Finite Group Databasebeta

← #16833568

Subgroups — #16833568

Order 3712.

1104 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#14271895C213yesmaximal, complemented
2Q8×Dic58C211yesmaximal, characteristic, complemented
3#14271871C213yesmaximal, complemented
4#14271832C211yesmaximal, characteristic, complemented
5#14271824C211yesmaximal, characteristic, complemented
6#14271817C213yesmaximal, complemented
7#14271813C213yesmaximal, complemented
8#14271786C213yesmaximal, complemented
9#14271782C213yesmaximal, complemented
10#14271708C213yesmaximal, complemented
11#14271693C213yesmaximal, complemented
12#14271646C211yesmaximal, characteristic, complemented
13#14271645C211yesmaximal, characteristic, complemented
14C2×Q8×Dic29C211yesmaximal, characteristic, complemented
15C29×#548C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C22×Q8×C29C2211yescharacteristic
18C2×#90809C2211yescharacteristic
19C2×#90807C2216yes—
20C29×#124C2213yes—
21C29×C22⋊Q8C2213yes—
22C29×#121C2213yes—
23C4×Q8×C29C2211yescharacteristic
24C4×D4×C29C2213yes—
25#11745309C2213yes—
26#11745308C2213yes—
27Q8×Dic29C2213yes—
28Q8×Dic29C2211yescharacteristic
29Q8×Dic29C2211yescharacteristic
30Q8×Dic29C2213yes—
31Q8×Dic29C2211yescharacteristic
32#11745302C2213yes—
33#11745301C2213yes—
34#11745300C2213yes—
35#11745297C2213yes—
36D4×Dic29C2213yes—
37D4×Dic29C2213yes—
38#11745292C2213yes—
39#11745292C2213yes—
40C4×C22⋊D29C2213yes—
41C4×C22⋊D29C2213yes—
42#11745288C2213yes—
43C2×C4⋊Dic29C2213yes—
44C2×C4⋊Dic29C2213yes—
45C2×C4×Dic29C2213yes—
46#11745260C2213yes—
47#11745259C2216yes—
48#11745255C2213yes—
49C4⋊C4×D29C2213yes—
50#11745251C2213yes—
51#11745250C2213yes—
52#11745250C2213yes—
53#11745248C2216yes—
54#11745248C2213yes—
55#11745247C2216yes—
56#11745247C2213yes—
57#11745245C2216yes—
58#11745243C2216yes—
59#11745241C2213yes—
60#11745240C2216yes—
61#11745240C2211yescharacteristic
62C4×D116C2213yes—
63#11745234C2213yes—
64#11745231C2213yes—
65C4×Dic58C2213yes—
66#41×C29—24no—
67C2×Q8×C29—24no—
68C2×Q8×C29C2313yes—
69C2×Q8×C29C2311yescharacteristic
70C2×D4×C29C2313yes—
71C2×D4×C29C2313yes—
72C22×C4×C29C2313yes—
73C22×C4×C29C2313yes—
74C2×C22⋊D29C2313yes—
75C2×C22⋊D29C2313yes—
76C2×C22⋊D29C2316yes—
77C22×Dic29C2311yescharacteristic
78C22×Dic29C2313yes—
79C22×Dic29C2316yes—
80#90809—24no—
81#90807—412no—
82C2×D116C2313yes—
83C2×C4×D29C2313yes—
84C2×C4×D29C2316yes—
85C2×Dic58C2311yescharacteristic
86C2×Dic58C2316yes—
87C4⋊C4×C29C2313yes—
88C4⋊C4×C29C2313yes—
89C22⋊C4×C29C2311yescharacteristic
90C22⋊C4×C29C2313yes—
91C22⋊C4×C29C2316yes—
92C42×C29C2313yes—
93C22⋊Dic29C2313yes—
94C22⋊Dic29C2313yes—
95C22⋊Dic29C2311yescharacteristic
96C22⋊Dic29C2313yes—
97#90781C2313yes—
98#90781C2313yes—
99#90781C2316yes—
100#90781C2313yes—
101#90781C2313yes—
102C4⋊Dic29C2313yes—
103C4⋊Dic29C2313yes—
104C4⋊Dic29C2313yes—
