Finite Group Databasebeta

← #16831003

Subgroups — #16831003

Order 3712.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745304C211yesmaximal, characteristic, complemented
2C22×#90784C211yesmaximal, characteristic, complemented
3#14271809C211yesmaximal, characteristic
4#14271555C211yesmaximal, characteristic, complemented
5#14271411C211yesmaximal, characteristic, complemented
6#14271402C211yesmaximal, characteristic, complemented
7#14271385C211yesmaximal, characteristic, complemented
8#14271365C211yesmaximal, characteristic, complemented
9#14271323C211yesmaximal, characteristic, complemented
10#14271319C211yesmaximal, characteristic, complemented
11#14271313C211yesmaximal, characteristic, complemented
12#14271302C211yesmaximal, characteristic, complemented
13#14271261C211yesmaximal, characteristic, complemented
14#14271235C211yesmaximal, characteristic, complemented
15D4×Q8×C29C211yesmaximal, characteristic, complemented
16C22×Q8×C29C2211yescharacteristic
17C22×Q8×C29C2211yescharacteristic, complemented
18C2×#90805C2211yescharacteristic
19C22×D116C2211yescharacteristic
20C29×C4⋊Q8—21no—
21C29×C4⋊Q8C2211yescharacteristic, complemented
22C29×C22⋊Q8—21no—
23C29×C22⋊Q8—21no—
24C29×C22⋊Q8C2211yescharacteristic
25C29×C22⋊Q8C2211yescharacteristic, complemented
26C4×Q8×C29C2211yescharacteristic, complemented
27C4×D4×C29—21no—
28C4×D4×C29C2211yescharacteristic
29C2×#90785C2211yescharacteristic, complemented
30C2×#90785C2211yescharacteristic, complemented
31C2×#90785C2211yescharacteristic, complemented
32#11745304—24no—
33C2×#90784—24no—
34C2×#90784C2211yescharacteristic
35C2×#90784C2211yescharacteristic
36C2×#90784C2211yescharacteristic
37C2×#90784C2211yescharacteristic
38C2×#90784C2211yescharacteristic
39C2×#90784C2211yescharacteristic
40#11745299—21no—
41#11745292C2211yescharacteristic
42#11745286C2211yescharacteristic
43C2×#90777C2211yescharacteristic, complemented
44C22×#6597C2211yescharacteristic
45#11745256—21no—
46#11745246—21no—
47C22⋊C4×D29—21no—
48#11745237C2211yescharacteristic
49C4×D116C2211yescharacteristic
50#11745183C2211yescharacteristic, complemented
51#11745183C2211yescharacteristic, complemented
52#11745183C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic
54C22⋊#6597C2211yescharacteristic, complemented
55#11745158C2211yescharacteristic, complemented
56#11745157C2211yescharacteristic
57#11745157C2211yescharacteristic
58#11745155C2211yescharacteristic, complemented
59C4⋊#6597C2211yescharacteristic, complemented
60#11745151C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×Q8×C29—21no—
63C2×Q8×C29—21no—
64C2×Q8×C29—22no—
65C2×Q8×C29—22no—
66C2×Q8×C29—41no—
67C2×Q8×C29—41no—
68C2×Q8×C29C2311yescharacteristic
69C2×Q8×C29C2311yescharacteristic
70C2×Q8×C29C2311yescharacteristic
71C2×D4×C29D411yescharacteristic
72C22×C4×C29—21no—
73C22×C4×C29—21no—
74C22×C4×C29C2311yescharacteristic
75C22×C4×C29C2311yescharacteristic
76C2×C22⋊D29—21no—
77C2×C22⋊D29—21no—
78#90805—22no—
79#90805—41no—
80C2×D116—22no—
81C2×D116—21no—
82C2×D116—41no—
83C2×D116C2311yescharacteristic
84C2×D116C2311yescharacteristic
85C2×C4×D29—21no—
86C2×C4×D29—21no—
87C2×Dic58C2311yescharacteristic
88C4⋊C4×C29—21no—
89C4⋊C4×C29—21no—
90C4⋊C4×C29—21no—
91C4⋊C4×C29—21no—
92C4⋊C4×C29—21no—
93C4⋊C4×C29D411yescharacteristic
94C4⋊C4×C29C2311yescharacteristic
95C22⋊C4×C29—21no—
96C22⋊C4×C29—21no—
97C22⋊C4×C29D411yescharacteristic
98C22⋊C4×C29D411yescharacteristic
99C42×C29—21no—
100C42×C29C2311yescharacteristic
101C22⋊Dic29—21no—
102#90785—22no—
103#90785—22no—
