Finite Group Databasebeta

← #16857450

Subgroups — #16857450

Order 3968.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11758527C211yesmaximal, characteristic, complemented
2C2×Q16×D31C211yesmaximal, characteristic, complemented
3#14521709C211yesmaximal, characteristic, complemented
4#14521706C211yesmaximal, characteristic, complemented
5#14521430C211yesmaximal, characteristic, complemented
6#14521288C211yesmaximal, characteristic, complemented
7#14521284C211yesmaximal, characteristic, complemented
8#14521273C211yesmaximal, characteristic, complemented
9#14521270C211yesmaximal, characteristic, complemented
10#14521213C211yesmaximal, characteristic, complemented
11#14521189C211yesmaximal, characteristic, complemented
12#14521186C211yesmaximal, characteristic, complemented
13#14521141C211yesmaximal, characteristic, complemented
14(C22⋊D31)⋊C8C211yesmaximal, characteristic, complemented
15C31×#452C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C2×Q8×D31C2211yescharacteristic
18C2×Q8×D31C2211yescharacteristic
19C2×#92495C2211yescharacteristic
20C2×Q16×C31C2211yescharacteristic
21C2×SD16×C31C2211yescharacteristic
22C31×C22⋊Q8C2211yescharacteristic
23#11758527—24no—
24#11758524C2211yescharacteristic
25Q8×Dic31C2211yescharacteristic
26C2×#92482C2211yescharacteristic
27C2×#92482C2211yescharacteristic
28C2×#92481C2211yescharacteristic
29#11758515—21no—
30#11758515C2211yescharacteristic
31#11758513—21no—
32#11758513C2211yescharacteristic
33C2×#92480C2211yescharacteristic
34C2×#92479C2211yescharacteristic
35C4×C22⋊D31—21no—
36C4×C22⋊D31C2211yescharacteristic
37#11758501—21no—
38#11758501C2211yescharacteristic
39C22×C31⋊C8C2211yescharacteristic
40Q16×D31—24no—
41C2×Dic124C2211yescharacteristic
42C2×#92470C2211yescharacteristic
43C2×C8×D31C2211yescharacteristic
44#11758473—21no—
45#11758472C2211yescharacteristic
46#11758470—21no—
47#11758464C2211yescharacteristic
48#11758462—21no—
49#11758461—21no—
50C31×#103C2211yescharacteristic
51C31×#102C2211yescharacteristic
52C31×C22⋊C8C2211yescharacteristic
53#11758423C2211yescharacteristic
54#11758409C2211yescharacteristic
55(C2×D31)⋊C8C2211yescharacteristic
56#11758404C2211yescharacteristic
57Dic31⋊C8C2211yescharacteristic
58C8×Dic31C2211yescharacteristic
59#11758398C2211yescharacteristic
60#11758395C2211yescharacteristic
61#41×C31—22no—
62#41×C31—41no—
63C2×Q8×C31C2311yescharacteristic
64C2×Q8×C31C2311yescharacteristic
65C2×D4×C31—21no—
66C2×D4×C31C2311yescharacteristic
67C22×C4×C31—21no—
68C22×C4×C31C2311yescharacteristic
69C2×C22⋊D31—21no—
70C2×C22⋊D31—21no—
71C2×C22⋊D31D411yescharacteristic
72Q8×D31—22no—
73Q8×D31—22no—
74Q8×D31—41no—
75Q8×D31—41no—
76#92495—22no—
77#92495—41no—
78C2×D124—21no—
79C2×D124C2311yescharacteristic
80C2×C4×D31—21no—
81C2×C4×D31—21no—
82C2×C4×D31—21no—
83C2×C4×D31C2311yescharacteristic
84C2×Dic62—21no—
85C2×Dic62—21no—
86C2×Dic62C2311yescharacteristic
87C2×Dic62C2311yescharacteristic
88Q16×C31—22no—
89SD16×C31—41no—
90C2×C8×C31C2311yescharacteristic
91C2×C8×C31C2311yescharacteristic
92C4⋊C4×C31—21no—
93C4⋊C4×C31C2311yescharacteristic
94C22⋊C4×C31—21no—
95C22⋊Dic31—21no—
96C22⋊Dic31—21no—
97C22⋊Dic31—21no—
