Finite Group Databasebeta

← #16859301

Subgroups — #16859301

Order 3968.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521781C213yesmaximal, complemented
2#14521775C211yesmaximal, characteristic, complemented
3#14521770C213yesmaximal, complemented
4#14521751C213yesmaximal, complemented
5#14521745C213yesmaximal, complemented
6#14521718C211yesmaximal, characteristic, complemented
7#14521710C211yesmaximal, characteristic, complemented
8#14521594C213yesmaximal, complemented
9#14521585C213yesmaximal, complemented
10#14521578C213yesmaximal, complemented
11#41×Dic31C211yesmaximal, characteristic, complemented
12C4×#92499C211yesmaximal, characteristic, complemented
13C4×#92497C213yesmaximal, complemented
14C2×Q8×Dic31C211yesmaximal, characteristic, complemented
15C31×#548C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic
18C2×#92499C2211yescharacteristic
19C2×#92497C2213yes—
20C31×#124C2213yes—
21C31×C22⋊Q8C2213yes—
22C31×#121C2213yes—
23C4×Q8×C31C2211yescharacteristic
24C4×D4×C31C2213yes—
25#11758524C2213yes—
26#11758523C2213yes—
27Q8×Dic31C2211yescharacteristic
28Q8×Dic31C2211yescharacteristic
29Q8×Dic31C2213yes—
30Q8×Dic31C2211yescharacteristic
31Q8×Dic31C2213yes—
32#11758517C2213yes—
33#11758516C2216yes—
34#11758515C2213yes—
35D4×Dic31C2213yes—
36D4×Dic31C2213yes—
37D4×Dic31C2213yes—
38#11758507C2213yes—
39#11758507C2213yes—
40C4×C22⋊D31C2213yes—
41C4×C22⋊D31C2213yes—
42#11758503C2213yes—
43C2×C4⋊Dic31C2213yes—
44C2×C4×Dic31C2213yes—
45C2×C4×Dic31C2213yes—
46#11758475C2216yes—
47#11758470C2213yes—
48#11758469C2213yes—
49#11758469C2213yes—
50#11758467C2213yes—
51#11758465C2213yes—
52#11758464C2213yes—
53#11758463C2216yes—
54#11758462C2216yes—
55#11758460C2216yes—
56(C22⋊D31)⋊C4C2213yes—
57(C22⋊D31)⋊C4C2213yes—
58(C22⋊D31)⋊C4C2216yes—
59#11758456C2216yes—
60#11758454C2211yescharacteristic
61#11758454C2213yes—
62C4×D124C2213yes—
63C42×D31C2213yes—
64#11758447C2213yes—
65C4×Dic62C2213yes—
66#41×C31—24no—
67C2×Q8×C31—24no—
68C2×Q8×C31C2313yes—
69C2×Q8×C31C2311yescharacteristic
70C2×D4×C31C2313yes—
71C2×D4×C31C2313yes—
72C22×C4×C31C2313yes—
73C22×C4×C31C2313yes—
74C2×C22⋊D31C2313yes—
75C2×C22⋊D31C2313yes—
76C2×C22⋊D31C2316yes—
77C22×Dic31C2313yes—
78C22×Dic31C2313yes—
79C22×Dic31C2313yes—
80C22×Dic31C2311yescharacteristic
81#92499—24no—
82#92497—212no—
83C2×D124C2313yes—
84C2×C4×D31C2313yes—
85C2×C4×D31C2313yes—
86C2×C4×D31C2313yes—
87C2×Dic62C2313yes—
88C4⋊C4×C31C2313yes—
89C4⋊C4×C31C2313yes—
90C22⋊C4×C31C2311yescharacteristic
91C22⋊C4×C31C2313yes—
92C22⋊C4×C31C2316yes—
93C42×C31C2313yes—
94C22⋊Dic31C2313yes—
95C22⋊Dic31C2311yescharacteristic
96C22⋊Dic31C2316yes—
97(C2×D31)⋊C4C2316yes—
98(C2×D31)⋊C4C2316yes—
99(C2×D31)⋊C4C2313yes—
100(C2×D31)⋊C4C2313yes—
101C4⋊Dic31C2313yes—
102C4⋊Dic31C2313yes—
103C4⋊Dic31C2313yes—
104C4⋊Dic31C2313yes—
105C4⋊Dic31C2316yes—
106Dic31⋊C4C2311yescharacteristic
107Dic31⋊C4C2313yes—