104#90785—41no—
105#90784—22no—
106#90784—22no—
107#90784—22no—
108#90784—22no—
109#90784—41no—
110#90784D414yescomplemented
111#90781—21no—
112#90781—21no—
113#90781—21no—
114#90781—21no—
115C4⋊Dic29C2311yescharacteristic
116#90779—21no—
117#90777—22no—
118C2×#6597—22no—
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122C2×#6597C2311yescharacteristic
123C23×C29C2×D411yescharacteristic
124C23×C29C2×D411yescharacteristic
125C22×D29—21no—
126C22×D29—21no—
127C22×D29—42no—
128C22×D29—42no—
129C22×D29—81no—
130Q8×C29—21no—
131Q8×C29—21no—
132Q8×C29—21no—
133Q8×C29—21no—
134Q8×C29—21no—
135Q8×C29—42no—
136Q8×C29—42no—
137Q8×C29C2×D412yes—
138D4×C29—22no—
139C2×C4×C29—21no—
140C2×C4×C29—21no—
141C2×C4×C29—21no—
142C2×C4×C29—21no—
143C2×C4×C29—21no—
144C2×C4×C29—21no—
145C2×C4×C29—21no—
146C2×C4×C29—21no—
147C2×C4×C29—42no—
148C2×C4×C29—42no—
149C2×C4×C29C2×D411yescharacteristic
150C2×C4×C29C2×D411yescharacteristic
151C2×C4×C29C2×D411yescharacteristic
152C2×C4×C29C2×D411yescharacteristic
153C2×C4×C29C2411yescharacteristic
154C22⋊D29—42no—
155C22⋊D29—81no—
156C2×Dic29—21no—
157C2×Dic29—21no—
158D116—21no—
159D116—21no—
160D116—21no—
161D116—42no—
162D116C2×D412yes—
163C4×D29—42no—
164C4×D29—81no—
165Dic58—21no—
166Dic58—21no—
167#6597—21no—
168#6597—21no—
169#6597—21no—
170#6597C2×D412yes—
171#3266—291nomaximal, Sylow
172C22×C29—21no—
173C22×C29—21no—
174C22×C29—21no—
175C22×C29—21no—
176C22×C29—21no—
177C22×C29#13712yescomplemented
178C22×C29C22×D411yescharacteristic
179C2×D29—22no—
180C2×D29—41no—
181C2×D29—41no—
182C2×D29—41no—
183C2×D29—41no—
184C2×D29—41no—
185C2×D29—84no—
186C4×C29—21no—
187C4×C29—21no—
188C4×C29—22no—
189C4×C29—21no—
190C4×C29—21no—
191C4×C29—41no—
192C4×C29—41no—
193C4×C29—41no—
194C4×C29C22×D411yescharacteristic
195C4×C29H(5,𝔽2)11yescharacteristic
196Dic29—41no—
197Dic29—41no—
198C2×#137—291no—
199C22×SD16—291no—
200D4×Q8D2911yescharacteristic, complemented
201#546—291no—
202#497—291no—
203#468—291no—
204#467—291no—
205#464—291no—
206#461—291no—
207#452—291no—
208#451—291no—
209#450—291no—
210#448—291no—
211#440—291no—
212#435—291no—
213C2×C29—22no—
214C2×C29—41no—
215C2×C29D4211yescharacteristic
216C2×C29C2×#13711yescharacteristic
217C2×C29#57711yescharacteristic
218D29—42no—
219D29—81no—
220C2×#41—291no—
221C22×Q8C2×D2911yescharacteristic
222C22×Q8C2×D2911yescharacteristic, complemented
223C22×D4—291no—
224#137—584no—
225C2×Q16—291no—
226C2×Q16—291no—
227C2×Q16—291no—
228C2×SD16—291no—
229C2×SD16—291no—
230C2×SD16—291no—
231C2×SD16—291no—
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—584no—
235C2×M16—291no—
236C22×C8—291no—
237C4⋊Q8—21no—
238C4⋊Q8C2×D2911yescharacteristic, complemented
239#124—291no—
240#123—581no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—291no—
244C22⋊Q8C2×D2911yescharacteristic
245C22⋊Q8C2×D2911yescharacteristic, complemented
246#121—291no—
247#121—581no—
248#120—581no—
249C4×Q8C2×D2911yescharacteristic, complemented
250C4×D4—21no—
251C4×D4—291no—
252C4×D4C2×D2911yescharacteristic
253C2×C22⋊C4—581no—
254#107—291no—
255C4⋊C8—291no—
256#103—291no—
257#103—291no—
258#103—291no—
259#103—291no—
260#102—291no—
261#102—291no—
262C22⋊C8—291no—
263C22⋊C8—291no—
264#97—291no—
265C29#326611yescharacteristic, complemented, Sylow
266C24—581no—
267#41—582no—
268#41—1161no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—291no—
276C2×Q8C22×D2911yescharacteristic
277C2×Q8C22×D2911yescharacteristic
278C2×Q8C22×D2911yescharacteristic
279C2×D4—291no—
280C2×D4—291no—
281C2×D4—582no—
