacteristic
41C131⋊#95C2211yescharacteristic
42C131⋊#95C2211yescharacteristic
43C131⋊#95C2211yescharacteristic
44C131⋊#95C2211yescharacteristic
45C131⋊#95C2211yescharacteristic
46C2×C4⋊C4×C131C2211yescharacteristic
47C2×C4⋊C4×C131C2211yescharacteristic
48C2×C42×C131C2211yescharacteristic
49#95×C131C2211yescharacteristic
50#95×C131C2211yescharacteristic
51#95×C131C2211yescharacteristic
52#95×C131C2211yescharacteristic
53C23×D131C2311yescharacteristic
54C23×D131C2311yescharacteristic
55C2×Dic262—22no—
56C2×Dic262—41no—
57C2×Dic262—41no—
58C2×D524—22no—
59C2×D524—22no—
60C2×D524Q814yescomplemented
61C22×Dic131C2311yescharacteristic
62C22×Dic131C2311yescharacteristic
63C22×Dic131C2311yescharacteristic
64C22×Dic131C2311yescharacteristic
65C22×Dic131C2311yescharacteristic
66C22×Dic131C2311yescharacteristic
67C2×C4×D131—22no—
68C2×C4×D131—22no—
69C2×C4×D131—22no—
70C2×C4×D131—22no—
71C2×C4×D131—22no—
72C2×C4×D131—41no—
73C2×C4×D131—41no—
74C2×C4×D131—41no—
75C2×C4×D131—41no—
76C2×C4×D131—41no—
77C2×C4×D131—41no—
78C2×C4×D131D414yes—
79C4⋊Dic131—22no—
80C4⋊Dic131—41no—
81Dic131⋊C4—22no—
82Dic131⋊C4—22no—
83Dic131⋊C4—22no—
84Dic131⋊C4—41no—
85(C2×D131)⋊C4—22no—
86(C2×D131)⋊C4—22no—
87(C2×D131)⋊C4—22no—
88(C2×D131)⋊C4—22no—
89(C2×D131)⋊C4—22no—
90(C2×D131)⋊C4—22no—
91(C2×D131)⋊C4—41no—
92(C2×D131)⋊C4—41no—
93C22×C4×C131C2311yescharacteristic
94C22×C4×C131C2311yescharacteristic
95C22×C4×C131C2311yescharacteristic
96C22×C4×C131C2311yescharacteristic
97C22×C4×C131C2311yescharacteristic
98C22×C4×C131C2311yescharacteristic
99C22×C4×C131C2311yescharacteristic
100C4⋊C4×C131—22no—
101C4⋊C4×C131—22no—
102C42×C131—22no—
103C22×D131—21no—
104C22×D131—21no—
105C22×D131—21no—
106C22×D131—21no—
107C22×D131—21no—
108C22×D131—21no—
109C22×D131—21no—
110C22×D131—21no—
111C22×D131—21no—
112C22×D131—21no—
113C22×D131C2×D412yes—
114C22×D131C2×Q812yes—
115C22×D131C2×Q812yes—
116C22×D131#4112yes—
117Dic262—44no—
118D524—28no—
119C2×Dic131—21no—
120C2×Dic131—21no—
121C2×Dic131—21no—
122C2×Dic131—21no—
123C2×Dic131—21no—
124C2×Dic131—21no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—21no—
131C2×Dic131—21no—
132C2×Dic131—21no—
133C2×Dic131—21no—
134C2×Dic131—21no—
135C2×Dic131C2×D412yes—
136C2×Dic131#4112yes—
137C4×D131—44no—
138C4×D131—44no—
139C4×D131—44no—
140C4×D131—44no—
141C23×C131C2411yescharacteristic
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×C131—21no—
150C2×C4×C131—21no—
151C2×C4×C131—21no—
152C2×C4×C131—21no—
153C2×C4×C131—21no—
154C2×C4×C131—21no—
155C2×C4×C131—21no—
156C2×C4×C131—21no—
157C2×C4×C131—21no—
158C2×C4×C131—21no—
159C2×C4×C131—21no—
160C2×C4×C131C2×D412yes—
161C2×C4×C131C2×Q812yes—
162C2×C4×C131#4112yes—
163C22×C131C22×D411yescharacteristic