108Dic31⋊C4C2316yes—
109Dic31⋊C4C2313yes—
110Dic31⋊C4C2311yescharacteristic
111C4×Dic31C2313yes—
112C4×Dic31C2313yes—
113C4×Dic31C2311yescharacteristic
114C4×Dic31C2313yes—
115C4×Dic31C2313yes—
116C4×Dic31C2313yes—
117C4×Dic31C2311yescharacteristic
118C4×Dic31C2313yes—
119C4×Dic31C2313yes—
120C23×C31C2413yes—
121C23×C31C2411yescharacteristic
122C22×D31C2413yes—
123Q8×C31—26no—
124Q8×C31#4114yescomplemented
125D4×C31—26no—
126D4×C31—26no—
127C2×C4×C31—26no—
128C2×C4×C31—26no—
129C2×C4×C31C2411yescharacteristic
130C2×C4×C31C2413yes—
131C2×C4×C31C2411yescharacteristic
132C2×C4×C31C2413yes—
133C2×C4×C31C2413yes—
134C22⋊D31—26no—
135C22⋊D31—26no—
136C22⋊D31—46no—
137C2×Dic31—26no—
138C2×Dic31—43no—
139C2×Dic31—43no—
140C2×Dic31#4114yescomplemented
141C2×Dic31C2411yescharacteristic
142C2×Dic31C2411yescharacteristic
143C2×Dic31C2413yes—
144C2×Dic31C2413yes—
145C2×Dic31C2413yes—
146C2×Dic31C2411yescharacteristic
147C2×Dic31C2411yescharacteristic
148D124—26no—
149C4×D31—26no—
150C4×D31—26no—
151C4×D31—43no—
152Dic62—26no—
153#3480—311nomaximal, Sylow
154C22×C31—21no—
155C22×C31—21no—
156C22×C31—23no—
157C22×C31—23no—
158C22×C31—23no—
159C22×C31C2×#4112yes—
160C22×C31C2511yescharacteristic
161C2×D31—23no—
162C2×D31—23no—
163C2×D31—23no—
164C4×C31—23no—
165C4×C31—21no—
166C4×C31—23no—
167C4×C31—21no—
168C4×C31C2×#4116yes—
169Dic31—23no—
170Dic31—21no—
171Dic31—23no—
172Dic31—21no—
173Dic31—23no—
174Dic31C2×#4112yes—
175Dic31C2×#4112yes—
176#559—313no—
177#557—311no—
178#556—313no—
179#553—313no—
180#553—313no—
181#550—311no—
182#548—311no—
183#548D3111yescharacteristic, complemented
184#532—313no—
185#529—313no—
186#529—313no—
187C4×#41—311no—
188C4×#41—311no—
189C4×#41—313no—
190C2×C4×Q8—311no—
191C2×C31—22no—
192C2×C31—43no—
193C2×C31C22×#4111yescharacteristic
194C2×C31C22×#4111yescharacteristic
195C2×C31#58511yescharacteristic
196D31—43no—
197C2×#41—311no—
198C2×#41—313no—
199C2×#41C2×D3111yescharacteristic
200C22×Q8C2×D3111yescharacteristic
201C4⋊Q8—313no—
202#127—313no—
203#126—316no—
204#126—316no—
205#125—313no—
206#125—313no—
207#124—313no—
208#124—316no—
209#124C2×D3113yes—
210#123—316no—
211#123—316no—
212C22⋊Q8—313no—
213C22⋊Q8C2×D3113yes—
214#121—313no—
215#121—316no—
216#121—313no—
217#121—313no—
218#121C2×D3113yes—
219C4×Q8—311no—
220C4×Q8—313no—
221C4×Q8—311no—
222C4×Q8—313no—
223C4×Q8—311no—
224C4×Q8—313no—
225C4×Q8—313no—
226C4×Q8C2×D3111yescharacteristic
227C4×D4—313no—
228C4×D4—313no—
229C4×D4—313no—
230C4×D4—313no—
231C4×D4—313no—
232C4×D4—313no—
233C4×D4—313no—
234C4×D4—313no—
235C4×D4—316no—
236C4×D4—313no—
237C4×D4C2×D3113yes—
238#117—313no—
239#117—313no—
240#117—313no—
241#117—311no—
242#117—313no—
243C2×C4⋊C4—313no—
244C2×C42—313no—
245C2×C42—313no—
246C2×C42—313no—
247C31#348011yescharacteristic, complemented, Sylow
248#41—24no—
249#41—6212no—
250#41—624no—
251C2×Q8—24no—
252C2×Q8—313no—
