Finite Group Databasebeta

← #16848697

Subgroups — #16848697

Order 3968.

652 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×#11758512C211yesmaximal, characteristic
2C2×#11758454C211yesmaximal, characteristic, complemented
3#14520944C211yesmaximal, characteristic, complemented
4#14520943C211yesmaximal, characteristic, complemented
5#14520937C211yesmaximal, characteristic, complemented
6#14520933C211yesmaximal, characteristic, complemented
7#14520930C211yesmaximal, characteristic, complemented
8#14520929C211yesmaximal, characteristic
9#14520922C211yesmaximal, characteristic, complemented
10#14520922C211yesmaximal, characteristic, complemented
11C4×Dic31⋊C4C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic31C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic31C211yesmaximal, characteristic, complemented
14C2×C31⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C31C211yesmaximal, characteristic
16C22×D4×C31C2211yescharacteristic
17C23×C4×C31C2211yescharacteristic
18C23×C4×C31C2211yescharacteristic
19C23×Dic31C2211yescharacteristic
20C23×Dic31C2211yescharacteristic
21C4×D4×C31—24no—
22C2×C4⋊C4×C31C2211yescharacteristic
23C2×C22⋊C4×C31C2211yescharacteristic
24C2×C22⋊C4×C31C2211yescharacteristic
25C2×C42×C31C2211yescharacteristic
26C2×C22⋊Dic31C2211yescharacteristic
27C2×C22⋊Dic31C2211yescharacteristic
28C2×C22⋊Dic31C2211yescharacteristic
29C2×C22⋊Dic31C2211yescharacteristic
30C2×C22⋊Dic31C2211yescharacteristic
31C2×C22⋊Dic31C2211yescharacteristic
32#11758512C418yescomplemented
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×Dic31⋊C4C2211yescharacteristic
36C2×Dic31⋊C4C2211yescharacteristic
37C2×Dic31⋊C4C2211yescharacteristic
38C2×Dic31⋊C4C2211yescharacteristic
39C2×C4×Dic31C2211yescharacteristic
40C2×C4×Dic31C2211yescharacteristic
41C2×C4×Dic31C2211yescharacteristic
42C2×C4×Dic31C2211yescharacteristic
43#11758454—24no—
44C31⋊#95C2211yescharacteristic
45C31⋊#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C31⋊#95C2211yescharacteristic
53C31⋊#95C2211yescharacteristic
54C24×C31C2311yescharacteristic
55C24×C31C2311yescharacteristic
56C2×D4×C31—22no—
57C2×D4×C31—22no—
58C2×D4×C31C2×C414yes—
59C22×C4×C31—22no—
60C22×C4×C31—22no—
61C22×C4×C31—22no—
62C22×C4×C31—22no—
63C22×C4×C31—41no—
64C22×C4×C31—41no—
65C22×C4×C31C2311yescharacteristic
66C22×C4×C31C2311yescharacteristic
67C22×C4×C31C2311yescharacteristic
68C22×C4×C31C2311yescharacteristic
69C22×C4×C31C2311yescharacteristic
70C22×Dic31—22no—
71C22×Dic31—22no—
72C22×Dic31—22no—
73C22×Dic31—22no—
74C22×Dic31—41no—
75C22×Dic31C2×C414yescomplemented
76C22×Dic31C2311yescharacteristic
77C22×Dic31C2311yescharacteristic
78C22×Dic31C2311yescharacteristic
79C22×Dic31C2311yescharacteristic
80C22×Dic31C2311yescharacteristic
81C22×Dic31C2311yescharacteristic
82C22×Dic31C2311yescharacteristic
83C22×Dic31C2311yescharacteristic
84C4⋊C4×C31—22no—
85C22⋊C4×C31—22no—
86C22⋊C4×C31D414yes—
87C42×C31—22no—
88C22⋊Dic31—22no—
89C22⋊Dic31—22no—
90C22⋊Dic31—41no—
91C22⋊Dic31C2×C414yescomplemented
92C22⋊Dic31C2×C414yescomplemented
93C22⋊Dic31C2×C414yescomplemented
94Dic31⋊C4—22no—
95Dic31⋊C4—22no—
96Dic31⋊C4—22no—
97Dic31⋊C4—41no—
98Dic31⋊C4C2×C414yescomplemented
99Dic31⋊C4C2×C414yescomplemented
100C4×Dic31—22no—
101C4×Dic31—22no—
102C4×Dic31—22no—
