Finite Group Databasebeta

← #16833138

Subgroups — #16833138

Order 3712.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745365C211yesmaximal, characteristic, complemented
2C22×Q8×D29C211yesmaximal, characteristic, complemented
3Q8×D116C211yesmaximal, characteristic, complemented
4D4×Dic58C213yesmaximal, complemented
5Q8×C22⋊D29C211yesmaximal, characteristic, complemented
6#14271826C213yesmaximal, complemented
7#14271821C213yesmaximal, complemented
8#14271820C211yesmaximal, characteristic, complemented
9#14271819C213yesmaximal, complemented
10#14271736C213yesmaximal, complemented
11#14271733C213yesmaximal, complemented
12#14271673C213yesmaximal, complemented
13#14271672C213yesmaximal, complemented
14#14271639C211yesmaximal, characteristic, complemented
15C29×#548C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic, complemented
17C22×Q8×C29C2211yescharacteristic
18#11745365—28no—
19C2×Q8×D29—28no—
20C2×Q8×D29C2211yescharacteristic
21C2×Q8×D29C2213yes—
22C2×Q8×D29C2213yes—
23C2×Q8×D29C2211yescharacteristic
24C2×Q8×D29C2211yescharacteristic, complemented
25C2×#90807C2213yes—
26C2×#90807C2213yescomplemented
27C2×#90805C2213yes—
28C22×C4×D29C2213yes—
29C22×Dic58C2213yes—
30C22×Dic58C2213yescomplemented
31C29×#124C2213yescomplemented
32C29×C22⋊Q8C2213yescomplemented
33C29×#121C2213yes—
34C4×Q8×C29C2211yescharacteristic, complemented
35C4×D4×C29C2213yes—
36#11745308C2213yescomplemented
37Q8×Dic29C2211yescharacteristic, complemented
38#11745306C2213yescomplemented
39#11745301C2213yes—
40#11745300C2213yes—
41#11745299C2213yes—
42#11745298C2213yescomplemented
43D4×Dic29C2216yes—
44C4×C22⋊D29C2213yes—
45C4×C22⋊D29C2213yes—
46#11745286C2213yes—
47#11745286C2213yescomplemented
48#11745259C2213yes—
49#11745259C2213yescomplemented
50#11745258C2213yes—
51#11745256C2213yes—
52#11745254C2213yes—
53#11745250C2216yescomplemented
54#11745249C2213yescomplemented
55#11745247C2216yescomplemented
56#11745246C2213yes—
57#11745245C2213yes—
58#11745245C2216yes—
59#11745244C2211yescharacteristic
60#11745243C2216yes—
61#11745240C2213yes—
62#11745240C2216yescomplemented
63#11745240C2216yescomplemented
64C4×D116C2213yes—
65#11745234C2213yes—
66#11745231C2213yescomplemented
67C4×Dic58C2213yescomplemented
68C23×D29C2311yescharacteristic
69#41×C29—24no—
70C2×Q8×C29—24no—
71C2×Q8×C29C2313yes—
72C2×Q8×C29C2311yescharacteristic
73C2×D4×C29C2313yes—
74C2×D4×C29C2313yes—
75C22×C4×C29C2313yes—
76C22×C4×C29C2313yes—
77C2×C22⋊D29C2311yescharacteristic
78C2×C22⋊D29C2313yes—
79C2×C22⋊D29C2313yes—
80C2×C22⋊D29C2313yes—
81C22×Dic29C2313yes—
82C22×Dic29C2316yes—
83Q8×D29—24no—
84Q8×D29—24no—