253C2×Q8C22×D3113yes—
254C2×Q8C22×D3111yescharacteristic
255C2×D4—313no—
256C2×D4—313no—
257C2×D4—316no—
258C2×D4—313no—
259C2×D4C22×D3113yes—
260C2×D4C22×D3113yes—
261C22×C4—311no—
262C22×C4—313no—
263C22×C4—313no—
264C22×C4—313no—
265C22×C4—313no—
266C22×C4—313no—
267C22×C4—313no—
268C22×C4C22×D3113yes—
269C22×C4C22×D3113yes—
270C4⋊C4—311no—
271C4⋊C4—311no—
272C4⋊C4—313no—
273C4⋊C4—313no—
274C4⋊C4—316no—
275C4⋊C4—313no—
276C4⋊C4—313no—
277C4⋊C4—316no—
278C4⋊C4—313no—
279C4⋊C4—313no—
280C4⋊C4C22×D3113yes—
281C4⋊C4C22×D3113yes—
282C22⋊C4—311no—
283C22⋊C4—313no—
284C22⋊C4—316no—
285C22⋊C4—313no—
286C22⋊C4—316no—
287C22⋊C4—313no—
288C22⋊C4—316no—
289C22⋊C4C22×D3111yescharacteristic
290C22⋊C4C22×D3113yes—
291C22⋊C4C22×D3116yes—
292C42—311no—
293C42—311no—
294C42—313no—
295C42—313no—
296C42—313no—
297C42—313no—
298C42—313no—
299C42—313no—
300C42—313no—
301C42C22×D3113yes—
302C23—313no—
303C23C23×D3113yes—
304C23C23×D3111yescharacteristic
305Q8—26no—
306Q8—626no—
307Q8#9249714yescomplemented
308D4—26no—
309D4—26no—
310D4—626no—
311D4—626no—
312D4—626no—
313D4—1246no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—314no—
317C2×C4—311no—
318C2×C4—311no—
319C2×C4—311no—
320C2×C4—311no—
321C2×C4—313no—
322C2×C4—313no—
323C2×C4—313no—
324C2×C4—626no—
325C2×C4—626no—
326C2×C4—626no—
327C2×C4—1243no—
328C2×C4—1243no—
329C2×C4—1243no—
330C2×C4C23×D3111yescharacteristic
331C2×C4C23×D3113yes—
332C2×C4C23×D3111yescharacteristic
333C2×C4C23×D3113yes—
334C2×C4C23×D3113yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—623no—
341C22—623no—
342C22—623no—
343C22#41×D3112yes—
344C22C24×D3111yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—312no—
350C4—312no—
351C4—623no—
352C4—621no—
353C4—623no—
354C4—621no—
355C4—623no—
356C4C2×#9249716yes—
357C2—22no—
358C2—43no—
359C2—1243no—
360C2C2×#41×D3111yescharacteristic
361C2C22×#9249711yescharacteristic
362C2#1452188611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D31
commutator, socle, cosocleC22×C31C25
FittingC31×#548C2
hypercenter#548D31
hypocenterC31#3480
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #14521781C2semidirect
2. #14521775C2semidirect
3. #14521770C2semidirect
4. #14521751C2semidirect
5. #14521745C2semidirect
6. #14521718C2semidirect
7. #14521710C2semidirect
8. #14521594C2semidirect
9. #14521585C2semidirect
10. #14521578C2semidirect
11. #41×Dic31C2semidirect
12. C4×#92499C2semidirect
13. C4×#92497C2semidirect
14. C2×Q8×Dic31C2semidirect
15. C31×#548C2semidirect
124. Q8×C31#41semidirect
140. C2×Dic31#41semidirect
183. #548D31semidirect
247. C31#3480semidirect
307. Q8#92497semidirect

Maximal subgroups

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

subgroupindex
#145217812
#145217752
#145217702
#145217512
#145217452
#145217182
#145217102
#145215942
#145215852
#145215782
#41×Dic312
C4×#924992
C4×#924972
C2×Q8×Dic312
C31×#5482
#348031