Finite Group Databasebeta

← #16857439

Subgroups — #16857439

Order 3968.

494 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758527C211yesmaximal, characteristic, complemented
2C2×#11758489C211yesmaximal, characteristic, complemented
3#14521717C211yesmaximal, characteristic, complemented
4#14521698C211yesmaximal, characteristic, complemented
5#14521420C211yesmaximal, characteristic, complemented
6#14521281C211yesmaximal, characteristic, complemented
7#14521276C211yesmaximal, characteristic, complemented
8#14521258C211yesmaximal, characteristic, complemented
9#14521242C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521208C211yesmaximal, characteristic, complemented
12#14521195C211yesmaximal, characteristic, complemented
13#14521140C211yesmaximal, characteristic, complemented
14(C22⋊D31)⋊C8C211yesmaximal, characteristic, complemented
15C31×#449C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×#92497C2211yescharacteristic
18C2×#92497C2211yescharacteristic
19C2×#92495C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C2×D8×C31C2211yescharacteristic
22C31×#121C2211yescharacteristic
23#11758527—24no—
24#11758523C2211yescharacteristic
25#11758521C2211yescharacteristic
26C2×#92482C2211yescharacteristic
27C2×#92481C2211yescharacteristic
28#11758517—21no—
29#11758517C2211yescharacteristic
30#11758516—21no—
31#11758516—21no—
32#11758512—21no—
33D4×Dic31C2211yescharacteristic
34C2×#92480C2211yescharacteristic
35C2×#92480C2211yescharacteristic
36C2×#92479C2211yescharacteristic
37#11758506—21no—
38C4×C22⋊D31C2211yescharacteristic
39#11758501C2211yescharacteristic
40C2×Dic31⋊C4—21no—
41C22×C31⋊C8C2211yescharacteristic
42#11758489—24no—
43C2×Dic124C2211yescharacteristic
44C2×#92470C2211yescharacteristic
45C2×C8×D31C2211yescharacteristic
46#11758472C2211yescharacteristic
47#11758464C2211yescharacteristic
48#11758461—21no—
49(C22⋊D31)⋊C4—21no—
50C31×#103C2211yescharacteristic
51C31×#102C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic
53#11758420C2211yescharacteristic
54#11758409C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic
56#11758404C2211yescharacteristic
57Dic31⋊C8C2211yescharacteristic
58C8×Dic31C2211yescharacteristic
59#11758398C2211yescharacteristic
60#11758396C2211yescharacteristic
61#41×C31—22no—
62#41×C31—41no—
63C2×Q8×C31C2311yescharacteristic
64C2×D4×C31—21no—
65C2×D4×C31—21no—
66C2×D4×C31C2311yescharacteristic
67C2×D4×C31C2311yescharacteristic
68C22×C4×C31—21no—
69C22×C4×C31C2311yescharacteristic
70C2×C22⋊D31—21no—
71C2×C22⋊D31—21no—
72C2×C22⋊D31—21no—
73C2×C22⋊D31—21no—
74C2×C22⋊D31—21no—
75C2×C22⋊D31D411yescharacteristic
76C22×Dic31—21no—
77C22×Dic31—21no—
78C22×Dic31—21no—
79#92497—22no—
80#92497—22no—
81#92497—41no—
82#92497—41no—
83#92495—22no—
84#92495—41no—
85C2×D124—21no—
86C2×D124C2311yescharacteristic
87C2×C4×D31—21no—
88C2×C4×D31C2311yescharacteristic
89C2×Dic62C2311yescharacteristic
90C2×Dic62C2311yescharacteristic
91SD16×C31—41no—
92D8×C31—22no—
93C2×C8×C31C2311yescharacteristic
94C2×C8×C31C2311yescharacteristic
95C4⋊C4×C31C2311yescharacteristic
96C22⋊C4×C31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31—21no—
99C22⋊Dic31D411yescharacteristic
100#92482—22no—
101#92481—22no—
102#92480—22no—
103#92480D414yescomplemented