85Q8×D29—212no—
86Q8×D29—212no—
87Q8×D29D418yescomplemented
88#90807—212no—
89#90807—212no—
90#90805—212no—
91C2×D116C2311yescharacteristic
92C2×D116C2311yescharacteristic
93C2×C4×D29—212no—
94C2×C4×D29C2313yes—
95C2×C4×D29C2313yes—
96C2×C4×D29C2313yes—
97C2×C4×D29C2313yes—
98C2×C4×D29C2313yes—
99C2×Dic58—212no—
100C2×Dic58—212no—
101C2×Dic58C2313yes—
102C2×Dic58C2313yes—
103C2×Dic58C2316yes—
104C2×Dic58C2313yes—
105C2×Dic58C2313yes—
106C4⋊C4×C29C2313yes—
107C4⋊C4×C29C2313yes—
108C22⋊C4×C29C2311yescharacteristic
109C22⋊C4×C29C2313yes—
110C22⋊C4×C29C2316yes—
111C42×C29C2313yes—
112C22⋊Dic29C2313yes—
113C22⋊Dic29C2316yes—
114C22⋊Dic29C2313yes—
115#90781C2311yescharacteristic
116#90781C2313yes—
117#90781C2313yes—
118#90781C2313yes—
119#90781C2313yes—
120#90781C2311yescharacteristic
121C4⋊Dic29C2313yes—
122C4⋊Dic29C2316yes—
123C4⋊Dic29C2313yes—
124#90779C2316yes—
125#90779C2313yes—
126#90779C2313yes—
127#90779C2313yes—
128#90779C2313yes—
129C4×Dic29C2313yes—
130C4×Dic29C2316yes—
131C23×C29C2413yes—
132C23×C29C2411yescharacteristic
133C22×D29—22no—
134C22×D29—22no—
135C22×D29—22no—
136C22×D29C2411yescharacteristic
137C22×D29C2411yescharacteristic
138C22×D29C2411yescharacteristic
139Q8×C29—26no—
140Q8×C29C2×D414yes—
141D4×C29—26no—
142D4×C29—26no—
143C2×C4×C29—26no—
144C2×C4×C29—26no—
145C2×C4×C29C2413yes—
146C2×C4×C29C2411yescharacteristic
147C2×C4×C29C2413yes—
148C2×C4×C29C2411yescharacteristic
149C2×C4×C29C2413yes—
150C22⋊D29—22no—
151C22⋊D29—26no—
152C22⋊D29—26no—
153C22⋊D29—26no—
154C2×Dic29—26no—
155C2×Dic29—212no—
156C2×Dic29C2411yescharacteristic
157C2×Dic29C2413yes—
158C2×Dic29C2413yes—
159C2×Dic29C2413yes—
160C2×Dic29C2413yes—
161D116—22no—
162D116—22no—
163C4×D29—26no—
164C4×D29—26no—
165C4×D29—26no—
166C4×D29—26no—
167C4×D29C2×D4112yes—
168Dic58—26no—
169Dic58—26no—
170Dic58—26no—
171Dic58—212no—
172Dic58C2×D4112yes—
173#3486—291nomaximal, Sylow
174C22×C29—21no—
175C22×C29—21no—
176C22×C29—23no—
177C22×C29—23no—
178C22×C29—23no—
179C22×C292−1+412yescomplemented
180C22×C29C2511yescharacteristic
181C2×D29—28no—
182C2×D29—21no—
183C2×D29—21no—
184C2×D29—21no—
185C2×D29—21no—
186C2×D29—21no—
187C2×D29—21no—
188C2×D29—21no—
189C2×D29C22×D412yes—
190C2×D292−1+412yescomplemented
191C4×C29—21no—
192C4×C29—23no—
193C4×C29—21no—
194C4×C29—23no—
195C4×C29C22×D416yes—
196Dic29—23no—
197Dic29—23no—
198Dic29—23no—
199Dic29—21no—
200Dic29C22×D416yes—
201C2×2−1+4—291no—
202C23×Q8—291no—
203D4×Q8—291no—
204D4×Q8—293no—
205D4×Q8—291no—
206D4×Q8—293no—
207#548—291no—
208#548—293no—
209#548—293no—
210#548D2911yescharacteristic, complemented
211#536—293no—
212#536—293no—
213#525—293no—