98C22⋊Dic31—21no—
99C22⋊Dic31D411yescharacteristic
100#92482—22no—
101#92482D414yescomplemented
102#92481—22no—
103#92480—22no—
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—
110C4⋊Dic31C2311yescharacteristic
111Dic31⋊C4—21no—
112Dic31⋊C4—21no—
113Dic31⋊C4D411yescharacteristic
114C4×Dic31—21no—
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×C31C2×D411yescharacteristic
125C22×D31—21no—
126C22×D31C2×D411yescharacteristic
127Q8×C31—21no—
128Q8×C31—21no—
129Q8×C31—21no—
130Q8×C31C2×D412yes—
131D4×C31—21no—
132D4×C31—21no—
133D4×C31—42no—
134C2×C4×C31—21no—
135C2×C4×C31—21no—
136C2×C4×C31—21no—
137C2×C4×C31—21no—
138C2×C4×C31—21no—
139C2×C4×C31—42no—
140C2×C4×C31C2×D411yescharacteristic
141C2×C4×C31C2411yescharacteristic
142C22⋊D31—22no—
143C22⋊D31—42no—
144C22⋊D31—42no—
145C2×Dic31—21no—
146C2×Dic31—21no—
147C2×Dic31—21no—
148C2×Dic31C2×D411yescharacteristic
149C2×Dic31C2×D411yescharacteristic
150C2×Dic31C2×D411yescharacteristic
151D124—21no—
152D124—21no—
153D124—42no—
154C4×D31—21no—
155C4×D31—21no—
156C4×D31—42no—
157C4×D31—42no—
158C4×D31—42no—
159Dic62—21no—
160Dic62—21no—
161Dic62—21no—
162Dic62—42no—
163Dic62—42no—
164Dic62C2×D412yes—
165C8×C31—21no—
166C8×C31—21no—
167C31⋊C8—21no—
168C31⋊C8C2×D412yes—
169#3285—311nomaximal, Sylow
170C22×C31—21no—
171C22×C31—21no—
172C22×C31—21no—
173C22×C31—41no—
174C22×C31—41no—
175C22×C31—41no—
176C22×C31C22×D411yescharacteristic
177C2×D31—21no—
178C2×D31—21no—
179C2×D31—41no—
180C2×D31—41no—
181C2×D31—41no—
182C2×D31#13512yescomplemented
183C4×C31—22no—
184C4×C31—21no—
185C4×C31—41no—
186C4×C31—41no—
187C4×C31C22×D411yescharacteristic
188C4×C31H(5,𝔽2)11yescharacteristic
189Dic31—22no—
190Dic31—21no—
191Dic31—21no—
192Dic31—21no—
193Dic31—41no—
194Dic31—41no—
195C2×#135—311no—
196C22×Q16—311no—
197#548—311no—
198#548—311no—
199#495—311no—
200#467—311no—
201#466—311no—
202#464—311no—
203#464—311no—
204#452—311no—
205#452D3111yescharacteristic, complemented
206#448—311no—
207#448—311no—
208C4×Q16—311no—
209C8×D4—311no—
210C2×C31—41no—
211C2×C31—81no—
212C2×C31D4211yescharacteristic
213C2×C31C2×#13511yescharacteristic
214C2×C31#57811yescharacteristic
215D31—22no—
216D31—81no—
217C2×#41—311no—
218C2×#41C2×D3111yescharacteristic
219C22×Q8—311no—
220C22×Q8—311no—
221#135—624no—
222C2×Q16—311no—
223C2×Q16—311no—
224C2×Q16—311no—
225C2×Q16—624no—
226C2×Q16C2×D3111yescharacteristic
227C2×SD16—311no—
228C2×SD16—311no—
229C2×SD16—311no—
230C2×SD16C2×D3111yescharacteristic
231C2×D8—311no—
232C22×C8—311no—
233C22×C8—311no—
234#124—311no—
235#124—311no—
236#124—621no—
237#124—621no—
238C22⋊Q8—311no—
239C22⋊Q8—621no—
240C22⋊Q8—621no—
241C22⋊Q8C2×D3111yescharacteristic
242#121—311no—
243#121—311no—
244#121—621no—
245#121—621no—
246C4×Q8—311no—
247C4×Q8—311no—
248C4×D4—311no—
249C4×D4—621no—
250C4×D4—621no—
251#107—311no—
252C4⋊C8—311no—
253#103—311no—
254#103—311no—
255#103—311no—
256#103C2×D3111yescharacteristic
257#102—311no—
258#102C2×D3111yescharacteristic
259C22⋊C8—311no—
260C22⋊C8C2×D3111yescharacteristic
261C4×C8—311no—
262C31#328511yescharacteristic, complemented, Sylow
263#41—22no—
264#41—41no—
265#41—622no—
266#41—1241no—
