Finite Group Databasebeta

← #16859327

Subgroups — #16859327

Order 3968.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521775C211yesmaximal, characteristic, complemented
2#14521768C213yesmaximal, complemented
3#14521739C213yesmaximal, complemented
4#14521733C213yesmaximal, complemented
5Q8×C22⋊D31C211yesmaximal, characteristic, complemented
6#14521706C211yesmaximal, characteristic, complemented
7#14521699C213yesmaximal, complemented
8#14521601C213yesmaximal, complemented
9#14521580C213yesmaximal, complemented
10#14521578C213yesmaximal, complemented
11#41×Dic31C211yesmaximal, characteristic, complemented
12C4×#92497C213yesmaximal, complemented
13C2×Q8×Dic31C211yesmaximal, characteristic, complemented
14C4×Q8×D31C211yesmaximal, characteristic, complemented
15C31×#548C211yesmaximal, characteristic, complemented
16C2×#41×C31C2211yescharacteristic
17C22×Q8×C31C2211yescharacteristic
18C2×Q8×D31C2211yescharacteristic
19C2×#92497C2213yes—
20C31×#124C2213yes—
21C31×C22⋊Q8C2213yes—
22C31×#121C2213yes—
23C4×Q8×C31C2211yescharacteristic
24C4×D4×C31C2213yes—
25#11758523C2213yes—
26Q8×Dic31C2213yes—
27Q8×Dic31C2211yescharacteristic
28Q8×Dic31C2211yescharacteristic
29Q8×Dic31C2213yes—
30Q8×Dic31C2211yescharacteristic
31#11758521C2213yes—
32#11758515C2213yes—
33#11758513C2213yes—
34#11758512C2216yes—
35D4×Dic31C2213yes—
36D4×Dic31C2213yes—
37D4×Dic31C2213yes—
38C4×C22⋊D31C2213yes—
39C4×C22⋊D31C2213yes—
40#11758503C2213yes—
41C2×C4⋊Dic31C2213yes—
42#11758501C2213yes—
43#11758501C2213yes—
44C2×C4×Dic31C2213yes—
45C2×C4×Dic31C2213yes—
46#11758469C2213yes—
47C4⋊C4×D31C2213yes—
48#11758467C2213yes—
49#11758466C2216yes—
50#11758465C2213yes—
51#11758464C2213yes—
52#11758464C2213yes—
53#11758460C2216yes—
54(C22⋊D31)⋊C4C2211yescharacteristic
55(C22⋊D31)⋊C4C2213yes—
56#11758456C2216yes—
57#11758456C2216yes—
58#11758455C2216yes—
59#11758454C2216yes—
60#11758454C2213yes—
61#11758454C2213yes—
62C42×D31C2213yes—
63#11758447C2213yes—
64C4×Dic62C2213yes—
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⋊D31C2311yescharacteristic
76C22×Dic31C2313yes—
77C22×Dic31C2313yes—
78C22×Dic31C2313yes—
79C22×Dic31C2311yescharacteristic
80Q8×D31—24no—
81#92497—212no—
82C2×C4×D31C2311yescharacteristic
83C2×C4×D31C2313yes—
84C2×C4×D31C2313yes—
85C2×Dic62C2313yes—
86C2×Dic62C2313yes—
87C4⋊C4×C31C2313yes—
88C4⋊C4×C31C2313yes—
89C22⋊C4×C31C2311yescharacteristic
90C22⋊C4×C31C2316yes—
91C22⋊C4×C31C2313yes—
92C42×C31C2313yes—
93C22⋊Dic31C2316yes—
94C22⋊Dic31C2313yes—
95C22⋊Dic31C2313yes—
96C22⋊Dic31C2316yes—
97(C2×D31)⋊C4C2313yes—
98(C2×D31)⋊C4C2311yescharacteristic
99C4⋊Dic31C2313yes—
100C4⋊Dic31C2313yes—
101C4⋊Dic31C2313yes—
102C4⋊Dic31C2313yes—
103C4⋊Dic31C2316yes—
104Dic31⋊C4C2311yescharacteristic
105Dic31⋊C4C2313yes—
106Dic31⋊C4C2316yes—