214#525—293no—
215#519—291no—
216C2×C29—22no—
217C2×C29—43no—
218C2×C29C23×D411yescharacteristic
219C2×C29C2×2−1+411yescharacteristic
220C2×C29C2×2−1+411yescharacteristic
221D29—22no—
222D29—22no—
223D29—41no—
2242−1+4—588no—
225C2×#41—293no—
226C2×#41—293no—
227C2×#41—293no—
228C2×#41C2×D2911yescharacteristic, complemented
229C22×Q8—291no—
230C22×Q8—293no—
231C22×Q8—293no—
232C22×Q8—291no—
233C22×Q8—291no—
234C22×Q8—293no—
235C22×Q8—293no—
236C22×Q8—588no—
237C22×Q8C2×D2911yescharacteristic
238C23×C4—293no—
239C4⋊Q8—293no—
240C4⋊Q8—296no—
241C4⋊Q8—293no—
242#124—293no—
243#124—296no—
244#124C2×D2913yescomplemented
245#123—293no—
246#123—296no—
247#123—293no—
248C22⋊Q8—293no—
249C22⋊Q8—293no—
250C22⋊Q8—293no—
251C22⋊Q8—293no—
252C22⋊Q8—293no—
253C22⋊Q8—293no—
254C22⋊Q8—293no—
255C22⋊Q8—296no—
256C22⋊Q8—296no—
257C22⋊Q8C2×D2913yescomplemented
258#121—293no—
259#121—293no—
260#121—293no—
261#121C2×D2913yes—
262#120—293no—
263#120—291no—
264C4×Q8—291no—
265C4×Q8—293no—
266C4×Q8—293no—
267C4×Q8C2×D2911yescharacteristic, complemented
268C4×D4—296no—
269C4×D4—293no—
270C4×D4—293no—
271C4×D4—296no—
272C4×D4—293no—
273C4×D4C2×D2913yes—
274#117—293no—
275#117—293no—
276C29#348611yescharacteristic, complemented, Sylow
277C24—291no—
278#41—24no—
279#41—5812no—
280#41—5812no—
281#41—5812no—
282C2×Q8—24no—
283C2×Q8—298no—
284C2×Q8—293no—
285C2×Q8—296no—
286C2×Q8—293no—
287C2×Q8—293no—
288C2×Q8—293no—
289C2×Q8—5812no—
290C2×Q8—5812no—
291C2×Q8—5812no—
292C2×Q8—5812no—
293C2×Q8—584no—
294C2×Q8—584no—
295C2×Q8C22×D2913yes—
296C2×Q8C22×D2911yescharacteristic
297C2×D4—293no—
298C2×D4—291no—
299C2×D4—293no—
300C2×D4—293no—
301C2×D4—291no—
302C2×D4—291no—
303C2×D4C22×D2913yes—
304C2×D4C22×D2913yes—
305C22×C4—293no—
306C22×C4—293no—
307C22×C4—296no—
308C22×C4—293no—
309C22×C4—293no—
310C22×C4—293no—
311C22×C4—293no—
312C22×C4—5812no—
313C22×C4C22×D2913yes—
314C22×C4C22×D2913yes—
315C4⋊C4—293no—
316C4⋊C4—293no—
317C4⋊C4—293no—
318C4⋊C4—293no—
319C4⋊C4—293no—
320C4⋊C4—296no—
321C4⋊C4—296no—
322C4⋊C4—293no—
323C4⋊C4C22×D2913yes—
324C4⋊C4C22×D2913yes—
325C22⋊C4—293no—
326C22⋊C4—296no—
327C22⋊C4—293no—
328C22⋊C4—293no—
329C22⋊C4—291no—
330C22⋊C4—291no—
331C22⋊C4—293no—
332C22⋊C4—293no—
333C22⋊C4—293no—
334C22⋊C4C22×D2916yes—
335C22⋊C4C22×D2911yescharacteristic
336C22⋊C4C22×D2913yes—
337C42—296no—
338C42—293no—
339C42C22×D2913yes—
340C23—291no—
341C23—291no—
342C23—291no—
343C23—582no—
344C23—582no—
345C23—582no—
346C23C23×D2913yes—
347C23C23×D2911yescharacteristic
348Q8—26no—
349Q8—2912no—
350Q8—5812no—
351Q8—586no—
352Q8—586no—
353Q8—586no—
354Q8D4×D2914yes—
355D4—26no—
356D4—26no—
