Finite Group Databasebeta

← #16833579

Subgroups — #16833579

Order 3712.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14271889C211yesmaximal, characteristic, complemented
2#14271888C213yesmaximal, complemented
3#14271884C213yesmaximal, complemented
4#14271855C213yesmaximal, complemented
5#14271853C213yesmaximal, complemented
6Q8×C22⋊D29C211yesmaximal, characteristic, complemented
7Q8×C22⋊D29C211yesmaximal, characteristic, complemented
8#14271774C213yesmaximal, complemented
9#14271773C213yesmaximal, complemented
10#14271715C213yesmaximal, complemented
11#14271693C213yesmaximal
12#14271641C211yesmaximal, characteristic, complemented
13#14271639C211yesmaximal, characteristic, complemented
14C2×Q8×Dic29C211yesmaximal, characteristic, complemented
15D4×Q8×C29C211yesmaximal, characteristic, complemented
16C22×Q8×C29C2211yescharacteristic
17C22×Q8×C29C2211yescharacteristic
18C2×Q8×D29C2211yescharacteristic
19C29×C4⋊Q8C2213yes—
20C29×C22⋊Q8C2213yes—
21C29×C22⋊Q8C2213yes—
22C4×Q8×C29C2211yescharacteristic
23C4×D4×C29C2213yes—
24#11745308C2213yes—
25#11745308C2213yes—
26Q8×Dic29C2213yes—
27Q8×Dic29C2213yes—
28Q8×Dic29C2211yescharacteristic
29Q8×Dic29C2211yescharacteristic
30Q8×Dic29C2213yes—
31Q8×Dic29C2213yes—
32Q8×Dic29C2211yescharacteristic
33#11745306C2213yes—
34#11745306C2213yes—
35#11745301C2211yescharacteristic
36#11745297C2213yes—
37C4×C22⋊D29C2213yes—
38C4×C22⋊D29C2213yes—
39#11745288C2213yes—
40#11745288C2213yes—
41C2×C4⋊Dic29C2213yes—
42#11745286C2213yes—
43#11745286C2213yes—
44C2×C4×Dic29C2213yes—
45#11745260C2213yes—
46#11745260C2216yes—
47#11745254C2213yes—
48#11745252C2216yes—
49#11745252C2213yes—
50#11745251C2216yes—
51#11745251C2216yes—
52#11745251C2213yes—
53#11745250C2213yes—
54#11745249C2213yes—
55#11745249C2216yes—
56#11745248C2213yes—
57#11745247C2213yes—
58#11745241C2213yes—
59#11745241C2216yes—
60#11745240C2213yes—
61#11745239C2216yes—
62#11745234C2213yes—
63#11745232C2213yes—
64C4×Dic58C2213yes—
65C4×Dic58C2213yes—
66C2×Q8×C29—24no—
67C2×Q8×C29—24no—
68C2×Q8×C29C2313yes—
69C2×Q8×C29C2313yes—
70C2×Q8×C29C2311yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C22×C4×C29C2313yes—
73C22×C4×C29C2313yes—
74C2×C22⋊D29C2311yescharacteristic
75C2×C22⋊D29C2311yescharacteristic
76C22×Dic29C2311yescharacteristic
77C22×Dic29C2313yes—
78Q8×D29—24no—
79C2×C4×D29C2313yes—
80C2×Dic58C2313yes—
81C2×Dic58C2313yes—
82C4⋊C4×C29C2313yes—
83C4⋊C4×C29C2313yes—
84C4⋊C4×C29C2316yes—
85C22⋊C4×C29C2313yes—
86C22⋊C4×C29C2313yes—
87C42×C29C2313yes—
88C22⋊Dic29C2313yes—
89C22⋊Dic29C2313yes—
90C22⋊Dic29C2313yes—
91C22⋊Dic29C2311yescharacteristic
92#90781C2313yes—
93#90781C2313yes—
94#90781C2311yescharacteristic
95#90781C2313yes—
96C4⋊Dic29C2313yes—
97C4⋊Dic29C2313yes—
98C4⋊Dic29C2313yes—
99C4⋊Dic29C2316yes—