107Dic31⋊C4C2316yes—
108Dic31⋊C4C2313yes—
109Dic31⋊C4C2316yes—
110Dic31⋊C4C2313yes—
111C4×Dic31C2313yes—
112C4×Dic31C2313yes—
113C4×Dic31C2313yes—
114C4×Dic31C2313yes—
115C4×Dic31C2313yes—
116C4×Dic31C2313yes—
117C4×Dic31C2313yes—
118C4×Dic31C2313yes—
119C4×Dic31C2311yescharacteristic
120C23×C31C2413yes—
121C23×C31C2411yescharacteristic
122C22×D31C2411yescharacteristic
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×C31C2411yescharacteristic
131C2×C4×C31C2413yes—
132C2×C4×C31C2413yes—
133C2×C4×C31C2413yes—
134C22⋊D31—26no—
135C22⋊D31#4114yescomplemented
136C2×Dic31—26no—
137C2×Dic31—26no—
138C2×Dic31—22no—
139C2×Dic31—26no—
140C2×Dic31C2411yescharacteristic
141C2×Dic31C2413yes—
142C2×Dic31C2413yes—
143C2×Dic31C2413yes—
144C2×Dic31C2413yes—
145C2×Dic31C2411yescharacteristic
146C2×Dic31C2411yescharacteristic
147C4×D31—26no—
148C4×D31—26no—
149C4×D31—22no—
150Dic62—26no—
151Dic62—26no—
152#3482—311nomaximal, Sylow
153C22×C31—21no—
154C22×C31—21no—
155C22×C31—23no—
156C22×C31—23no—
157C22×C31—23no—
158C22×C31C2×#4112yes—
159C22×C31C2511yescharacteristic
160C2×D31—21no—
161C2×D31—21no—
162C2×D31C2×#4112yes—
163C4×C31—23no—
164C4×C31—21no—
165C4×C31—23no—
166C4×C31—21no—
167C4×C31C2×#4116yes—
168Dic31—23no—
169Dic31—23no—
170Dic31—23no—
171Dic31—21no—
172Dic31—23no—
173Dic31C2×#4112yes—
174Dic312−1+412yes—
175#557—311no—
176#556—313no—
177#552—313no—
178#552—313no—
179D4×Q8—311no—
180#548—311no—
181#548D3111yescharacteristic, complemented
182#547—313no—
183#533—313no—
184#529—313no—
185#529—313no—
186C4×#41—311no—
187C4×#41—313no—
188C2×C4×Q8—311no—
189C2×C4×Q8—311no—
190C2×C31—22no—
191C2×C31—43no—
192C2×C31C22×#4111yescharacteristic
193C2×C31C22×#4111yescharacteristic
194C2×C31C2×2−1+411yescharacteristic
195D31—22no—
196C2×#41—313no—
197C2×#41C2×D3111yescharacteristic
198C22×Q8—311no—
199C22×Q8C2×D3111yescharacteristic
200C4⋊Q8—313no—
201C4⋊Q8—313no—
202#126—316no—
203#126—316no—
204#125—313no—
205#125—316no—
206#125—313no—
207#124—313no—
208#124C2×D3113yes—
209#123—316no—
210#123—316no—
211C22⋊Q8—313no—
212C22⋊Q8—313no—
213C22⋊Q8—313no—
214C22⋊Q8—316no—
215C22⋊Q8C2×D3113yes—
216#121—313no—
217#121C2×D3113yes—
218C4×Q8—311no—
219C4×Q8—313no—
220C4×Q8—311no—
221C4×Q8—313no—
222C4×Q8—311no—
223C4×Q8—313no—
224C4×Q8—313no—
225C4×Q8—313no—
226C4×Q8—313no—
227C4×Q8C2×D3111yescharacteristic
228C4×D4—313no—
229C4×D4—313no—
230C4×D4—313no—
231C4×D4—313no—
232C4×D4—313no—
233C4×D4—311no—
234C4×D4—313no—
235C4×D4C2×D3113yes—
236#117—313no—
237#117—313no—
238#117—313no—
239#117—313no—
240#117—316no—
241C2×C4⋊C4—313no—
242C2×C4⋊C4—313no—
243C2×C42—313no—
244C2×C42—313no—
245C2×C42—313no—
246C31#348211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—6212no—
249C2×Q8—24no—
250C2×Q8—313no—
251C2×Q8—313no—