164C22×C131C22×Q811yescharacteristic
165C22×C131C2×#4111yescharacteristic
166C22×C131C2×#4111yescharacteristic
167C22×C131C2×#4111yescharacteristic
168C22×C131H(5,𝔽2)11yescharacteristic
169C22×C131H(5,𝔽2)11yescharacteristic
170C2×D131—22no—
171C2×D131—22no—
172C2×D131—22no—
173C2×D131—22no—
174C2×D131—22no—
175C2×D131—22no—
176C2×D131—22no—
177C2×D131—22no—
178C2×D131—22no—
179C2×D131—41no—
180C2×D131—41no—
181C2×D131—41no—
182C2×D131—41no—
183C2×D131C22⋊Q814yes—
184C4×C131—22no—
185C4×C131—22no—
186C4×C131—22no—
187C4×C131—41no—
188C4×C131—41no—
189C4×C131—41no—
190C4×C131—41no—
191Dic131—22no—
192Dic131—22no—
193Dic131—41no—
194Dic131—41no—
195Dic131—41no—
196Dic131—41no—
197C2×C131C2×C22⋊Q811yescharacteristic
198C2×C131#53211yescharacteristic
199C2×C131#53411yescharacteristic
200C2×C131#55111yescharacteristic
201C2×C131#55111yescharacteristic
202C2×C131#55411yescharacteristic
203C2×C131#55411yescharacteristic
204D131—24no—
205D131—42no—
206C131#256411yescharacteristic, complemented, Sylow
207#2564—1311nomaximal, Sylow
208C2×C22⋊Q8—1311no—
209C2×C4×D4—1311no—
210#400—1311no—
211#399—1311no—
212#395—1311no—
213#393—1311no—
214#387—1311no—
215#387D13111yescharacteristic, complemented
216#386—1311no—
217#386—1311no—
218#385—1311no—
219#385—1311no—
220#380—1311no—
221#380—1311no—
222C2×#95—1311no—
223C22×Q8—1311no—
224C22×D4—1311no—
225C23×C4—1311no—
226C23×C4—1311no—
227C23×C4—1311no—
228C23×C4—1311no—
229C22⋊Q8—2624no—
230C4×D4—2624no—
231C2×C4⋊C4—1311no—
232C2×C4⋊C4—1311no—
233C2×C4⋊C4—1311no—
234C2×C4⋊C4—1311no—
235C2×C4⋊C4—1311no—
236C2×C4⋊C4—1311no—
237C2×C4⋊C4C2×D13111yescharacteristic
238C2×C4⋊C4C2×D13111yescharacteristic
239C2×C22⋊C4—1311no—
240C2×C22⋊C4—1311no—
241C2×C22⋊C4—1311no—
242C2×C22⋊C4—1311no—
243C2×C22⋊C4—1311no—
244C2×C22⋊C4—1311no—
245C2×C22⋊C4—1311no—
246C2×C22⋊C4—1311no—
247C2×C42C2×D13111yescharacteristic
248#95—1311no—
249#95—1311no—
250#95—1311no—
251#95—1311no—
252#95—1311no—
253#95—1311no—
254#95—1311no—
255#95—1311no—
256#95C2×D13111yescharacteristic
257#95C2×D13111yescharacteristic
258#95C2×D13111yescharacteristic
259#95C2×D13111yescharacteristic
260C24—1311no—
261C24—1311no—
262C2×Q8—2622no—
263C2×Q8—5241no—
264C2×Q8—5241no—
265C2×D4—2622no—
266C2×D4—2622no—
267C2×D4—1314no—
268C22×C4—2622no—
269C22×C4—2622no—
270C22×C4—2622no—
271C22×C4—2622no—
272C22×C4—2622no—
273C22×C4—5241no—
274C22×C4—5241no—
275C22×C4—5241no—
276C22×C4—5241no—
277C22×C4—5241no—
278C22×C4—5241no—
279C22×C4—1311no—
280C22×C4—1311no—
281C22×C4—1311no—
282C22×C4—1311no—
283C22×C4—1311no—
284C22×C4—1311no—
285C22×C4—1314no—
286C22×C4C22×D13111yescharacteristic
287C22×C4C22×D13111yescharacteristic
288C22×C4C22×D13111yescharacteristic
289C22×C4C22×D13111yescharacteristic