104#92479—22no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4D411yescharacteristic
108C4⋊Dic31C2311yescharacteristic
109Dic31⋊C4—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4—21no—
114Dic31⋊C4D411yescharacteristic
115C4×Dic31—21no—
116C4×Dic31C2311yescharacteristic
117C2×C31⋊C8—22no—
118C2×C31⋊C8C2311yescharacteristic
119C2×C31⋊C8C2311yescharacteristic
120Dic124—22no—
121#92470—41no—
122C8×D31—22no—
123C23×C31—21no—
124C23×C31—21no—
125C23×C31C2×D411yescharacteristic
126C22×D31—21no—
127C22×D31C2×D411yescharacteristic
128Q8×C31—21no—
129Q8×C31—21no—
130D4×C31—21no—
131D4×C31—21no—
132D4×C31—21no—
133D4×C31—42no—
134D4×C31—81no—
135D4×C31C2×D412yes—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—21no—
140C2×C4×C31—42no—
141C2×C4×C31C2×D411yescharacteristic
142C2×C4×C31C2411yescharacteristic
143C22⋊D31—22no—
144C22⋊D31—42no—
145C22⋊D31—42no—
146C22⋊D31—42no—
147C22⋊D31—42no—
148C22⋊D31—81no—
149C2×Dic31—21no—
150C2×Dic31—21no—
151C2×Dic31—21no—
152C2×Dic31—42no—
153C2×Dic31—42no—
154C2×Dic31—42no—
155C2×Dic31C2×D411yescharacteristic
156C2×Dic31C2×D411yescharacteristic
157C2×Dic31C2×D411yescharacteristic
158D124—21no—
159D124—21no—
160D124—81no—
161C4×D31—21no—
162C4×D31—21no—
163C4×D31—42no—
164Dic62—21no—
165Dic62—21no—
166Dic62—21no—
167Dic62C2×D412yes—
168C8×C31—21no—
169C8×C31—21no—
170C31⋊C8—21no—
171C31⋊C8C2×D412yes—
172#3284—311nomaximal, Sylow
173C22×C31—22no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—21no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31—41no—
180C22×C31—41no—
181C22×C31—41no—
182C22×C31C22×D411yescharacteristic
183C2×D31—21no—
184C2×D31—21no—
185C2×D31—21no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C4×C31—21no—
190C4×C31—41no—
191C4×C31—41no—
192C4×C31C22×D411yescharacteristic
193C4×C31H(5,𝔽2)11yescharacteristic
194Dic31—22no—
195Dic31—21no—
196Dic31—21no—
197Dic31—41no—
198Dic31—41no—
199Dic31#13512yescomplemented
200C2×#135—311no—
201C2×#135—311no—
202#550—311no—
203#547—311no—
204#492—311no—
205#465—311no—
206#465—311no—
207#462—311no—
208#459—311no—
209#452—311no—
210#452—311no—
211#449—311no—
212#449D3111yescharacteristic, complemented
213C4×SD16—311no—
214C8×D4—311no—
215C2×C31—41no—
216C2×C31—42no—
217C2×C31—81no—
218C2×C31D4211yescharacteristic
219C2×C31C2×#13511yescharacteristic
220C2×C31#57711yescharacteristic
221D31—41no—
222D31—81no—
223C2×#41—311no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41C2×D3111yescharacteristic
227#135—624no—
228#135—624no—
229C2×Q16—311no—
230C2×Q16—311no—
231C2×SD16—311no—
232C2×SD16—311no—
233C2×SD16—311no—
234C2×SD16—311no—
235C2×SD16C2×D3111yescharacteristic
236C2×D8—311no—
237C2×D8C2×D3111yescharacteristic
238C22×C8—311no—
239C22×C8—311no—
240C4⋊Q8—311no—
241#127—311no—
242#127—621no—
243#123—621no—
244#123—621no—
245C22⋊Q8—311no—
246C22⋊Q8—311no—
247#121—311no—
248#121—621no—
249#121—621no—
250#121—621no—
251#121C2×D3111yescharacteristic
252C4×Q8—311no—
253C4×D4—311no—
254C4×D4—311no—
255C4×D4—621no—
256C2×C4⋊C4—621no—
257#106—311no—
258C4⋊C8—311no—
259#103—311no—
260#103—311no—
261#103C2×D3111yescharacteristic
262#102—311no—
263#102—311no—
264#102C2×D3111yescharacteristic
265C22⋊C8—311no—
266C22⋊C8C2×D3111yescharacteristic
267C4×C8—311no—