252C2×Q8—624no—
253C2×Q8C22×D3113yes—
254C2×Q8C22×D3111yescharacteristic
255C2×D4—311no—
256C2×D4—313no—
257C2×D4C22×D3113yes—
258C2×D4C22×D3113yes—
259C22×C4—311no—
260C22×C4—313no—
261C22×C4—313no—
262C22×C4—313no—
263C22×C4—311no—
264C22×C4—313no—
265C22×C4—313no—
266C22×C4C22×D3113yes—
267C22×C4C22×D3113yes—
268C4⋊C4—316no—
269C4⋊C4—311no—
270C4⋊C4—313no—
271C4⋊C4—313no—
272C4⋊C4—313no—
273C4⋊C4—316no—
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—313no—
283C22⋊C4—316no—
284C22⋊C4—313no—
285C22⋊C4—316no—
286C22⋊C4—311no—
287C22⋊C4—313no—
288C22⋊C4C22×D3111yescharacteristic
289C22⋊C4C22×D3113yes—
290C22⋊C4C22×D3116yes—
291C42—311no—
292C42—313no—
293C42—313no—
294C42—313no—
295C42—313no—
296C42—313no—
297C42—313no—
298C42—313no—
299C42—313no—
300C42C22×D3113yes—
301C23—311no—
302C23C23×D3113yes—
303C23C23×D3111yescharacteristic
304Q8—26no—
305Q8—626no—
306Q8—626no—
307Q8#9249714yescomplemented
308D4—26no—
309D4—26no—
310D4—314no—
311D4—626no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—311no—
315C2×C4—311no—
316C2×C4—311no—
317C2×C4—313no—
318C2×C4—313no—
319C2×C4—313no—
320C2×C4—313no—
321C2×C4—622no—
322C2×C4—626no—
323C2×C4—622no—
324C2×C4—626no—
325C2×C4—626no—
326C2×C4—626no—
327C2×C4—626no—
328C2×C4C23×D3111yescharacteristic
329C2×C4C23×D3111yescharacteristic
330C2×C4C23×D3113yes—
331C2×C4C23×D3113yes—
332C2×C4C23×D3113yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—312no—
339C22—621no—
340C22—621no—
341C22#41×D3112yes—
342C22C24×D3111yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—312no—
348C4—312no—
349C4—623no—
350C4—623no—
351C4—623no—
352C4—621no—
353C4—623no—
354C4C2×#9249716yes—
355C2—22no—
356C2—43no—
357C2—622no—
358C2C2×#41×D3111yescharacteristic
359C2C22×#9249711yescharacteristic
360C22−1+4×D3111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D31
commutator, socle, cosocleC22×C31C25
FittingC31×#548C2
hypercenter#548D31
hypocenterC31#3482
LCSG ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #14521775C2semidirect
2. #14521768C2semidirect
3. #14521739C2semidirect
4. #14521733C2semidirect
5. Q8×C22⋊D31C2semidirect
6. #14521706C2semidirect
7. #14521699C2semidirect
8. #14521601C2semidirect
9. #14521580C2semidirect
10. #14521578C2semidirect
11. #41×Dic31C2semidirect
12. C4×#92497C2semidirect
13. C2×Q8×Dic31C2semidirect
14. C4×Q8×D31C2semidirect
15. C31×#548C2semidirect
124. Q8×C31#41semidirect
135. C22⋊D31#41semidirect
181. #548D31semidirect
246. C31#3482semidirect
307. Q8#92497semidirect

Maximal subgroups

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

subgroupindex
#145217752
#145217682
#145217392
#145217332
Q8×C22⋊D312
#145217062
#145216992
#145216012
#145215802
#145215782
#41×Dic312
C4×#924972
C2×Q8×Dic312
C4×Q8×D312
C31×#5482
#348231