105C4⋊Dic29C2313yes—
106C4⋊Dic29C2316yes—
107C4⋊Dic29C2311yescharacteristic
108C4⋊Dic29C2313yes—
109#90779C2311yescharacteristic
110#90779C2313yes—
111#90779C2313yes—
112#90779C2313yes—
113#90779C2313yes—
114#90779C2311yescharacteristic
115C4×Dic29C2313yes—
116C4×Dic29C2313yes—
117C4×Dic29C2313yes—
118C4×Dic29C2313yes—
119C4×Dic29C2313yes—
120C23×C29C2413yes—
121C23×C29C2411yescharacteristic
122C22×D29C2413yes—
123Q8×C29—26no—
124Q8×C29#4114yescomplemented
125D4×C29—26no—
126D4×C29—26no—
127C2×C4×C29—26no—
128C2×C4×C29—26no—
129C2×C4×C29C2413yes—
130C2×C4×C29C2413yes—
131C2×C4×C29C2411yescharacteristic
132C2×C4×C29C2411yescharacteristic
133C2×C4×C29C2413yes—
134C22⋊D29—26no—
135C22⋊D29—26no—
136C22⋊D29—46no—
137C2×Dic29—26no—
138C2×Dic29—22no—
139C2×Dic29—46no—
140C2×Dic29C2411yescharacteristic
141C2×Dic29C2411yescharacteristic
142C2×Dic29C2413yes—
143C2×Dic29C2413yes—
144C2×Dic29C2413yes—
145C2×Dic29C2411yescharacteristic
146C2×Dic29C2411yescharacteristic
147D116—26no—
148C4×D29—26no—
149C4×D29—46no—
150Dic58—22no—
151Dic58—46no—
152#3522—291nomaximal, Sylow
153C22×C29—21no—
154C22×C29—21no—
155C22×C29—23no—
156C22×C29—23no—
157C22×C29—23no—
158C22×C29H(5,𝔽2)12yes—
159C22×C29C2511yescharacteristic
160C2×D29—23no—
161C2×D29—23no—
162C2×D29—23no—
163C4×C29—21no—
164C4×C29—23no—
165C4×C29—23no—
166C4×C29—21no—
167C4×C29C2×#4116yes—
168Dic29—21no—
169Dic29—23no—
170Dic29—23no—
171Dic29—23no—
172Dic29—21no—
173Dic29—21no—
174Dic292−1+412yes—
175#559—293no—
176Q82—291no—
177#554—293no—
178#550—291no—
179#548—291no—
180#548D2911yescharacteristic, complemented
181#547—293no—
182#547—293no—
183#542—293no—
184#542—293no—
185#532—293no—
186#529—293no—
187#520—291no—
188#520—291no—
189C2×C4×Q8—291no—
190C2×C29—22no—
191C2×C29—43no—
192C2×C29C22×#4111yescharacteristic
193C2×C29C2×H(5,𝔽2)11yescharacteristic
194C2×C29C2×2−1+411yescharacteristic
195D29—43no—
196C2×#41—291no—
197C2×#41—296no—
198C2×#41C2×D2911yescharacteristic
199C22×Q8C2×D2911yescharacteristic
200C4⋊Q8—293no—
201C4⋊Q8—293no—
202C4⋊Q8—293no—
203#127—293no—
204#126—293no—
205#126—293no—
206#125—293no—
207#124—293no—
208#124—293no—
209#124—296no—
210#124C2×D2913yes—
211#123—293no—
212#123—293no—
213#123—296no—
214#123—296no—
215C22⋊Q8—293no—
216C22⋊Q8—296no—
217C22⋊Q8—291no—
218C22⋊Q8—296no—
219C22⋊Q8C2×D2913yes—
220#121—293no—
221#121—293no—
222#121—293no—
223#121—293no—
224#121C2×D2913yes—
225C4×Q8—291no—
226C4×Q8—293no—
227C4×Q8—291no—
228C4×Q8—293no—
229C4×Q8—291no—
230C4×Q8—293no—
231C4×Q8C2×D2911yescharacteristic
232C4×D4—293no—
233C4×D4—293no—
234C4×D4—293no—
235C4×D4—293no—
236C4×D4—293no—
237C4×D4—296no—
238C4×D4—293no—
239C4×D4C2×D2913yes—
240#117—293no—
241#117—293no—
242C2×C4⋊C4—293no—
243C2×C4⋊C4—293no—
244C2×C4⋊C4—293no—
245C2×C42—293no—
246C29#352211yescharacteristic, complemented, Sylow
247#41—24no—
248#41—584no—