103C4×Dic31—41no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31—21no—
111C23×C31—21no—
112C23×C31—21no—
113C23×C31—21no—
114C23×C31C22×C412yes—
115C23×C31C22×C412yes—
116C23×C31C2×D412yes—
117C23×C31#4112yes—
118C23×C31C2411yescharacteristic
119D4×C31—28no—
120C2×C4×C31—21no—
121C2×C4×C31—21no—
122C2×C4×C31—21no—
123C2×C4×C31—21no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31—21no—
127C2×C4×C31—21no—
128C2×C4×C31—21no—
129C2×C4×C31—21no—
130C2×C4×C31—21no—
131C2×C4×C31—44no—
132C2×C4×C31—44no—
133C2×C4×C31C22×C412yes—
134C2×C4×C31C2×D412yes—
135C2×C4×C31C2×D412yes—
136C2×C4×C31#4112yes—
137C2×Dic31—28no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31—21no—
141C2×Dic31—21no—
142C2×Dic31—21no—
143C2×Dic31—21no—
144C2×Dic31—21no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—21no—
149C2×Dic31—21no—
150C2×Dic31—21no—
151C2×Dic31—21no—
152C2×Dic31—21no—
153C2×Dic31—21no—
154C2×Dic31—21no—
155C2×Dic31—21no—
156C2×Dic31—21no—
157C2×Dic31—44no—
158C2×Dic31C22×C412yes—
159C2×Dic31C22×C412yes—
160C2×Dic31C22×C412yes—
161C2×Dic31C22×C412yes—
162C2×Dic31#4112yes—
163#2359—311nomaximal, Sylow
164C22×C31—22no—
165C22×C31—22no—
166C22×C31—22no—
167C22×C31—22no—
168C22×C31—22no—
169C22×C31—22no—
170C22×C31—22no—
171C22×C31—22no—
172C22×C31—22no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31—41no—
177C22×C31C4×D414yes—
178C22×C31C23×C411yescharacteristic
179C22×C31C22×D411yescharacteristic
180C22×C31C2×#4111yescharacteristic
181C22×C31C2×#4111yescharacteristic
182C22×C31C2×#4111yescharacteristic
183C22×C31H(5,𝔽2)11yescharacteristic
184C22×C312−1+411yescharacteristic
185C4×C31—22no—
186C4×C31—22no—
187C4×C31—22no—
188C4×C31—22no—
189C4×C31—41no—
190Dic31—22no—
191Dic31—22no—
192Dic31—22no—
193Dic31—22no—
194Dic31—22no—
195Dic31—41no—
196Dic31—41no—
197Dic31—41no—
198C2×#123—311no—
199C2×C4×D4D3111yescharacteristic
200C2×#117—311no—
201#389—311no—
202#388—311no—
203#388—311no—
204#387—311no—
205#387—311no—
206#386—311no—
207#385—311no—
208#385—311no—
209C4×C4⋊C4—311no—
210C4×C22⋊C4—311no—
211C4×C22⋊C4—311no—
212C2×#95—311no—
213C2×C31—24no—
214C2×C31—42no—
215C2×C31C2×C4×D411yescharacteristic
216C2×C31C4×#4111yescharacteristic
217C2×C31#52011yescharacteristic
218C2×C31#54611yescharacteristic
219C2×C31#54711yescharacteristic
220C2×C31#55111yescharacteristic
221C2×C31#55211yescharacteristic
222C22×D4C2×D3111yescharacteristic
223C23×C4—311no—
224C23×C4—311no—
225C23×C4C2×D3111yescharacteristic
226C23×C4C2×D3111yescharacteristic
227#123—318no—
228C4×D4—24no—
229#117—624no—
230C2×C4⋊C4—311no—
231C2×C4⋊C4—311no—
232C2×C4⋊C4—311no—
233C2×C4⋊C4—311no—
234C2×C4⋊C4—311no—
235C2×C4⋊C4—311no—
236C2×C4⋊C4C2×D3111yescharacteristic
237C2×C22⋊C4—311no—
238C2×C22⋊C4—311no—
239C2×C22⋊C4—311no—
240C2×C22⋊C4—311no—
241C2×C22⋊C4—311no—
242C2×C22⋊C4—311no—
243C2×C22⋊C4C2×D3111yescharacteristic
244C2×C22⋊C4C2×D3111yescharacteristic
245C2×C42—311no—
246C2×C42—311no—
247C2×C42—311no—
248C2×C42—311no—
249C2×C42C2×D3111yescharacteristic
250#95—311no—
251#95—311no—
252#95—311no—
253#95—311no—
254#95—311no—
255#95—311no—
256#95—311no—
257#95—311no—
258#95—311no—
259#95—311no—
260C31#235911yescharacteristic, complemented, Sylow
261C24C22×D3111yescharacteristic
262C24C22×D3111yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C4×D3114yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—314no—
