Finite Group Databasebeta

← #16829171

Subgroups — #16829171

Order 3712.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745273C211yesmaximal, characteristic, complemented
2C2×#11745263C211yesmaximal, characteristic, complemented
3#14271742C211yesmaximal, characteristic, complemented
4#14271729C211yesmaximal, characteristic, complemented
5#14271414C211yesmaximal, characteristic, complemented
6#14271413C211yesmaximal, characteristic, complemented
7#14271410C211yesmaximal, characteristic, complemented
8#14271407C211yesmaximal, characteristic, complemented
9#14271398C211yesmaximal, characteristic, complemented
10#14271392C211yesmaximal, characteristic, complemented
11#14271391C211yesmaximal, characteristic, complemented
12#14271204C211yesmaximal, characteristic, complemented
13#14271156C211yesmaximal, characteristic, complemented
14#102×D29C211yesmaximal, characteristic, complemented
15C29×#466C211yesmaximal, characteristic, complemented
16C2×#90807C2211yescharacteristic
17C2×D4×D29C2211yescharacteristic
18C2×#90805C2211yescharacteristic
19C2×D8×C29C2211yescharacteristic
20C22×C8×C29C2211yescharacteristic, complemented
21C29×#121C2211yescharacteristic
22C29×#121C2211yescharacteristic, complemented
23#11745302—21no—
24#11745301—21no—
25#11745300—21no—
26#11745300C2211yescharacteristic
27#11745300C2211yescharacteristic, complemented
28#11745299—21no—
29C2×#90783C2211yescharacteristic
30C2×#90782C2211yescharacteristic
31#11745292C2211yescharacteristic
32#11745286C2211yescharacteristic, complemented
33C2×#90777C2211yescharacteristic, complemented
34#11745273—24no—
35C2×#90773C2211yescharacteristic
36#11745263—24no—
37C2×#90772C2211yescharacteristic
38C2×#90772C2211yescharacteristic, complemented
39C2×C8×D29C2211yescharacteristic
40C2×C8×D29C2211yescharacteristic, complemented
41#11745259C2211yescharacteristic, complemented
42#11745257C2211yescharacteristic
43#11745254C2211yescharacteristic
44C4⋊C4×D29C2211yescharacteristic
45#11745247—21no—
46#11745246—21no—
47#11745244—21no—
48#11745240—21no—
49C29×#107C2211yescharacteristic, complemented
50C29×#102C2211yescharacteristic
51C29×#102C2211yescharacteristic, complemented
52#11745180C2211yescharacteristic
53#11745180C2211yescharacteristic, complemented
54#11745169C2211yescharacteristic
55#11745165C2211yescharacteristic, complemented
56#11745164C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745157C2211yescharacteristic
59#11745156C2211yescharacteristic, complemented
60#11745155C2211yescharacteristic, complemented
61C23×D29—21no—
62C2×D4×C29—21no—
63C2×D4×C29—21no—
64C2×D4×C29C2311yescharacteristic
65C2×D4×C29C2311yescharacteristic
66C22×C4×C29C2311yescharacteristic
67C2×C22⋊D29—21no—
68C2×C22⋊D29—21no—
69C2×C22⋊D29—21no—
70C2×C22⋊D29—21no—
71C2×C22⋊D29—21no—
72C2×C22⋊D29D411yescharacteristic
73C2×C22⋊D29D411yescharacteristic
74C22×Dic29—21no—
75#90807—22no—
76#90807—22no—
77D4×D29—22no—
78D4×D29—22no—
79#90805—22no—
80#90805—22no—
81C2×D116—21no—
82C2×D116D411yescharacteristic
83C2×D116C2311yescharacteristic
84C2×C4×D29—21no—
85C2×C4×D29C2311yescharacteristic
86C2×C4×D29C2311yescharacteristic
87C2×Dic58D411yescharacteristic
88C2×Dic58C2311yescharacteristic
89D8×C29—22no—
90C2×C8×C29—22no—
91C2×C8×C29C2311yescharacteristic
92C2×C8×C29C2311yescharacteristic
93C4⋊C4×C29C2311yescharacteristic
94C4⋊C4×C29C2311yescharacteristic
95C22⋊C4×C29—21no—
96C22⋊C4×C29—21no—
97C22⋊Dic29—21no—
98C22⋊Dic29—21no—
99C22⋊Dic29—21no—
100#90783—22no—
101#90782—22no—
102#90781—21no—
103#90781—21no—
