Finite Group Databasebeta

← #16830997

Subgroups — #16830997

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×#11745280C211yesmaximal, characteristic, complemented
2C2×SD16×D29C211yesmaximal, characteristic, complemented
3Q8×D116C211yesmaximal, characteristic, complemented
4#14271806C211yesmaximal, characteristic
5#14271558C211yesmaximal, characteristic, complemented
6#14271408C211yesmaximal, characteristic, complemented
7#14271403C211yesmaximal, characteristic, complemented
8#14271369C211yesmaximal, characteristic, complemented
9#14271328C211yesmaximal, characteristic, complemented
10#14271317C211yesmaximal, characteristic, complemented
11#14271315C211yesmaximal, characteristic, complemented
12#14271300C211yesmaximal, characteristic, complemented
13#14271261C211yesmaximal, characteristic, complemented
14#14271233C211yesmaximal, characteristic, complemented
15C29×#464C211yesmaximal, characteristic, complemented
16C2×#90809C2211yescharacteristic
17C2×Q8×D29C2211yescharacteristic
18C2×Q8×D29C2211yescharacteristic, complemented
19C2×D4×D29C2211yescharacteristic
20C2×Q16×C29C2211yescharacteristic, complemented
21C2×SD16×C29C2211yescharacteristic
22C29×#124C2211yescharacteristic
23C4×Q8×C29C2211yescharacteristic, complemented
24#11745308C2211yescharacteristic
25C2×#90785C2211yescharacteristic, complemented
26C2×#90785C2211yescharacteristic, complemented
27C2×#90784C2211yescharacteristic
28#11745300C2211yescharacteristic
29C2×#90783C2211yescharacteristic
30C2×#90783C2211yescharacteristic
31#11745280—24no—
32SD16×D29—24no—
33C2×#90773C2211yescharacteristic
34C2×#90773C2211yescharacteristic
35C2×#90772C2211yescharacteristic, complemented
36C2×C8×D29C2211yescharacteristic
37#11745259—21no—
38#11745259—21no—
39#11745259C2211yescharacteristic
40#11745259C2211yescharacteristic, complemented
41#11745246—21no—
42#11745245—21no—
43#11745244—21no—
44C22⋊C4×D29—21no—
45C4×D116—21no—
46C4×D116C2211yescharacteristic
47C4×D116C2211yescharacteristic
48#11745231—21no—
49#11745231C2211yescharacteristic, complemented
50C29×C4⋊C8C2211yescharacteristic, complemented
51C29×#103C2211yescharacteristic, complemented
52C29×#102C2211yescharacteristic
53#11745183C2211yescharacteristic, complemented
54#11745168C2211yescharacteristic
55#11745168C2211yescharacteristic, complemented
56#11745164C2211yescharacteristic, complemented
57#11745164C2211yescharacteristic, complemented
58#11745157C2211yescharacteristic
59#11745155C2211yescharacteristic, complemented
60#11745151C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×Q8×C29C2311yescharacteristic
63C2×Q8×C29C2311yescharacteristic
64C2×D4×C29C2311yescharacteristic
65C2×C22⋊D29—21no—
66C2×C22⋊D29—21no—
67#90809—22no—
68#90809—41no—
69Q8×D29—22no—
70Q8×D29—22no—
71Q8×D29—41no—
72Q8×D29—41no—
73D4×D29—22no—
74D4×D29—41no—
75C2×D116—21no—
76C2×D116D411yescharacteristic
77C2×D116C2311yescharacteristic
78C2×C4×D29—21no—
79C2×C4×D29—21no—
80C2×C4×D29—21no—
81C2×C4×D29—21no—
82C2×C4×D29C2311yescharacteristic
83C2×C4×D29C2311yescharacteristic
84C2×Dic58—21no—
85C2×Dic58—21no—
86C2×Dic58C2311yescharacteristic
87C2×Dic58C2311yescharacteristic
88Q16×C29—22no—
89SD16×C29—22no—
90C2×C8×C29C2311yescharacteristic
91C2×C8×C29C2311yescharacteristic
92C4⋊C4×C29—21no—
93C4⋊C4×C29C2311yescharacteristic
94C22⋊C4×C29—21no—
95C22⋊C4×C29—21no—
96C42×C29—21no—
97C42×C29C2311yescharacteristic
98C22⋊Dic29—21no—
99#90785—22no—
100#90785—41no—
101#90784D414yescomplemented
102#90783—22no—
103#90783—41no—
104#90781—21no—
105#90781—21no—
106#90781—21no—
