Finite Group Databasebeta

← #16857438

Subgroups — #16857438

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×#11758481C211yesmaximal, characteristic, complemented
2C2×#11758493C211yesmaximal, characteristic, complemented
3#14521717C211yesmaximal, characteristic, complemented
4#14521701C211yesmaximal, characteristic, complemented
5#14521420C211yesmaximal, characteristic, complemented
6#14521279C211yesmaximal, characteristic, complemented
7#14521258C211yesmaximal, characteristic, complemented
8#14521243C211yesmaximal, characteristic, complemented
9#14521212C211yesmaximal, characteristic, complemented
10#14521209C211yesmaximal, characteristic, complemented
11#14521197C211yesmaximal, characteristic, complemented
12#14521192C211yesmaximal, characteristic, complemented
13#14521136C211yesmaximal, characteristic, complemented
14C8×C22⋊D31C211yesmaximal, characteristic, complemented
15C31×#465C211yesmaximal, characteristic, complemented
16C2×#92499C2211yescharacteristic
17C2×#92497C2211yescharacteristic
18C2×#92495C2211yescharacteristic
19C2×#92495C2211yescharacteristic
20C2×SD16×C31C2211yescharacteristic
21C22×C8×C31C2211yescharacteristic
22C31×C22⋊Q8C2211yescharacteristic
23C31×#121C2211yescharacteristic
24#11758521C2211yescharacteristic
25C2×#92482C2211yescharacteristic
26C2×#92481C2211yescharacteristic
27#11758517—21no—
28#11758517C2211yescharacteristic
29#11758516—21no—
30C2×#92480C2211yescharacteristic
31C2×#92479C2211yescharacteristic
32C4×C22⋊D31C2211yescharacteristic
33#11758493—24no—
34C2×Dic124C2211yescharacteristic
35#11758481—24no—
36C2×D248C2211yescharacteristic
37C2×#92470C2211yescharacteristic
38C2×#92470C2211yescharacteristic
39C2×C8×D31C2211yescharacteristic
40#11758474C2211yescharacteristic
41#11758472C2211yescharacteristic
42#11758471—21no—
43#11758470C2211yescharacteristic
44C4⋊C4×D31—21no—
45#11758464C2211yescharacteristic
46#11758461—21no—
47#11758461—21no—
48#11758460—21no—
49(C22⋊D31)⋊C4—21no—
50C31×#106C2211yescharacteristic
51C31×#103C2211yescharacteristic
52C31×#102C2211yescharacteristic
53C22⋊(C31⋊C8)C2211yescharacteristic
54(C2×D31)⋊C8C2211yescharacteristic
55Dic31⋊C8C2211yescharacteristic
56C8×Dic31C2211yescharacteristic
57#11758398C2211yescharacteristic
58#11758398C2211yescharacteristic
59#11758397C2211yescharacteristic
60#11758397C2211yescharacteristic
61C2×Q8×C31C2311yescharacteristic
62C2×D4×C31—21no—
63C2×D4×C31C2311yescharacteristic
64C22×C4×C31C2311yescharacteristic
65C2×C22⋊D31—21no—
66C2×C22⋊D31—21no—
67C2×C22⋊D31—21no—
68C2×C22⋊D31—21no—
69C2×C22⋊D31—21no—
70C2×C22⋊D31D411yescharacteristic
71C22×Dic31—21no—
72#92499—22no—
73#92499—41no—
74#92497—22no—
75#92497—41no—
76#92495—22no—
77#92495—22no—
78#92495—41no—
79#92495—41no—
80C2×D124—21no—
81C2×D124—21no—
82C2×D124C2311yescharacteristic
83C2×D124C2311yescharacteristic
84C2×C4×D31—21no—
85C2×C4×D31—21no—
86C2×C4×D31—21no—
87C2×C4×D31—21no—
88C2×C4×D31C2311yescharacteristic
89C2×Dic62C2311yescharacteristic
90C2×Dic62C2311yescharacteristic
91SD16×C31—22no—
92C2×C8×C31—22no—
93C2×C8×C31C2311yescharacteristic
94C2×C8×C31C2311yescharacteristic
95C4⋊C4×C31—21no—