273C22×C4—311no—
274C22×C4—311no—
275C22×C4—311no—
276C22×C4—311no—
277C22×C4—311no—
278C22×C4—311no—
279C22×C4—311no—
280C22×C4—311no—
281C22×C4—622no—
282C22×C4—622no—
283C22×C4—622no—
284C22×C4—622no—
285C22×C4—1241no—
286C22×C4C22×D3111yescharacteristic
287C22×C4C22×D3111yescharacteristic
288C22×C4C22×D3111yescharacteristic
289C22×C4C22×D3111yescharacteristic
290C22×C4C22×D3111yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—314no—
293C4⋊C4—314no—
294C4⋊C4—622no—
295C4⋊C4—622no—
296C4⋊C4—622no—
297C4⋊C4—1241no—
298C22⋊C4—22no—
299C22⋊C4—314no—
300C22⋊C4—314no—
301C22⋊C4—314no—
302C22⋊C4—622no—
303C22⋊C4—622no—
304C22⋊C4—1241no—
305C22⋊C4C22⋊D3114yes—
306C42—22no—
307C42—622no—
308C42—622no—
309C42—622no—
310C42—1241no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×D3112yes—
322C23C2×C4×D3112yes—
323C23#9249512yes—
324C23C2×C22⋊D3112yes—
325C23C23×D3111yescharacteristic
326D4—28no—
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—44no—
339C2×C4—44no—
340C2×C4—312no—
341C2×C4—312no—
342C2×C4—312no—
343C2×C4—312no—
344C2×C4—312no—
345C2×C4—628no—
346C2×C4—621no—
347C2×C4—621no—
348C2×C4—621no—
349C2×C4—621no—
350C2×C4—621no—
351C2×C4—621no—
352C2×C4—621no—
353C2×C4—621no—
354C2×C4—621no—
355C2×C4—621no—
356C2×C4—621no—
357C2×C4—621no—
358C2×C4—621no—
359C2×C4—621no—
360C2×C4—621no—
361C2×C4—621no—
362C2×C4—621no—
363C2×C4—621no—
364C2×C4—621no—
365C2×C4—1244no—
366C2×C4C2×C4×D3112yes—
367C2×C4#9249712yes—
368C2×C4C2×C22⋊D3112yes—
369C2×C4C2×C22⋊D3112yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C4×C22⋊D3114yes—
384C22C22×C4×D3111yescharacteristic
385C22C2×#9249511yescharacteristic
386C22C2×#9249711yescharacteristic
387C22#1175856111yescharacteristic
388C22#41×D3111yescharacteristic
389C22#1175856711yescharacteristic
390C22C22×C22⋊D3111yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—622no—
397C4—622no—
398C4—622no—
399C4—622no—
400C4—622no—
401C4—1241no—
402C4—1241no—
403C4—1241no—
404C2—24no—
405C2—42no—
406C2C2×C4×C22⋊D3111yescharacteristic
407C2C4×#9249711yescharacteristic
408C2#1452152811yescharacteristic
409C2#1452169311yescharacteristic
410C2#1452169811yescharacteristic
411C2#1452172811yescharacteristic
412C2#1452173911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC2×C4×D4×C31C2
socle, cosocleC23×C31C24
prime radicalC22×D4×C31C22
hypercenterC2×C4×D4D31
hypocenterC31#2359
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#11758454C2semidirect
3. #14520944C2semidirect
4. #14520943C2semidirect
5. #14520937C2semidirect
6. #14520933C2semidirect
7. #14520930C2semidirect
9. #14520922C2semidirect
10. #14520922C2semidirect
11. C4×Dic31⋊C4C2semidirect
12. C22⋊C4×Dic31C2semidirect
13. C4×C22⋊Dic31C2semidirect
14. C2×C31⋊#95C2semidirect
32. #11758512C4semidirect
75. C22×Dic31C2×C4semidirect
91. C22⋊Dic31C2×C4semidirect
92. C22⋊Dic31C2×C4semidirect
93. C22⋊Dic31C2×C4semidirect
98. Dic31⋊C4C2×C4semidirect
99. Dic31⋊C4C2×C4semidirect
260. C31#2359semidirect

Maximal subgroups

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

subgroupindex
C2×#117585122
C2×#117584542
#145209442
#145209432
#145209372
#145209332
#145209302
#145209292
#145209222
C4×Dic31⋊C42
C22⋊C4×Dic312
C4×C22⋊Dic312
C2×C31⋊#952
C2×C4×D4×C312
#235931