Finite Group Databasebeta

← #16828330

Subgroups — #16828330

Order 3712.

470 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
1C2×#11745280C211yesmaximal, characteristic
2#14271645C211yesmaximal, characteristic
3#14271639C211yesmaximal, characteristic, complemented
4#14271270C211yesmaximal, characteristic, complemented
5#14271268C211yesmaximal, characteristic, complemented
6#14271262C211yesmaximal, characteristic, complemented
7#14271261C211yesmaximal, characteristic, complemented
8#14271255C211yesmaximal, characteristic, complemented
9C4×#90773C211yesmaximal, characteristic, complemented
10#14271248C211yesmaximal, characteristic, complemented
11#14271204C211yesmaximal, characteristic, complemented
12#14271158C211yesmaximal, characteristic
13#103×D29C211yesmaximal, characteristic, complemented
14#14271117C211yesmaximal, characteristic, complemented
15C4×Q16×C29C211yesmaximal, characteristic, complemented
16C2×#90809C2211yescharacteristic
17C2×Q8×D29C2211yescharacteristic
18C2×Q16×C29C2211yescharacteristic
19C4×Q8×C29C2211yescharacteristic
20C4×Q8×C29C2211yescharacteristic
21Q8×Dic29C2211yescharacteristic
22Q8×Dic29C2211yescharacteristic
23C2×#90785C2211yescharacteristic
24C2×#90784C2211yescharacteristic
25#11745280C418yescomplemented
26C2×#90773C2211yescharacteristic
27C2×#90772C2211yescharacteristic
28C2×C8×D29C2211yescharacteristic
29#11745255—21no—
30#11745255C2211yescharacteristic
31#11745254—21no—
32#11745254C2211yescharacteristic
33C4⋊C4×D29—21no—
34C4⋊C4×D29C2211yescharacteristic
35#11745249—21no—
36#11745249C2211yescharacteristic
37C4×D116—21no—
38C4×D116C2211yescharacteristic
39#11745234—21no—
40#11745234—21no—
41#11745234C2211yescharacteristic
42C4×Dic58—21no—
43C4×Dic58C2211yescharacteristic
44C29×#107C2211yescharacteristic
45C29×#103C2211yescharacteristic
46C29×#103C2211yescharacteristic
47C4×C8×C29C2211yescharacteristic
48#11745183C2211yescharacteristic
49#11745183C2211yescharacteristic
50#11745169C2211yescharacteristic
51#11745165C2211yescharacteristic
52#11745164C2211yescharacteristic
53#11745163C2211yescharacteristic
54C8×Dic29C2211yescharacteristic
55#11745158C2211yescharacteristic
56#11745157C2211yescharacteristic
57#11745156C2211yescharacteristic
58#11745155C2211yescharacteristic
59#11745151C2211yescharacteristic
60C2×Q8×C29C2311yescharacteristic
61C2×Q8×C29C2311yescharacteristic
62#90809—22no—
63#90809C2×C414yes—
64Q8×D29—22no—
65Q8×D29C2×C414yescomplemented
66C2×D116—21no—
67C2×D116C2311yescharacteristic
68C2×C4×D29—21no—
69C2×C4×D29—21no—
70C2×C4×D29—21no—
71C2×C4×D29—21no—
72C2×C4×D29—21no—
73C2×C4×D29C2311yescharacteristic
74C2×Dic58—21no—
75C2×Dic58C2311yescharacteristic
76Q16×C29C2×C414yescomplemented
77C2×C8×C29C2311yescharacteristic
78C2×C8×C29C2311yescharacteristic
79C4⋊C4×C29—21no—
80C4⋊C4×C29—21no—
81C4⋊C4×C29C2311yescharacteristic
82C4⋊C4×C29C2311yescharacteristic
83C42×C29—21no—
84C42×C29—21no—
85C42×C29C2311yescharacteristic
86#90785C2×C414yescomplemented
87#90784C2×C414yescomplemented
88#90781—21no—
89#90781—21no—