249#41—11612no—
250C2×Q8—24no—
251C2×Q8—291no—
252C2×Q8—296no—
253C2×Q8C22×D2913yes—
254C2×Q8C22×D2911yescharacteristic
255C2×D4—293no—
256C2×D4—293no—
257C2×D4—296no—
258C2×D4—293no—
259C2×D4C22×D2913yes—
260C2×D4C22×D2913yes—
261C22×C4—291no—
262C22×C4—293no—
263C22×C4—296no—
264C22×C4—293no—
265C22×C4—296no—
266C22×C4C22×D2913yes—
267C22×C4C22×D2913yes—
268C4⋊C4—293no—
269C4⋊C4—293no—
270C4⋊C4—293no—
271C4⋊C4—293no—
272C4⋊C4—293no—
273C4⋊C4—296no—
274C4⋊C4—291no—
275C4⋊C4—291no—
276C4⋊C4—291no—
277C4⋊C4—293no—
278C4⋊C4—293no—
279C4⋊C4—293no—
280C4⋊C4—293no—
281C4⋊C4C22×D2913yes—
282C4⋊C4C22×D2913yes—
283C22⋊C4—293no—
284C22⋊C4—293no—
285C22⋊C4—291no—
286C22⋊C4—293no—
287C22⋊C4—293no—
288C22⋊C4—293no—
289C22⋊C4—293no—
290C22⋊C4—296no—
291C22⋊C4—293no—
292C22⋊C4C22×D2916yes—
293C22⋊C4C22×D2911yescharacteristic
294C22⋊C4C22×D2913yes—
295C42—293no—
296C42—293no—
297C42—293no—
298C42—293no—
299C42—293no—
300C42C22×D2913yes—
301C23—293no—
302C23C23×D2913yes—
303C23C23×D2911yescharacteristic
304Q8—26no—
305Q8—582no—
306Q8—1166no—
307Q8#9080714yescomplemented
308D4—26no—
309D4—26no—
310D4—586no—
311D4—586no—
312D4—586no—
313D4—1166no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—291no—
317C2×C4—291no—
318C2×C4—291no—
319C2×C4—293no—
320C2×C4—293no—
321C2×C4—293no—
322C2×C4—291no—
323C2×C4—582no—
324C2×C4—586no—
325C2×C4—586no—
326C2×C4—1166no—
327C2×C4—1166no—
328C2×C4C23×D2913yes—
329C2×C4C23×D2911yescharacteristic
330C2×C4C23×D2913yes—
331C2×C4C23×D2911yescharacteristic
332C2×C4C23×D2913yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—583no—
339C22—583no—
340C22—583no—
341C22#1174536412yes—
342C22C24×D2911yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—292no—
348C4—581no—
349C4—583no—
350C4—583no—
351C4—583no—
352C4—581no—
353C4—581no—
354C4C2×#9080716yes—
355C2—22no—
356C2—43no—
357C2—1163no—
358C2C22×#9080711yescharacteristic
359C2C2×#1174536411yescharacteristic
360C22−1+4×D2911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingC29×#548C2
hypercenter#548D29
hypocenterC29#3522
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #14271895C2semidirect
2. Q8×Dic58C2semidirect
3. #14271871C2semidirect
4. #14271832C2semidirect
5. #14271824C2semidirect
6. #14271817C2semidirect
7. #14271813C2semidirect
8. #14271786C2semidirect
9. #14271782C2semidirect
10. #14271708C2semidirect
11. #14271693C2semidirect
12. #14271646C2semidirect
13. #14271645C2semidirect
14. C2×Q8×Dic29C2semidirect
15. C29×#548C2semidirect
124. Q8×C29#41semidirect
180. #548D29semidirect
246. C29#3522semidirect
307. Q8#90807semidirect

Maximal subgroups

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

subgroupindex
#142718952
Q8×Dic582
#142718712
#142718322
#142718242
#142718172
#142718132
#142717862
#142717822
#142717082
#142716932
#142716462
#142716452
C2×Q8×Dic292
C29×#5482
#352229