96C4⋊C4×C31C2311yescharacteristic
97C4⋊C4×C31C2311yescharacteristic
98C22⋊C4×C31—21no—
99C22⋊C4×C31—21no—
100C22⋊Dic31D411yescharacteristic
101#92482—22no—
102#92481—41no—
103#92480—41no—
104#92479—22no—
105(C2×D31)⋊C4—21no—
106(C2×D31)⋊C4—21no—
107(C2×D31)⋊C4—21no—
108(C2×D31)⋊C4D411yescharacteristic
109C4⋊Dic31—21no—
110Dic31⋊C4—21no—
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
115C4×Dic31C2311yescharacteristic
116C2×C31⋊C8C2311yescharacteristic
117C2×C31⋊C8C2311yescharacteristic
118Dic124—22no—
119D248—22no—
120#92470—22no—
121#92470D414yescomplemented
122C8×D31—22no—
123C23×C31—21no—
124C23×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31—21no—
127C22×D31C2×D411yescharacteristic
128Q8×C31—21no—
129Q8×C31—21no—
130D4×C31—21no—
131D4×C31—21no—
132D4×C31—81no—
133C2×C4×C31—21no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31C2×D411yescharacteristic
139C2×C4×C31C2411yescharacteristic
140C22⋊D31—22no—
141C22⋊D31—42no—
142C22⋊D31—42no—
143C22⋊D31—42no—
144C22⋊D31—42no—
145C22⋊D31—81no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31—42no—
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151C2×Dic31C2×D411yescharacteristic
152D124—21no—
153D124—21no—
154D124—21no—
155D124—42no—
156D124—81no—
157D124C2×D412yes—
158C4×D31—21no—
159C4×D31—21no—
160C4×D31—42no—
161C4×D31—42no—
162C4×D31—42no—
163C4×D31—42no—
164Dic62—21no—
165Dic62—21no—
166Dic62—21no—
167Dic62C2×D412yes—
168C8×C31—21no—
169C8×C31C2×D412yes—
170C31⋊C8—21no—
171C31⋊C8—21no—
172#3284—311nomaximal, Sylow
173C22×C31—21no—
174C22×C31—21no—
175C22×C31—21no—
176C22×C31—41no—
177C22×C31—41no—
178C22×C31—41no—
179C22×C31C22×D411yescharacteristic
180C2×D31—21no—
181C2×D31—22no—
182C2×D31—21no—
183C2×D31—21no—
184C2×D31—41no—
185C2×D31—41no—
186C2×D31—41no—
187C2×D31—41no—
188C2×D31—41no—
189C4×C31—21no—
190C4×C31—41no—
191C4×C31—41no—
192C4×C31—41no—
193C4×C31C22×D411yescharacteristic
194C4×C31H(5,𝔽2)11yescharacteristic
195Dic31—22no—
196Dic31—21no—
197Dic31—21no—
198Dic31—41no—
199Dic31#13512yescomplemented
200C2×#135—311no—
201C2×#135—311no—
202#550—311no—
203#547—311no—
204#492—311no—
205#465—311no—
206#465D3111yescharacteristic, complemented
207#462—311no—
208#459—311no—
209#452—311no—
210#452—311no—
211#449—311no—
212#449—311no—
213C4×SD16—311no—
214C8×D4—311no—
215C2×C31—41no—
216C2×C31—81no—
217C2×C31D4211yescharacteristic
218C2×C31C2×#13511yescharacteristic
219C2×C31#57711yescharacteristic
220D31—41no—
221D31—42no—
222D31—81no—
223C2×#41—311no—
224C2×#41—311no—
225C2×#41—311no—
226C2×#41—311no—
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×D8—311no—
238C22×C8—311no—
239C22×C8C2×D3111yescharacteristic
240C4⋊Q8—311no—
241#127—311no—
242#127—621no—
243#123—621no—
244#123—621no—
245C22⋊Q8—311no—
246C22⋊Q8C2×D3111yescharacteristic
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#106C2×D3111yescharacteristic
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⋊C8—311no—
267C4×C8—311no—
268C31#328411yescharacteristic, complemented, Sylow
269#41—622no—
270#41—622no—
271#41—622no—
272#41—622no—
273#41—1241no—
274#41—1241no—
275#41—1241no—
276#41—1241no—