104#90781—21no—
105#90781—21no—
106C4⋊Dic29—21no—
107C4⋊Dic29C2311yescharacteristic
108C4⋊Dic29C2311yescharacteristic
109#90779—21no—
110#90779—21no—
111C4×Dic29—21no—
112#90777—22no—
113C2×#6597C2311yescharacteristic
114C2×#6597C2311yescharacteristic
115#90773—22no—
116#90772—22no—
117#90772D414yescomplemented
118C8×D29—22no—
119C8×D29—22no—
120C23×C29—21no—
121C23×C29—21no—
122C23×C29C2×D411yescharacteristic
123C22×D29—21no—
124C22×D29—42no—
125C22×D29—81no—
126C22×D29—81no—
127C22×D29C2×D411yescharacteristic
128C22×D29C2×D411yescharacteristic
129D4×C29—21no—
130D4×C29—21no—
131D4×C29—21no—
132D4×C29—21no—
133D4×C29—81no—
134D4×C29—81no—
135C2×C4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—21no—
139C2×C4×C29C2×D411yescharacteristic
140C2×C4×C29C2411yescharacteristic
141C22⋊D29—22no—
142C22⋊D29—22no—
143C22⋊D29—42no—
144C22⋊D29—42no—
145C22⋊D29—81no—
146C22⋊D29—81no—
147C22⋊D29—81no—
148C2×Dic29—21no—
149C2×Dic29—21no—
150C2×Dic29—21no—
151C2×Dic29—42no—
152C2×Dic29C2×D411yescharacteristic
153C2×Dic29C2×D411yescharacteristic
154D116—22no—
155D116—21no—
156D116—21no—
157D116—81no—
158C4×D29—21no—
159C4×D29—21no—
160C4×D29—81no—
161C4×D29C2×D412yes—
162C4×D29#4112yes—
163Dic58—22no—
164Dic58—21no—
165Dic58—21no—
166C8×C29—21no—
167C8×C29C2×D412yes—
168#6597—21no—
169#6597C2×D412yes—
170#3062—291nomaximal, Sylow
171C22×C29—21no—
172C22×C29—21no—
173C22×C29—21no—
174C22×C29—41no—
175C22×C29—41no—
176C22×C29—41no—
177C22×C29—41no—
178C22×C29—41no—
179C22×C29—41no—
180C22×C29C22×D411yescharacteristic
181C2×D29—21no—
182C2×D29—21no—
183C2×D29—21no—
184C2×D29—21no—
185C2×D29—21no—
186C2×D29—41no—
187C2×D29—41no—
188C2×D29—41no—
189C2×D29—84no—
190C2×D29#12112yes—
191C4×C29—21no—
192C4×C29—41no—
193C4×C29—41no—
194C4×C29C22×D411yescharacteristic
195C4×C29C2×#4111yescharacteristic
196Dic29—21no—
197Dic29—41no—
198Dic29—41no—
199Dic29—41no—
200Dic29#12112yes—
201C2×AGL(1,ℤ/8)—291no—
202C2×#131—291no—
203#537—291no—
204#535—291no—
205#469—291no—
206#469—291no—
207#468—291no—
208#468—291no—
209#466—291no—
210#466D2911yescharacteristic, complemented
211#465—291no—
212#465—291no—
213#428—291no—
214#416—291no—
215C2×#102—291no—
216C2×C29—41no—
217C2×C29—81no—
218C2×C29—81no—
219C2×C29C2×#12111yescharacteristic
220C2×C29#57611yescharacteristic
221C2×C29#57711yescharacteristic
222D29—22no—
223D29—41no—
224D29—81no—
225C2×#41—291no—
226C2×#41—291no—
227C22×D4—291no—
228AGL(1,ℤ/8)—584no—
229C2×SD16—291no—
230C2×SD16—291no—
231C2×D8—291no—
232C2×D8C2×D2911yescharacteristic
233#131—584no—
234C2×M16—291no—
235C2×M16—291no—
236C2×M16—291no—
237C22×C8—291no—
238C22×C8—291no—
239C22×C8C2×D2911yescharacteristic, complemented
240#127—581no—
241#124—581no—
242C22⋊Q8—291no—
243C22⋊Q8—291no—
244C22⋊Q8—581no—
245#121—291no—
246#121—291no—
247#121—291no—
248#121—291no—
249#121—581no—
250#121—581no—
251#121—581no—
252#121C2×D2911yescharacteristic
253#121C2×D2911yescharacteristic, complemented
254#120—581no—
255#120—581no—
256#117—291no—
257C2×C4⋊C4—291no—
258#107—291no—
259#107C2×D2911yescharacteristic, complemented
260#106—291no—
261#106—291no—
262#103—291no—
263#103—291no—
264#102—291no—
265#102—291no—
266#102—291no—
267#102—291no—
268#102C2×D2911yescharacteristic
269#102C2×D2911yescharacteristic, complemented
270C29#306211yescharacteristic, complemented, Sylow
271C24—581no—
272#41—582no—
273#41—582no—
274#41—582no—
275#41—582no—
276C2×Q8—291no—
277C2×Q8—291no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—291no—
281C2×D4—291no—
282C2×D4—291no—
