Finite Group Databasebeta

← #16848520

Subgroups — #16848520

Order 3968.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758513C211yesmaximal, characteristic
2#14520953C211yesmaximal, characteristic, complemented
3#14520953C211yesmaximal, characteristic, complemented
4#14520943C211yesmaximal, characteristic, complemented
5#14520943C211yesmaximal, characteristic, complemented
6#14520937C211yesmaximal, characteristic, complemented
7#14520937C211yesmaximal, characteristic, complemented
8#14520929C211yesmaximal, characteristic
9#14520922C211yesmaximal, characteristic, complemented
10#14520921C211yesmaximal, characteristic, complemented
11#14520916C211yesmaximal, characteristic, complemented
12#14520900C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic31C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic31C211yesmaximal, characteristic, complemented
15C31×#386C211yesmaximal, characteristic
16C22×D4×C31C2211yescharacteristic
17C23×C4×C31C2211yescharacteristic
18C23×Dic31C2211yescharacteristic
19C22×Dic62C2211yescharacteristic
20C2×C22⋊C4×C31C2211yescharacteristic
21C2×C22⋊C4×C31C2211yescharacteristic
22C2×C22⋊C4×C31C2211yescharacteristic
23C2×C22⋊Dic31C2211yescharacteristic
24C2×C22⋊Dic31C2211yescharacteristic
25C2×C22⋊Dic31C2211yescharacteristic
26C2×C22⋊Dic31C2211yescharacteristic
27C2×C22⋊Dic31C2211yescharacteristic
28C2×C22⋊Dic31C2211yescharacteristic
29C2×C22⋊Dic31C2211yescharacteristic
30#11758513C418yescomplemented
31C2×Dic31⋊C4C2211yescharacteristic
32C2×Dic31⋊C4C2211yescharacteristic
33C2×Dic31⋊C4C2211yescharacteristic
34C2×Dic31⋊C4C2211yescharacteristic
35C2×C4×Dic31C2211yescharacteristic
36C2×C4×Dic31C2211yescharacteristic
37C2×C4×Dic31C2211yescharacteristic
38C2×C4×Dic31C2211yescharacteristic
39C2×C4×Dic31C2211yescharacteristic
40C31×#95C2211yescharacteristic
41C31×#95C2211yescharacteristic
42C31⋊#95C2211yescharacteristic
43C31⋊#95C2211yescharacteristic
44C31⋊#95C2211yescharacteristic
45C31⋊#95C2211yescharacteristic
46C31⋊#95C2211yescharacteristic
47C31⋊#95C2211yescharacteristic
48C31⋊#95C2211yescharacteristic
49C31⋊#95C2211yescharacteristic
50C31⋊#95C2211yescharacteristic
51C31⋊#95C2211yescharacteristic
52C24×C31C2311yescharacteristic
53C24×C31C2311yescharacteristic
54C2×D4×C31—41no—
55C2×D4×C31—41no—
56C2×D4×C31C2×C414yes—
57C22×C4×C31—22no—
58C22×C4×C31—41no—
59C22×C4×C31—41no—
60C22×C4×C31C2311yescharacteristic
61C22×C4×C31C2311yescharacteristic
62C22×C4×C31C2311yescharacteristic
63C22×C4×C31C2311yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C22×Dic31—22no—
66C22×Dic31—41no—
67C22×Dic31—41no—
68C22×Dic31C2311yescharacteristic
69C22×Dic31C2311yescharacteristic
70C22×Dic31C2311yescharacteristic
71C22×Dic31C2311yescharacteristic
72C22×Dic31C2311yescharacteristic
73C22×Dic31C2311yescharacteristic
74C22×Dic31C2311yescharacteristic
75C22×Dic31C2311yescharacteristic
76C2×Dic62—41no—
77C2×Dic62—41no—
78C2×Dic62C2×C414yescomplemented
79C22⋊C4×C31—22no—
80C22⋊C4×C31—41no—
81C22⋊C4×C31—41no—
82C22⋊Dic31—22no—
83C22⋊Dic31—41no—
84C22⋊Dic31—41no—
85C22⋊Dic31C2×C414yescomplemented
86C22⋊Dic31C2×C414yescomplemented
87C22⋊Dic31C2×C414yescomplemented
88C22⋊Dic31C2×C414yescomplemented