107#90781—21no—
108#90781D411yescharacteristic
109#90781D411yescharacteristic
110C4⋊Dic29—21no—
111C4⋊Dic29—21no—
112C4⋊Dic29—21no—
113C4⋊Dic29—21no—
114C4⋊Dic29D411yescharacteristic
115C4⋊Dic29C2311yescharacteristic
116#90779—21no—
117C2×#6597C2311yescharacteristic
118C2×#6597C2311yescharacteristic
119#90773—22no—
120#90773—22no—
121#90772—22no—
122C8×D29—22no—
123C23×C29—21no—
124C22×D29—21no—
125C22×D29—42no—
126C22×D29—42no—
127C22×D29—81no—
128C22×D29C2×D411yescharacteristic
129C22×D29C2×D411yescharacteristic
130Q8×C29—21no—
131Q8×C29—21no—
132Q8×C29—21no—
133Q8×C29C2×D412yes—
134D4×C29—21no—
135D4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—21no—
139C2×C4×C29C2×D411yescharacteristic
140C2×C4×C29C2×D411yescharacteristic
141C2×C4×C29C2411yescharacteristic
142C22⋊D29—42no—
143C22⋊D29—81no—
144C2×Dic29—21no—
145C2×Dic29—21no—
146C2×Dic29—21no—
147C2×Dic29C2×D411yescharacteristic
148C2×Dic29C2×D411yescharacteristic
149D116—22no—
150D116—21no—
151D116—42no—
152D116C2×D412yes—
153C4×D29—21no—
154C4×D29—21no—
155C4×D29—21no—
156C4×D29—21no—
157C4×D29—42no—
158C4×D29—42no—
159C4×D29—42no—
160C4×D29—81no—
161Dic58—21no—
162Dic58—21no—
163Dic58—21no—
164Dic58—21no—
165Dic58—42no—
166Dic58—42no—
167C8×C29—21no—
168C8×C29—21no—
169#6597—21no—
170#6597C2×D412yes—
171#3266—291nomaximal, Sylow
172C22×C29—41no—
173C22×C29—41no—
174C22×C29—41no—
175C22×C29C22×D411yescharacteristic
176C2×D29—21no—
177C2×D29—21no—
178C2×D29—21no—
179C2×D29—22no—
180C2×D29—21no—
181C2×D29—21no—
182C2×D29—41no—
183C2×D29—41no—
184C2×D29—84no—
185C2×D29#13712yescomplemented
186C4×C29—22no—
187C4×C29—21no—
188C4×C29—21no—
189C4×C29—41no—
190C4×C29—41no—
191C4×C29C22×D411yescharacteristic
192C4×C29H(5,𝔽2)11yescharacteristic
193Dic29—21no—
194Dic29—21no—
195Dic29—41no—
196Dic29—41no—
197Dic29—41no—
198C2×#137—291no—
199C22×SD16—291no—
200D4×Q8—291no—
201#546—291no—
202#497—291no—
203#468—291no—
204#467—291no—
205#464D2911yescharacteristic, complemented
206#461—291no—
207#452—291no—
208#451—291no—
209#450—291no—
210#448—291no—
211#440—291no—
212#435—291no—
213C2×C29—81no—
214C2×C29D4211yescharacteristic
215C2×C29C2×#13711yescharacteristic
216C2×C29#57711yescharacteristic
217D29—22no—
218D29—41no—
219D29—42no—
220C2×#41—291no—
221C22×Q8—291no—
222C22×Q8—291no—
223C22×D4—291no—
224#137—584no—
225C2×Q16—291no—
226C2×Q16—291no—
227C2×Q16C2×D2911yescharacteristic, complemented
228C2×SD16—291no—
229C2×SD16—291no—
230C2×SD16—291no—
231C2×SD16—291no—
232C2×SD16—291no—
233C2×SD16—584no—
234C2×SD16C2×D2911yescharacteristic
235C2×M16—291no—
236C22×C8—291no—
237C4⋊Q8—291no—
238C4⋊Q8—581no—
239#124C2×D2911yescharacteristic
240#123—581no—
241C22⋊Q8—291no—
242C22⋊Q8—291no—
243C22⋊Q8—291no—
244C22⋊Q8—581no—
245C22⋊Q8—581no—
246#121—291no—
247#121—581no—
248#120—581no—
249C4×Q8C2×D2911yescharacteristic, complemented
250C4×D4—291no—
251C4×D4—291no—
252C4×D4—581no—
253C2×C22⋊C4—581no—
254#107—291no—
255C4⋊C8C2×D2911yescharacteristic, complemented
256#103—291no—
257#103—291no—
258#103—291no—
259#103C2×D2911yescharacteristic, complemented
260#102—291no—
261#102C2×D2911yescharacteristic
262C22⋊C8—291no—
263C22⋊C8—291no—
264#97—291no—
265C29#326611yescharacteristic, complemented, Sylow
266C24—581no—
267#41—582no—
268#41—1161no—
269C2×Q8—291no—
270C2×Q8—291no—
271C2×Q8—581no—
272C2×Q8—581no—
273C2×Q8—582no—
274C2×Q8—582no—
275C2×Q8—1161no—
276C2×Q8—1161no—
277C2×Q8C22×D2911yescharacteristic
278C2×Q8C22×D2911yescharacteristic
279C2×D4—291no—
280C2×D4—291no—
281C2×D4—581no—
282C2×D4—581no—