277C2×Q8—311no—
278C2×Q8—311no—
279C2×Q8C22×D3111yescharacteristic
280C2×D4—21no—
281C2×D4—311no—
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×D4—621no—
291C2×D4C22×D3111yescharacteristic
292C22×C4—311no—
293C22×C4—621no—
294C22×C4—621no—
295C22×C4—621no—
296C22×C4—621no—
297C22×C4—621no—
298C22×C4C22×D3111yescharacteristic
299Q16—622no—
300Q16—622no—
301SD16—22no—
302SD16—314no—
303SD16—622no—
304SD16—1241no—
305SD16—1241no—
306D8—622no—
307D8—622no—
308C2×C8—22no—
309C2×C8—311no—
310C2×C8—311no—
311C2×C8—622no—
312C2×C8C22×D3111yescharacteristic
313C2×C8C22×D3111yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—311no—
316C4⋊C4—621no—
317C4⋊C4—621no—
318C4⋊C4—621no—
319C4⋊C4—621no—
320C4⋊C4C22×D3111yescharacteristic
321C4⋊C4C22×D3111yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—311no—
325C22⋊C4—311no—
326C22⋊C4—621no—
327C22⋊C4—621no—
328C22⋊C4—621no—
329C42—311no—
330C42—621no—
331C23—21no—
332C23—311no—
333C23—621no—
334C23—621no—
335C23D4×D3111yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—312no—
339Q8—621no—
340Q8—621no—
341Q8—621no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—312no—
346D4—622no—
347D4—621no—
348D4—621no—
349D4—621no—
350D4—1242no—
351D4—1242no—
352D4—1242no—
353D4—1242no—
354D4—1242no—
355D4—2481no—
356D4—2481no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—311no—
363C2×C4—311no—
364C2×C4—311no—
365C2×C4—621no—
366C2×C4—621no—
367C2×C4—621no—
368C2×C4—621no—
369C2×C4—1242no—
370C2×C4—1242no—
371C2×C4—1242no—
372C2×C4—1242no—
373C2×C4—1242no—
374C2×C4D4×D3111yescharacteristic
375C2×C4C23×D3111yescharacteristic
376C8—21no—
377C8—621no—
378C8—621no—
379C8D4×D3112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—621no—
387C22—622no—
388C22—621no—
389C22—621no—
390C22—1241no—
391C22—1241no—
392C22—1241no—
393C22—1241no—
394C22—1241no—
395C22C2×D4×D3111yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—312no—
401C4—621no—
402C4—622no—
403C4—621no—
404C4—1241no—
405C4C2×D4×D3111yescharacteristic
406C4#1175856611yescharacteristic
407C2—41no—
408C2—81no—
409C2—1241no—
410C2—1242no—
411C2—2481no—
412C2#1452168311yescharacteristic
413C2#135×D3111yescharacteristic
414C2#1452185511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#465C2
socleC22×C31C22×D4
hypercenter#465D31
hypocenterC31#3284
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#11758481C2semidirect
2. C2×#11758493C2semidirect
3. #14521717C2semidirect
4. #14521701C2semidirect
5. #14521420C2semidirect
6. #14521279C2semidirect
7. #14521258C2semidirect
8. #14521243C2semidirect
9. #14521212C2semidirect
10. #14521209C2semidirect
11. #14521197C2semidirect
12. #14521192C2semidirect
13. #14521136C2semidirect
14. C8×C22⋊D31C2semidirect
15. C31×#465C2semidirect
121. #92470D4semidirect
199. Dic31#135semidirect
206. #465D31semidirect
268. C31#3284semidirect

Maximal subgroups

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

subgroupindex
C2×#117584812
C2×#117584932
#145217172
#145217012
#145214202
#145212792
#145212582
#145212432
#145212122
#145212092
#145211972
#145211922
#145211362
C8×C22⋊D312
C31×#4652
#328431