357D4—582no—
358D4—582no—
359D4—586no—
360D4—586no—
361D4—586no—
362D4—582no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—293no—
366C2×C4—2912no—
367C2×C4—293no—
368C2×C4—291no—
369C2×C4—293no—
370C2×C4—293no—
371C2×C4—586no—
372C2×C4—586no—
373C2×C4—586no—
374C2×C4—5812no—
375C2×C4—586no—
376C2×C4—586no—
377C2×C4C23×D2913yes—
378C2×C4C23×D2911yescharacteristic
379C2×C4C23×D2913yes—
380C2×C4C23×D2911yescharacteristic
381C2×C4C23×D2913yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—292no—
388C22—292no—
389C22—588no—
390C22—581no—
391C22—581no—
392C22—581no—
393C22—581no—
394C22—581no—
395C22—581no—
396C22—581no—
397C22#1174536512yescomplemented
398C22C24×D2911yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—296no—
404C4—583no—
405C4—583no—
406C4—583no—
407C4—581no—
408C4C2×D4×D2916yes—
409C2—22no—
410C2—43no—
411C2—582no—
412C2—582no—
413C2—1161no—
414C2C22×D4×D2911yescharacteristic
415C22−1+4×D2911yescharacteristic
416C2C2×#1174536511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D29
commutator, socle, cosocleC22×C29C25
FittingC29×#548C2
hypercenter#548D29
hypocenterC29#3486
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#11745365C2semidirect
2. C22×Q8×D29C2semidirect
3. Q8×D116C2semidirect
4. D4×Dic58C2semidirect
5. Q8×C22⋊D29C2semidirect
6. #14271826C2semidirect
7. #14271821C2semidirect
8. #14271820C2semidirect
9. #14271819C2semidirect
10. #14271736C2semidirect
11. #14271733C2semidirect
12. #14271673C2semidirect
13. #14271672C2semidirect
14. #14271639C2semidirect
15. C29×#548C2semidirect
16. C2×#41×C29C22semidirect
24. C2×Q8×D29C22semidirect
26. C2×#90807C22semidirect
30. C22×Dic58C22semidirect
31. C29×#124C22semidirect
32. C29×C22⋊Q8C22semidirect
34. C4×Q8×C29C22semidirect
36. #11745308C22semidirect
37. Q8×Dic29C22semidirect
38. #11745306C22semidirect
42. #11745298C22semidirect
47. #11745286C22semidirect
49. #11745259C22semidirect
53. #11745250C22semidirect
54. #11745249C22semidirect
55. #11745247C22semidirect
62. #11745240C22semidirect
63. #11745240C22semidirect
66. #11745231C22semidirect
67. C4×Dic58C22semidirect
87. Q8×D29D4semidirect
179. C22×C292−1+4semidirect
190. C2×D292−1+4semidirect
210. #548D29semidirect
228. C2×#41C2×D29semidirect
244. #124C2×D29semidirect
257. C22⋊Q8C2×D29semidirect
267. C4×Q8C2×D29semidirect
276. C29#3486semidirect
397. C22#11745365semidirect

Maximal subgroups

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

subgroupindex
C2×#117453652
C22×Q8×D292
Q8×D1162
D4×Dic582
Q8×C22⋊D292
#142718262
#142718212
#142718202
#142718192
#142717362
#142717332
#142716732
#142716722
#142716392
C29×#5482
#348629