89Dic31⋊C4—22no—
90Dic31⋊C4—22no—
91Dic31⋊C4—41no—
92Dic31⋊C4—41no—
93C4×Dic31—41no—
94C4×Dic31—41no—
95C4×Dic31—41no—
96C4×Dic31—41no—
97C4×Dic31C2×C414yescomplemented
98C23×C31—21no—
99C23×C31—21no—
100C23×C31—21no—
101C23×C31—21no—
102C23×C31—21no—
103C23×C31—21no—
104C23×C31—21no—
105C23×C31—21no—
106C23×C31—21no—
107C23×C31—21no—
108C23×C31—21no—
109C23×C31—21no—
110C23×C31C22×C412yes—
111C23×C31C22×C412yes—
112C23×C31C2411yescharacteristic
113D4×C31—44no—
114C2×C4×C31—21no—
115C2×C4×C31—21no—
116C2×C4×C31—21no—
117C2×C4×C31—21no—
118C2×C4×C31—21no—
119C2×C4×C31—21no—
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—44no—
128C2×C4×C31C22×C412yes—
129C2×C4×C31#4112yes—
130C2×Dic31—21no—
131C2×Dic31—21no—
132C2×Dic31—21no—
133C2×Dic31—21no—
134C2×Dic31—21no—
135C2×Dic31—21no—
136C2×Dic31—21no—
137C2×Dic31—21no—
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—44no—
150C2×Dic31C22×C412yes—
151C2×Dic31C22×C412yes—
152C2×Dic31C22×C412yes—
153C2×Dic31C22×C412yes—
154C2×Dic31#4112yes—
155Dic62—44no—
156#2337—311nomaximal, Sylow
157C22×C31—22no—
158C22×C31—22no—
159C22×C31—22no—
160C22×C31—22no—
161C22×C31—22no—
162C22×C31—22no—
163C22×C31—41no—
164C22×C31—41no—
165C22×C31—41no—
166C22×C31—41no—
167C22×C31—41no—
168C22×C31—41no—
169C22×C31—41no—
170C22×C31—41no—
171C22×C31C23×C411yescharacteristic
172C22×C31C2×#4111yescharacteristic
173C22×C31C2×#4111yescharacteristic
174C22×C31H(5,𝔽2)11yescharacteristic
175C22×C31H(5,𝔽2)11yescharacteristic
176C22×C31H(5,𝔽2)11yescharacteristic
177C22×C312−1+411yescharacteristic
178C4×C31—22no—
179C4×C31—22no—
180C4×C31—41no—
181C4×C31—41no—
182C4×C31—41no—
183Dic31—22no—
184Dic31—22no—
185Dic31—22no—
186Dic31—22no—
187Dic31—22no—
188Dic31—41no—
189Dic31—41no—
190Dic31—41no—
191C2×#124—311no—
192#391—311no—
193#391—311no—
194#388—311no—
195#388—311no—
196#388—311no—
197#388—311no—
198#386—311no—
199#386D3111yescharacteristic
200#385—311no—
201#385—311no—
202#383—311no—
203C4×C4⋊C4—311no—
204C4×C22⋊C4—311no—
205C4×C22⋊C4—311no—
206C2×C31—42no—
207C2×C31—42no—
208C2×C31C4×#4111yescharacteristic
209C2×C31#51811yescharacteristic
210C2×C31#52011yescharacteristic
211C2×C31#53811yescharacteristic
212C2×C31#53811yescharacteristic
213C2×C31#54211yescharacteristic
214C2×C31#54211yescharacteristic
215C22×Q8—311no—
216C22×D4C2×D3111yescharacteristic
217C23×C4—311no—
218C23×C4C2×D3111yescharacteristic
219#124—318no—
220C2×C4⋊C4—311no—
221C2×C4⋊C4—311no—
222C2×C4⋊C4—311no—
223C2×C4⋊C4—311no—
224C2×C22⋊C4—311no—
225C2×C22⋊C4—311no—
226C2×C22⋊C4—311no—
227C2×C22⋊C4—311no—
228C2×C22⋊C4—311no—
229C2×C22⋊C4—311no—
230C2×C22⋊C4—311no—
231C2×C22⋊C4C2×D3111yescharacteristic
232C2×C22⋊C4C2×D3111yescharacteristic
233C2×C22⋊C4C2×D3111yescharacteristic
234C2×C42—311no—
235C2×C42—311no—
236C2×C42—311no—
237C2×C42—311no—
238C2×C42—311no—
239#95—311no—
240#95—311no—
241#95—311no—
242#95—311no—
243#95—311no—
244#95—311no—
245#95—311no—
246#95—311no—
247#95—311no—
248#95—311no—
249#95C2×D3111yescharacteristic
250#95C2×D3111yescharacteristic
251C31#233711yescharacteristic, complemented, Sylow
252C24C22×D3111yescharacteristic
253C24C22×D3111yescharacteristic
254C2×Q8—314no—