267C2×Q8—311no—
268C2×Q8—311no—
269C2×Q8—621no—
270C2×Q8—621no—
271C2×Q8—622no—
272C2×Q8—622no—
273C2×Q8—1241no—
274C2×Q8—1241no—
275C2×Q8C22×D3111yescharacteristic
276C2×Q8C22×D3111yescharacteristic
277C2×D4—21no—
278C2×D4—311no—
279C2×D4—311no—
280C2×D4—621no—
281C2×D4—621no—
282C2×D4—621no—
283C2×D4C22×D3111yescharacteristic
284C22×C4—21no—
285C22×C4—311no—
286C22×C4—621no—
287C22×C4—621no—
288C22×C4—621no—
289C22×C4C22×D3111yescharacteristic
290Q16—22no—
291Q16—314no—
292Q16—622no—
293Q16—622no—
294SD16—41no—
295SD16—622no—
296SD16—622no—
297SD16—1241no—
298D8—622no—
299C2×C8—311no—
300C2×C8—311no—
301C2×C8—622no—
302C2×C8—622no—
303C2×C8C22×D3111yescharacteristic
304C2×C8C22×D3111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—311no—
307C4⋊C4—311no—
308C4⋊C4—621no—
309C4⋊C4—621no—
310C4⋊C4—621no—
311C4⋊C4C22×D3111yescharacteristic
312C22⋊C4—21no—
313C22⋊C4—311no—
314C22⋊C4—311no—
315C22⋊C4—621no—
316C22⋊C4—621no—
317C22⋊C4—621no—
318C22⋊C4—621no—
319C22⋊C4—621no—
320C22⋊C4—621no—
321C22⋊C4—621no—
322C42—311no—
323C42—621no—
324C42—621no—
325C23—21no—
326C23—311no—
327C23—621no—
328C23D4×D3111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—312no—
333Q8—621no—
334Q8—621no—
335Q8—621no—
336Q8—1242no—
337Q8—1242no—
338Q8D4×D3112yes—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—622no—
343D4—621no—
344D4—621no—
345D4—1242no—
346D4—1242no—
347D4—1242no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—311no—
355C2×C4—311no—
356C2×C4—311no—
357C2×C4—621no—
358C2×C4—621no—
359C2×C4—621no—
360C2×C4—621no—
361C2×C4—621no—
362C2×C4—1242no—
363C2×C4—1242no—
364C2×C4—1242no—
365C2×C4D4×D3111yescharacteristic
366C2×C4C23×D3111yescharacteristic
367C8—21no—
368C8—21no—
369C8—312no—
370C8—621no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—312no—
378C22—621no—
379C22—621no—
380C22—1241no—
381C22—1241no—
382C22—1241no—
383C22C2×D4×D3111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—621no—
389C4—621no—
390C4—622no—
391C4—621no—
392C4—1241no—
393C4—1241no—
394C4C2×D4×D3111yescharacteristic
395C4#1175856111yescharacteristic
396C2—41no—
397C2—81no—
398C2—622no—
399C2—2481no—
400C2#1452168111yescharacteristic
401C2#135×D3111yescharacteristic
402C2#1452186511yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#452C2
socleC22×C31C22×D4
hypercenter#452D31
hypocenterC31#3285
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#11758527C2semidirect
2. C2×Q16×D31C2semidirect
3. #14521709C2semidirect
4. #14521706C2semidirect
5. #14521430C2semidirect
6. #14521288C2semidirect
7. #14521284C2semidirect
8. #14521273C2semidirect
9. #14521270C2semidirect
10. #14521213C2semidirect
11. #14521189C2semidirect
12. #14521186C2semidirect
13. #14521141C2semidirect
14. (C22⋊D31)⋊C8C2semidirect
15. C31×#452C2semidirect
101. #92482D4semidirect
182. C2×D31#135semidirect
205. #452D31semidirect
262. C31#3285semidirect

Maximal subgroups

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

subgroupindex
C2×#117585272
C2×Q16×D312
#145217092
#145217062
#145214302
#145212882
#145212842
#145212732
#145212702
#145212132
#145211892
#145211862
#145211412
(C22⋊D31)⋊C82
C31×#4522
#328531