283C2×D4—582no—
284C2×D4—581no—
285C2×D4—1161no—
286C2×D4C22×D2911yescharacteristic
287C22×C4—291no—
288C22×C4—291no—
289C22×C4—581no—
290C22×C4—581no—
291C22×C4—581no—
292C22×C4—581no—
293Q16—22no—
294Q16—582no—
295Q16—1161no—
296SD16—22no—
297SD16—294no—
298SD16—582no—
299SD16—582no—
300SD16—582no—
301SD16—1161no—
302M16—582no—
303C2×C8—291no—
304C2×C8—291no—
305C2×C8—582no—
306C2×C8C22×D2911yescharacteristic
307C2×C8C22×D2911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—291no—
310C4⋊C4—291no—
311C4⋊C4—581no—
312C4⋊C4—581no—
313C4⋊C4—581no—
314C4⋊C4—581no—
315C4⋊C4—581no—
316C4⋊C4C22×D2911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—291no—
320C22⋊C4—291no—
321C22⋊C4—581no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C42—21no—
327C42C22×D2911yescharacteristic
328C23—21no—
329C23—291no—
330C23—291no—
331C23—581no—
332C23—1162no—
333C23—1162no—
334C23—2321no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—581no—
339Q8—581no—
340Q8—581no—
341Q8—581no—
342Q8—1162no—
343Q8—1162no—
344Q8D4×D2912yes—
345D4—21no—
346D4—21no—
347D4—292no—
348D4—582no—
349D4—581no—
350D4—1162no—
351D4—1162no—
352D4—2321no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—291no—
357C2×C4—291no—
358C2×C4—581no—
359C2×C4—581no—
360C2×C4—581no—
361C2×C4—581no—
362C2×C4—581no—
363C2×C4—581no—
364C2×C4—581no—
365C2×C4—1162no—
366C2×C4—1162no—
367C2×C4—1162no—
368C2×C4—2321no—
369C2×C4D4×D2911yescharacteristic
370C2×C4D4×D2911yescharacteristic
371C2×C4C23×D2911yescharacteristic
372C8—21no—
373C8—21no—
374C8—292no—
375C8—581no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—292no—
380C22—581no—
381C22—581no—
382C22—581no—
383C22—582no—
384C22—581no—
385C22—581no—
386C22—1161no—
387C22—1161no—
388C22—2324no—
389C22C2×D4×D2911yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—581no—
396C4—581no—
397C4—1161no—
398C4—1161no—
399C4—1161no—
400C4C2×D4×D2911yescharacteristic
401C4#1174536411yescharacteristic
402C2—81no—
403C2—582no—
404C2—1161no—
405C2—1162no—
406C2#1427179711yescharacteristic
407C2#137×D2911yescharacteristic
408C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#464C2
socleC22×C29C22×D4
prime radical#14271806C2
hypercenter#464D29
hypocenterC29#3266
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#11745280C2semidirect
2. C2×SD16×D29C2semidirect
3. Q8×D116C2semidirect
5. #14271558C2semidirect
6. #14271408C2semidirect
7. #14271403C2semidirect
8. #14271369C2semidirect
9. #14271328C2semidirect
10. #14271317C2semidirect
11. #14271315C2semidirect
12. #14271300C2semidirect
13. #14271261C2semidirect
14. #14271233C2semidirect
15. C29×#464C2semidirect
18. C2×Q8×D29C22semidirect
20. C2×Q16×C29C22semidirect
23. C4×Q8×C29C22semidirect
25. C2×#90785C22semidirect
26. C2×#90785C22semidirect
35. C2×#90772C22semidirect
40. #11745259C22semidirect
49. #11745231C22semidirect
50. C29×C4⋊C8C22semidirect
51. C29×#103C22semidirect
53. #11745183C22semidirect
55. #11745168C22semidirect
56. #11745164C22semidirect
57. #11745164C22semidirect
59. #11745155C22semidirect
60. #11745151C22semidirect
101. #90784D4semidirect
185. C2×D29#137semidirect
205. #464D29semidirect
227. C2×Q16C2×D29semidirect
249. C4×Q8C2×D29semidirect
255. C4⋊C8C2×D29semidirect
259. #103C2×D29semidirect
265. C29#3266semidirect

Maximal subgroups

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

subgroupindex
C2×#117452802
C2×SD16×D292
Q8×D1162
#142718062
#142715582
#142714082
#142714032
#142713692
#142713282
#142713172
#142713152
#142713002
#142712612
#142712332
C29×#4642
#326629