90#90781—21no—
91#90781D411yescharacteristic
92#90781D411yescharacteristic
93C4⋊Dic29—21no—
94C4⋊Dic29—21no—
95C4⋊Dic29C2311yescharacteristic
96C4⋊Dic29C2311yescharacteristic
97#90779—21no—
98#90779—21no—
99#90779—21no—
100#90779D411yescharacteristic
101#90779D411yescharacteristic
102C4×Dic29—21no—
103C4×Dic29—21no—
104C4×Dic29—21no—
105C4×Dic29—21no—
106C4×Dic29—21no—
107C4×Dic29C2311yescharacteristic
108C2×#6597C2311yescharacteristic
109C2×#6597C2311yescharacteristic
110#90773C2×C414yescomplemented
111#90772C2×C414yescomplemented
112C8×D29—22no—
113C22×D29—21no—
114C22×D29C2×D411yescharacteristic
115Q8×C29—21no—
116Q8×C29—21no—
117Q8×C29C22×C412yes—
118Q8×C29C22×C412yes—
119C2×C4×C29—21no—
120C2×C4×C29—21no—
121C2×C4×C29—21no—
122C2×C4×C29—21no—
123C2×C4×C29C2×D411yescharacteristic
124C2×C4×C29C2×D411yescharacteristic
125C2×C4×C29C2411yescharacteristic
126C2×Dic29—21no—
127C2×Dic29—21no—
128C2×Dic29—21no—
129C2×Dic29C2×D411yescharacteristic
130C2×Dic29C2×D411yescharacteristic
131C2×Dic29C2×D411yescharacteristic
132D116—24no—
133D116—21no—
134D116C22×C412yes—
135C4×D29—24no—
136C4×D29—24no—
137C4×D29—81no—
138C4×D29—81no—
139C4×D29—81no—
140C4×D29C22×C412yes—
141C4×D29#4112yes—
142Dic58—21no—
143Dic58—24no—
144Dic58C22×C412yes—
145C8×C29—21no—
146C8×C29C22×C412yes—
147#6597—21no—
148#6597C22×C412yes—
149#2947—291nomaximal, Sylow
150C22×C29C22×D411yescharacteristic
151C2×D29—21no—
152C2×D29—21no—
153C2×D29—22no—
154C2×D29—41no—
155C2×D29—41no—
156C2×D29C4×D412yes—
157C4×C29—22no—
158C4×C29—22no—
159C4×C29—21no—
160C4×C29—21no—
161C4×C29—41no—
162C4×C29—41no—
163C4×C29C23×C411yescharacteristic
164C4×C29C2×#4111yescharacteristic
165Dic29—22no—
166Dic29—21no—
167Dic29—21no—
168Dic29—41no—
169Dic29—41no—
170Dic29C4×D412yes—
171C2×#137—291no—
172#520—291no—
173#519—291no—
174#441—291no—
175#441—291no—
176#440—291no—
177#440—291no—
178C4×Q16—291no—
179C4×Q16D2911yescharacteristic, complemented
180C4×SD16—291no—
181C4×SD16—291no—
182#428—291no—
183#416—291no—
184C2×#103—291no—
185#405—291no—
186C2×C29C2×C4×D411yescharacteristic
187C2×C29#57711yescharacteristic
188C2×C29#57811yescharacteristic
189D29—22no—
190D29—42no—
191C2×#41—291no—
192C22×Q8—291no—
193#137—298no—
194C2×Q16—291no—
195C2×Q16C2×D2911yescharacteristic
196C2×SD16—291no—
197C2×SD16—291no—
198C2×M16—291no—
199C22×C8—291no—
200C4×Q8—291no—
201C4×Q8—291no—
202C4×Q8—291no—
203C4×Q8—291no—
204C4×Q8—581no—
205C4×Q8—581no—
206C4×Q8C2×D2911yescharacteristic
207C4×Q8C2×D2911yescharacteristic
208C4×D4—291no—
209C4×D4—291no—
210C4×D4—581no—
211C4×D4—581no—
212#117—291no—
213#117—291no—
214#117—581no—
215#117—581no—
216#117—581no—
217C2×C4⋊C4—291no—
218C2×C4⋊C4—581no—
219#107—291no—
220#107C2×D2911yescharacteristic
221#106—291no—
222#106—291no—
223#103—291no—
224#103—291no—
225#103—291no—
226#103—291no—
227#103C2×D2911yescharacteristic
228#103C2×D2911yescharacteristic
229#102—291no—
230#102—291no—
231#97—291no—
232#97—291no—
233C4×C8—291no—
234C4×C8C2×D2911yescharacteristic
235C29#294711yescharacteristic, complemented, Sylow