283C2×D4—291no—
284C2×D4—581no—
285C2×D4—582no—
286C2×D4—582no—
287C2×D4—581no—
288C2×D4—581no—
289C2×D4—581no—
290C2×D4—581no—
291C2×D4—581no—
292C2×D4C22×D2911yescharacteristic
293C2×D4C22×D2911yescharacteristic
294C22×C4—291no—
295C22×C4—291no—
296C22×C4—581no—
297C22×C4—581no—
298C22×C4C22×D2911yescharacteristic
299SD16—582no—
300SD16—582no—
301D8—22no—
302D8—582no—
303M16—294no—
304M16—582no—
305M16—582no—
306C2×C8—22no—
307C2×C8—291no—
308C2×C8—291no—
309C2×C8—582no—
310C2×C8—582no—
311C2×C8C22×D2911yescharacteristic
312C2×C8C22×D2911yescharacteristic
313C4⋊C4—291no—
314C4⋊C4—291no—
315C4⋊C4—581no—
316C4⋊C4—581no—
317C4⋊C4—581no—
318C4⋊C4C22×D2911yescharacteristic
319C4⋊C4C22×D2911yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C22⋊C4—581no—
327C22⋊C4—581no—
328C22⋊C4—581no—
329C42—581no—
330C23—21no—
331C23—21no—
332C23—291no—
333C23—291no—
334C23—581no—
335C23—1162no—
336C23—2321no—
337C23—2321no—
338C23D4×D2911yescharacteristic
339Q8—582no—
340Q8—581no—
341Q8—581no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—582no—
349D4—582no—
350D4—582no—
351D4—581no—
352D4—581no—
353D4—1162no—
354D4—1162no—
355D4—2321no—
356D4—2321no—
357D4—2321no—
358D4—2321no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—291no—
364C2×C4—292no—
365C2×C4—292no—
366C2×C4—291no—
367C2×C4—581no—
368C2×C4—581no—
369C2×C4—581no—
370C2×C4—581no—
371C2×C4—581no—
372C2×C4—1162no—
373C2×C4—2321no—
374C2×C4D4×D2911yescharacteristic
375C2×C4C23×D2911yescharacteristic
376C8—21no—
377C8—292no—
378C8—581no—
379C8D4×D2912yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—292no—
390C22—581no—
391C22—581no—
392C22—581no—
393C22—581no—
394C22—581no—
395C22—1161no—
396C22—1161no—
397C22—1161no—
398C22—2324no—
399C22C2×D4×D2911yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—292no—
404C4—581no—
405C4—1161no—
406C4—1161no—
407C4—1161no—
408C4C2×D4×D2911yescharacteristic
409C4#41×D2911yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—582no—
414C2—1161no—
415C2—2321no—
416C2#121×D2911yescharacteristic
417C2#1427196411yescharacteristic
418C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#466C2
socleC22×C29C22×D4
hypercenter#466D29
hypocenterC29#3062
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#11745273C2semidirect
2. C2×#11745263C2semidirect
3. #14271742C2semidirect
4. #14271729C2semidirect
5. #14271414C2semidirect
6. #14271413C2semidirect
7. #14271410C2semidirect
8. #14271407C2semidirect
9. #14271398C2semidirect
10. #14271392C2semidirect
11. #14271391C2semidirect
12. #14271204C2semidirect
13. #14271156C2semidirect
14. #102×D29C2semidirect
15. C29×#466C2semidirect
20. C22×C8×C29C22semidirect
22. C29×#121C22semidirect
27. #11745300C22semidirect
32. #11745286C22semidirect
33. C2×#90777C22semidirect
38. C2×#90772C22semidirect
40. C2×C8×D29C22semidirect
41. #11745259C22semidirect
49. C29×#107C22semidirect
51. C29×#102C22semidirect
53. #11745180C22semidirect
55. #11745165C22semidirect
56. #11745164C22semidirect
57. #11745158C22semidirect
59. #11745156C22semidirect
60. #11745155C22semidirect
117. #90772D4semidirect
210. #466D29semidirect
239. C22×C8C2×D29semidirect
253. #121C2×D29semidirect
259. #107C2×D29semidirect
269. #102C2×D29semidirect
270. C29#3062semidirect

Maximal subgroups

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

subgroupindex
C2×#117452732
C2×#117452632
#142717422
#142717292
#142714142
#142714132
#142714102
#142714072
#142713982
#142713922
#142713912
#142712042
#142711562
#102×D292
C29×#4662
#306229