100C4⋊Dic29C2313yes—
101C4⋊Dic29C2313yes—
102C4⋊Dic29C2313yes—
103#90779C2316yes—
104#90779C2316yes—
105#90779C2313yes—
106#90779C2313yes—
107#90779C2313yes—
108#90779C2313yes—
109#90779C2311yescharacteristic
110#90779C2316yes—
111#90779C2313yes—
112C4×Dic29C2313yes—
113C4×Dic29C2313yes—
114C4×Dic29C2313yes—
115C4×Dic29C2313yes—
116C4×Dic29C2313yes—
117C4×Dic29C2316yes—
118C4×Dic29C2316yes—
119C23×C29C2411yescharacteristic
120C23×C29C2411yescharacteristic
121C22×D29C2411yescharacteristic
122Q8×C29—26no—
123Q8×C29—26no—
124Q8×C29#4114yes—
125D4×C29—22no—
126C2×C4×C29—26no—
127C2×C4×C29—26no—
128C2×C4×C29C2413yes—
129C2×C4×C29C2413yes—
130C2×C4×C29C2413yes—
131C2×C4×C29C2411yescharacteristic
132C2×C4×C29C2413yes—
133C22⋊D29—22no—
134C22⋊D29—22no—
135C2×Dic29—26no—
136C2×Dic29—22no—
137C2×Dic29C2411yescharacteristic
138C2×Dic29C2413yes—
139C2×Dic29C2413yes—
140C2×Dic29C2411yescharacteristic
141C2×Dic29C2413yes—
142C2×Dic29C2413yes—
143C2×Dic29C2411yescharacteristic
144C4×D29—26no—
145Dic58—26no—
146Dic58—26no—
147#3523—291nomaximal, Sylow
148C22×C29—21no—
149C22×C29—21no—
150C22×C29—21no—
151C22×C29—21no—
152C22×C29—21no—
153C22×C292−1+412yes—
154C22×C29C2511yescharacteristic
155C2×D29—21no—
156C2×D29—21no—
157C2×D29—21no—
158C4×C29—23no—
159C4×C29—23no—
160C4×C29—23no—
161C4×C29—21no—
162C4×C29C2×#4116yes—
163Dic29—23no—
164Dic29—23no—
165Dic29—21no—
166Dic29—23no—
167Dic29—23no—
168Dic29—21no—
169Dic292−1+412yes—
170#557—293no—
171#557—291no—
172#556—293no—
173#552—293no—
174#552—293no—
175D4×Q8—291no—
176D4×Q8—291no—
177D4×Q8D2911yescharacteristic, complemented
178#541—293no—
179#541—293no—
180#533—293no—
181#529—293no—
182#519—291no—
183#519—291no—
184C2×C4×Q8—291no—
185C2×C29—22no—
186C2×C29—41no—
187C2×C29C22×#4111yescharacteristic
188C2×C29C2×2−1+411yescharacteristic
189C2×C29C2×2−1+411yescharacteristic
190D29—41no—
191C22×Q8—291no—
192C22×Q8C2×D2911yescharacteristic
193C22×Q8C2×D2911yescharacteristic
194C4⋊Q8—293no—
195C4⋊Q8—293no—
196C4⋊Q8—293no—
197C4⋊Q8C2×D2913yes—
198#126—293no—
199#126—296no—
200#126—293no—
201#126—296no—
202#125—293no—
203#125—296no—
204#125—296no—
205#125—296no—
206#125—293no—
207#125—293no—
208#124—293no—
209#123—293no—
210#123—293no—
211C22⋊Q8—293no—
212C22⋊Q8—293no—
213C22⋊Q8—293no—
214C22⋊Q8—293no—
215C22⋊Q8—293no—
216C22⋊Q8C2×D2913yes—
217C22⋊Q8C2×D2913yes—
218#121—291no—
219C4×Q8—291no—
220C4×Q8—293no—
221C4×Q8—291no—
222C4×Q8—293no—
223C4×Q8—293no—
224C4×Q8—291no—
225C4×Q8—293no—
226C4×Q8—293no—
227C4×Q8—296no—
228C4×Q8—293no—
229C4×Q8—293no—
230C4×Q8C2×D2911yescharacteristic
231C4×D4—293no—
232C4×D4—293no—
233C4×D4C2×D2913yes—
234#117—293no—
235#117—293no—
236#117—293no—
237#117—296no—
238#117—293no—
239C2×C4⋊C4—293no—
240C2×C42—293no—
241C29#352311yescharacteristic, complemented, Sylow