282C2×D4—581no—
283C2×D4—581no—
284C2×D4—581no—
285C2×D4—1161no—
286C2×D4C22⋊D2911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—581no—
290C22×C4—581no—
291C22×C4C22×D2911yescharacteristic
292C22×C4C22×D2911yescharacteristic
293Q16—582no—
294Q16—582no—
295Q16—1161no—
296SD16—294no—
297SD16—582no—
298SD16—582no—
299SD16—582no—
300SD16—582no—
301SD16—1161no—
302M16—582no—
303C2×C8—291no—
304C2×C8—291no—
305C2×C8—291no—
306C2×C8—291no—
307C2×C8—582no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—291no—
314C4⋊C4—581no—
315C4⋊C4C22⋊D2911yescharacteristic
316C4⋊C4C22×D2911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—581no—
320C22⋊C4—581no—
321C22⋊C4—581no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4C22⋊D2911yescharacteristic
325C22⋊C4C22⋊D2911yescharacteristic
326C42—21no—
327C42C22×D2911yescharacteristic
328C23—581no—
329C23—581no—
330C23—1162no—
331C23—1162no—
332C23—2321no—
333C23C2×C22⋊D2911yescharacteristic
334C23C2×C22⋊D2911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—581no—
343Q8—581no—
344Q8D4×D2912yes—
345D4—22no—
346D4—292no—
347D4—581no—
348D4—581no—
349D4—581no—
350D4—1162no—
351D4—1162no—
352D4—2321no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—581no—
364C2×C4—581no—
365C2×C4—1162no—
366C2×C4—2321no—
367C2×C4C2×C22⋊D2911yescharacteristic
368C2×C4C2×C22⋊D2911yescharacteristic
369C2×C4C2×C22⋊D2911yescharacteristic
370C2×C4C2×C22⋊D2911yescharacteristic
371C2×C4C23×D2911yescharacteristic
372C8—292no—
373C8—581no—
374C8—581no—
375C8—581no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—582no—
382C22—1161no—
383C22—1161no—
384C22—1161no—
385C22—1161no—
386C22—1161no—
387C22—2324no—
388C22#1174530412yescomplemented
389C22C22×C22⋊D2911yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—1161no—
399C4—1161no—
400C4C2×D4×D2911yescharacteristic
401C4#1174535911yescharacteristic
402C2—22no—
403C2—41no—
404C2—1162no—
405C2—2321no—
406C2D4×C22⋊D2911yescharacteristic
407C2C2×#1174530411yescharacteristic
408C2#1427197111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingD4×Q8×C29C2
socleC22×C29C22×D4
prime radical#14271809C2
hypercenterD4×Q8D29
hypocenterC29#3266
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11745304C2semidirect
2. C22×#90784C2semidirect
4. #14271555C2semidirect
5. #14271411C2semidirect
6. #14271402C2semidirect
7. #14271385C2semidirect
8. #14271365C2semidirect
9. #14271323C2semidirect
10. #14271319C2semidirect
11. #14271313C2semidirect
12. #14271302C2semidirect
13. #14271261C2semidirect
14. #14271235C2semidirect
15. D4×Q8×C29C2semidirect
17. C22×Q8×C29C22semidirect
21. C29×C4⋊Q8C22semidirect
25. C29×C22⋊Q8C22semidirect
26. C4×Q8×C29C22semidirect
29. C2×#90785C22semidirect
30. C2×#90785C22semidirect
31. C2×#90785C22semidirect
43. C2×#90777C22semidirect
50. #11745183C22semidirect
51. #11745183C22semidirect
52. #11745183C22semidirect
54. C22⋊#6597C22semidirect
55. #11745158C22semidirect
58. #11745155C22semidirect
59. C4⋊#6597C22semidirect
60. #11745151C22semidirect
110. #90784D4semidirect
177. C22×C29#137semidirect
200. D4×Q8D29semidirect
222. C22×Q8C2×D29semidirect
238. C4⋊Q8C2×D29semidirect
245. C22⋊Q8C2×D29semidirect
249. C4×Q8C2×D29semidirect
265. C29#3266semidirect
388. C22#11745304semidirect

Maximal subgroups

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

subgroupindex
C2×#117453042
C22×#907842
#142718092
#142715552
#142714112
#142714022
#142713852
#142713652
#142713232
#142713192
#142713132
#142713022
#142712612
#142712352
D4×Q8×C292
#326629