290C22×C4C22×D13111yescharacteristic
291C22×C4C22×D13111yescharacteristic
292C22×C4C22×D13111yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—2622no—
296C4⋊C4—2622no—
297C4⋊C4—2622no—
298C4⋊C4—2622no—
299C4⋊C4—5241no—
300C4⋊C4—5241no—
301C22⋊C4—2622no—
302C22⋊C4—2622no—
303C22⋊C4—2622no—
304C22⋊C4—2622no—
305C22⋊C4—2622no—
306C22⋊C4—2622no—
307C22⋊C4—5241no—
308C22⋊C4—5241no—
309C42—22no—
310C23—2621no—
311C23—2621no—
312C23—2621no—
313C23—2621no—
314C23—2621no—
315C23—2621no—
316C23—2621no—
317C23—2621no—
318C23—2621no—
319C23—2621no—
320C23—1312no—
321C23—1312no—
322C23—1312no—
323C23—1312no—
324C23C23×D13111yescharacteristic
325Q8—5244no—
326D4—2628no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—2621no—
346C2×C4—2621no—
347C2×C4—2621no—
348C2×C4—2621no—
349C2×C4—2621no—
350C2×C4—2621no—
351C2×C4—2621no—
352C2×C4—2621no—
353C2×C4—2621no—
354C2×C4—2621no—
355C2×C4—2621no—
356C2×C4—2621no—
357C2×C4—2621no—
358C2×C4—2621no—
359C2×C4—2621no—
360C2×C4—2621no—
361C2×C4—5244no—
362C2×C4—5244no—
363C2×C4—5244no—
364C2×C4—5244no—
365C2×C4—1312no—
366C2×C4—1312no—
367C2×C4D4×D13112yes—
368C2×C4Q8×D13112yes—
369C2×C4#1452351312yes—
370C22—2622no—
371C22—2622no—
372C22—2622no—
373C22—2622no—
374C22—2622no—
375C22—2622no—
376C22—2622no—
377C22—2622no—
378C22—2622no—
379C22—5241no—
380C22—5241no—
381C22—5241no—
382C22—5241no—
383C22—1314no—
384C22C2×D4×D13111yescharacteristic
385C22C2×Q8×D13111yescharacteristic
386C22#41×D13111yescharacteristic
387C22#41×D13111yescharacteristic
388C22C2×#1452351311yescharacteristic
389C22#1686229211yescharacteristic
390C22#1686229211yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—2622no—
399C4—2622no—
400C4—5241no—
401C4—5241no—
402C4—5241no—
403C4—5241no—
404C2—2624no—
405C2—5242no—
406C2C22⋊Q8×D13111yescharacteristic
407C2#2255385911yescharacteristic
408C2#2255387311yescharacteristic
409C2#2255398611yescharacteristic
410C2#2255398711yescharacteristic
411C2#2255401811yescharacteristic
412C2#2255402211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#387×C131C2
prime radicalC22×D524C22
hypercenter#387D131
hypocenterC131#2564
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#16862224C2semidirect
3. #22553254C2semidirect
4. #22553247C2semidirect
5. #22553233C2semidirect
6. #22553225C2semidirect
7. #22553196C2semidirect
10. #22553186C2semidirect
11. #22553183C2semidirect
12. #22553171C2semidirect
13. #22553169C2semidirect
14. #95×D131C2semidirect
15. #387×C131C2semidirect
60. C2×D524Q8semidirect
206. C131#2564semidirect
215. #387D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622242
C2×C4×D5242
#225532542
#225532472
#225532332
#225532252
#225531962
#225531902
#225531862
#225531832
#225531712
#225531692
#95×D1312
#387×C1312
#2564131