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—293no—
245C2×Q8—293no—
246C2×Q8—584no—
247C2×Q8C22×D2913yes—
248C2×Q8C22×D2913yes—
249C2×Q8C22×D2911yescharacteristic
250C2×D4—291no—
251C2×D4—291no—
252C2×D4C22×D2911yescharacteristic
253C22×C4—291no—
254C22×C4—293no—
255C22×C4—293no—
256C22×C4C22×D2913yes—
257C22×C4C22×D2913yes—
258C4⋊C4—293no—
259C4⋊C4—293no—
260C4⋊C4—296no—
261C4⋊C4—293no—
262C4⋊C4—293no—
263C4⋊C4—293no—
264C4⋊C4—293no—
265C4⋊C4—293no—
266C4⋊C4—296no—
267C4⋊C4—296no—
268C4⋊C4—293no—
269C4⋊C4—293no—
270C4⋊C4—291no—
271C4⋊C4—296no—
272C4⋊C4—293no—
273C4⋊C4—293no—
274C4⋊C4C22×D2913yes—
275C4⋊C4C22×D2913yes—
276C4⋊C4C22×D2916yes—
277C22⋊C4—291no—
278C22⋊C4—293no—
279C22⋊C4—293no—
280C22⋊C4—293no—
281C22⋊C4—291no—
282C22⋊C4—293no—
283C22⋊C4—293no—
284C22⋊C4—293no—
285C22⋊C4C22×D2913yes—
286C22⋊C4C22×D2913yes—
287C42—293no—
288C42—293no—
289C42—293no—
290C42—293no—
291C42—296no—
292C42—296no—
293C42—293no—
294C42C22×D2913yes—
295C23—291no—
296C23C23×D2911yescharacteristic
297C23C23×D2911yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—586no—
301Q8—586no—
302Q8#9080714yes—
303D4—22no—
304D4—582no—
305D4—582no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—291no—
309C2×C4—291no—
310C2×C4—293no—
311C2×C4—293no—
312C2×C4—291no—
313C2×C4—293no—
314C2×C4—293no—
315C2×C4—582no—
316C2×C4—586no—
317C2×C4—586no—
318C2×C4C23×D2913yes—
319C2×C4C23×D2913yes—
320C2×C4C23×D2913yes—
321C2×C4C23×D2911yescharacteristic
322C2×C4C23×D2913yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—581no—
329C22—581no—
330C22—581no—
331C22#1174536212yes—
332C22C24×D2911yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—292no—
338C4—583no—
339C4—581no—
340C4—583no—
341C4—583no—
342C4—583no—
343C4—581no—
344C4C2×#9080716yes—
345C2—22no—
346C2—41no—
347C2—1161no—
348C2C22×#9080711yescharacteristic
349C22−1+4×D2911yescharacteristic
350C2C2×#1174536211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingD4×Q8×C29C2
prime radical#11745301C22
hypercenterD4×Q8D29
hypocenterC29#3523
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #14271889C2semidirect
2. #14271888C2semidirect
3. #14271884C2semidirect
4. #14271855C2semidirect
5. #14271853C2semidirect
6. Q8×C22⋊D29C2semidirect
7. Q8×C22⋊D29C2semidirect
8. #14271774C2semidirect
9. #14271773C2semidirect
10. #14271715C2semidirect
12. #14271641C2semidirect
13. #14271639C2semidirect
14. C2×Q8×Dic29C2semidirect
15. D4×Q8×C29C2semidirect
177. D4×Q8D29semidirect
241. C29#3523semidirect

Maximal subgroups

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

subgroupindex
#142718892
#142718882
#142718842
#142718552
#142718532
Q8×C22⋊D292
#142717742
#142717732
#142717152
#142716932
#142716412
#142716392
C2×Q8×Dic292
D4×Q8×C292
#352329