236#41—294no—
237#41—582no—
238C2×Q8—294no—
239C2×Q8—291no—
240C2×Q8—581no—
241C2×Q8—582no—
242C2×Q8C22×D2911yescharacteristic
243C2×Q8C22×D2911yescharacteristic
244C2×D4—291no—
245C2×D4—581no—
246C22×C4—291no—
247C22×C4—581no—
248C22×C4—581no—
249C22×C4—581no—
250C22×C4—581no—
251C22×C4—581no—
252Q16—294no—
253Q16C4×D2914yescomplemented
254SD16—294no—
255SD16—294no—
256M16—294no—
257C2×C8—291no—
258C2×C8—291no—
259C2×C8—582no—
260C2×C8C22×D2911yescharacteristic
261C2×C8C22×D2911yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—291no—
265C4⋊C4—291no—
266C4⋊C4—291no—
267C4⋊C4—291no—
268C4⋊C4—581no—
269C4⋊C4—581no—
270C4⋊C4—581no—
271C4⋊C4—581no—
272C4⋊C4—581no—
273C4⋊C4C22×D2911yescharacteristic
274C4⋊C4C22×D2911yescharacteristic
275C22⋊C4—291no—
276C22⋊C4—291no—
277C22⋊C4—581no—
278C22⋊C4—581no—
279C22⋊C4—581no—
280C42—21no—
281C42—21no—
282C42—291no—
283C42—581no—
284C42—581no—
285C42—581no—
286C42—581no—
287C42—581no—
288C42C22×D2911yescharacteristic
289C23—291no—
290C23—581no—
291Q8—21no—
292Q8—21no—
293Q8—292no—
294Q8—584no—
295Q8—581no—
296Q8C2×C4×D2912yes—
297Q8C2×C4×D2912yes—
298D4—292no—
299D4—584no—
300D4—581no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—291no—
306C2×C4—292no—
307C2×C4—292no—
308C2×C4—291no—
309C2×C4—291no—
310C2×C4—584no—
311C2×C4—584no—
312C2×C4—581no—
313C2×C4—581no—
314C2×C4—581no—
315C2×C4—2321no—
316C2×C4—2321no—
317C2×C4—2321no—
318C2×C4D4×D2911yescharacteristic
319C2×C4D4×D2911yescharacteristic
320C2×C4C23×D2911yescharacteristic
321C8—21no—
322C8—292no—
323C8—581no—
324C8C2×C4×D2912yes—
325C22—292no—
326C22—581no—
327C22—581no—
328C22—582no—
329C22—1161no—
330C22—1161no—
331C22C2×D4×D2911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—292no—
339C4—582no—
340C4—581no—
341C4—581no—
342C4—1161no—
343C4—1161no—
344C4C22×C4×D2911yescharacteristic
345C4#41×D2911yescharacteristic
346C2—582no—
347C2—1162no—
348C2C4×D4×D2911yescharacteristic
349C2#1427197011yescharacteristic
350C2#1427197611yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutatorC4×C29C23×C4
FrattiniC2×C4C23×D29
FittingC4×Q16×C29C2
socleC22×C29C22×D4
prime radicalC2×#90809C22
cosocleC2×C4×C29C24
hypercenterC4×Q16D29
hypocenterC29#2947
LCSG ▷ C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
3. #14271639C2semidirect
4. #14271270C2semidirect
5. #14271268C2semidirect
6. #14271262C2semidirect
7. #14271261C2semidirect
8. #14271255C2semidirect
9. C4×#90773C2semidirect
10. #14271248C2semidirect
11. #14271204C2semidirect
13. #103×D29C2semidirect
14. #14271117C2semidirect
15. C4×Q16×C29C2semidirect
25. #11745280C4semidirect
65. Q8×D29C2×C4semidirect
76. Q16×C29C2×C4semidirect
86. #90785C2×C4semidirect
87. #90784C2×C4semidirect
110. #90773C2×C4semidirect
111. #90772C2×C4semidirect
179. C4×Q16D29semidirect
235. C29#2947semidirect
253. Q16C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452802
#142716452
#142716392
#142712702
#142712682
#142712622
#142712612
#142712552
C4×#907732
#142712482
#142712042
#142711582
#103×D292
#142711172
C4×Q16×C292
#294729