268C31#328411yescharacteristic, complemented, Sylow
269#41—22no—
270#41—41no—
271#41—622no—
272#41—622no—
273#41—622no—
274#41—1241no—
275#41—1241no—
276#41—1241no—
277C2×Q8—311no—
278C2×Q8—311no—
279C2×Q8C22×D3111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—311no—
283C2×D4—311no—
284C2×D4—621no—
285C2×D4—621no—
286C2×D4—621no—
287C2×D4—621no—
288C2×D4—621no—
289C2×D4—621no—
290C2×D4C22×D3111yescharacteristic
291C2×D4C22×D3111yescharacteristic
292C22×C4—21no—
293C22×C4—311no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4—621no—
298C22×C4C22×D3111yescharacteristic
299Q16—622no—
300Q16—622no—
301SD16—41no—
302SD16—314no—
303SD16—622no—
304SD16—622no—
305SD16—1241no—
306D8—22no—
307D8—622no—
308C2×C8—311no—
309C2×C8—311no—
310C2×C8—622no—
311C2×C8—622no—
312C2×C8C22×D3111yescharacteristic
313C2×C8C22×D3111yescharacteristic
314C4⋊C4—311no—
315C4⋊C4—311no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4—621no—
321C4⋊C4C22×D3111yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—311no—
324C22⋊C4—311no—
325C22⋊C4—621no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C42—311no—
330C42—621no—
331C23—21no—
332C23—21no—
333C23—311no—
334C23—621no—
335C23D4×D3111yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—312no—
339Q8—621no—
340Q8—621no—
341Q8—621no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—81no—
347D4—622no—
348D4—621no—
349D4—621no—
350D4—1242no—
351D4—1242no—
352D4—1242no—
353D4—1242no—
354D4—2481no—
355D4—2481no—
356D4D4×D3112yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—311no—
363C2×C4—311no—
364C2×C4—311no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—621no—
370C2×C4—1242no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4D4×D3111yescharacteristic
375C2×C4C23×D3111yescharacteristic
376C8—21no—
377C8—21no—
378C8—312no—
379C8—621no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—621no—
390C22—621no—
391C22—621no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22C2×D4×D3111yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—312no—
400C4—621no—
401C4—621no—
402C4—622no—
403C4—1241no—
404C4—1241no—
405C4C2×D4×D3111yescharacteristic
406C4#1175856111yescharacteristic
407C2—41no—
408C2—42no—
409C2—81no—
410C2—1241no—
411C2—2481no—
412C2#1452168111yescharacteristic
413C2#135×D3111yescharacteristic
414C2#1452185911yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#449C2
socleC22×C31C22×D4
hypercenter#449D31
hypocenterC31#3284
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. C2×#11758489C2semidirect
3. #14521717C2semidirect
4. #14521698C2semidirect
5. #14521420C2semidirect
6. #14521281C2semidirect
7. #14521276C2semidirect
8. #14521258C2semidirect
9. #14521242C2semidirect
10. #14521213C2semidirect
11. #14521208C2semidirect
12. #14521195C2semidirect
13. #14521140C2semidirect
14. (C22⋊D31)⋊C8C2semidirect
15. C31×#449C2semidirect
103. #92480D4semidirect
199. Dic31#135semidirect
212. #449D31semidirect
268. C31#3284semidirect

Maximal subgroups

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

subgroupindex
C2×#117585272
C2×#117584892
#145217172
#145216982
#145214202
#145212812
#145212762
#145212582
#145212422
#145212132
#145212082
#145211952
#145211402
(C22⋊D31)⋊C82
C31×#4492
#328431