255C2×Q8—1241no—
256C2×Q8—1241no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4C4×D3114yes—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—311no—
264C22×C4—311no—
265C22×C4—311no—
266C22×C4—311no—
267C22×C4—311no—
268C22×C4—311no—
269C22×C4—311no—
270C22×C4—311no—
271C22×C4—622no—
272C22×C4—1241no—
273C22×C4—1241no—
274C22×C4C22×D3111yescharacteristic
275C22×C4C22×D3111yescharacteristic
276C22×C4C22×D3111yescharacteristic
277C22×C4C22×D3111yescharacteristic
278C22×C4C22×D3111yescharacteristic
279C4⋊C4—622no—
280C4⋊C4—622no—
281C4⋊C4—1241no—
282C4⋊C4—1241no—
283C22⋊C4—22no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—314no—
287C22⋊C4—314no—
288C22⋊C4—314no—
289C22⋊C4—314no—
290C22⋊C4—622no—
291C22⋊C4—1241no—
292C22⋊C4—1241no—
293C42—314no—
294C42—1241no—
295C42—1241no—
296C42—1241no—
297C42—1241no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23C2×C4×D3112yes—
311C23C2×C4×D3112yes—
312C23C23×D3111yescharacteristic
313Q8—1244no—
314D4—44no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—44no—
329C2×C4—312no—
330C2×C4—312no—
331C2×C4—312no—
332C2×C4—312no—
333C2×C4—312no—
334C2×C4—621no—
335C2×C4—621no—
336C2×C4—621no—
337C2×C4—621no—
338C2×C4—621no—
339C2×C4—621no—
340C2×C4—621no—
341C2×C4—621no—
342C2×C4—621no—
343C2×C4—621no—
344C2×C4—621no—
345C2×C4—621no—
346C2×C4—621no—
347C2×C4—621no—
348C2×C4—621no—
349C2×C4—621no—
350C2×C4—621no—
351C2×C4—621no—
352C2×C4—621no—
353C2×C4—1244no—
354C2×C4C2×C4×D3112yes—
355C2×C4#9249712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C22×C4×D3111yescharacteristic
371C22C2×#9249711yescharacteristic
372C22#1175856111yescharacteristic
373C22#1175856111yescharacteristic
374C22#1175856111yescharacteristic
375C22#41×D3111yescharacteristic
376C22#1175856711yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—622no—
383C4—622no—
384C4—622no—
385C4—622no—
386C4—622no—
387C4—1241no—
388C4—1241no—
389C4—1241no—
390C2—42no—
391C2—42no—
392C2C4×#9249711yescharacteristic
393C2#1452152211yescharacteristic
394C2#1452152811yescharacteristic
395C2#1452163111yescharacteristic
396C2#1452163411yescharacteristic
397C2#1452166611yescharacteristic
398C2#1452166711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D31
commutatorC22×C31C23×C4
FittingC31×#386C2
socle, cosocleC23×C31C24
prime radicalC22×D4×C31C22
hypercenter#386D31
hypocenterC31#2337
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #14520953C2semidirect
3. #14520953C2semidirect
4. #14520943C2semidirect
5. #14520943C2semidirect
6. #14520937C2semidirect
7. #14520937C2semidirect
9. #14520922C2semidirect
10. #14520921C2semidirect
11. #14520916C2semidirect
12. #14520900C2semidirect
13. C22⋊C4×Dic31C2semidirect
14. C4×C22⋊Dic31C2semidirect
30. #11758513C4semidirect
78. C2×Dic62C2×C4semidirect
85. C22⋊Dic31C2×C4semidirect
86. C22⋊Dic31C2×C4semidirect
87. C22⋊Dic31C2×C4semidirect
88. C22⋊Dic31C2×C4semidirect
97. C4×Dic31C2×C4semidirect
251. C31#2337semidirect

Maximal subgroups

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

subgroupindex
C2×#117585132
#145209532
#145209432
#145209372
#145209292
#145209222
#145209212
#145209162
#145209002
C22⋊C4×Dic312
C4×C22⋊Dic312
C31×#3862
#233731