Finite Group Databasebeta

← #16831078

Subgroups — #16831078

Order 3712.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745304C211yesmaximal, characteristic, complemented
2C2×#11745312C211yesmaximal, characteristic, complemented
3#14271834C211yesmaximal, characteristic, complemented
4#14271553C211yesmaximal, characteristic, complemented
5#14271414C211yesmaximal, characteristic, complemented
6#14271394C211yesmaximal, characteristic, complemented
7#14271382C211yesmaximal, characteristic, complemented
8#14271363C211yesmaximal, characteristic, complemented
9#14271323C211yesmaximal, characteristic, complemented
10#14271308C211yesmaximal, characteristic, complemented
11#14271306C211yesmaximal, characteristic, complemented
12#14271302C211yesmaximal, characteristic, complemented
13#14271268C211yesmaximal, characteristic, complemented
14#14271235C211yesmaximal, characteristic, complemented
15C29×#548C211yesmaximal, characteristic, complemented
16C2×#41×C29C2211yescharacteristic
17C22×Q8×C29C2211yescharacteristic
18C2×#90805C2211yescharacteristic
19C2×#90805C2211yescharacteristic
20C29×#124—21no—
21C29×#124C2211yescharacteristic
22C29×C22⋊Q8—21no—
23C29×C22⋊Q8C2211yescharacteristic
24C29×#121—21no—
25C29×#121C2211yescharacteristic
26C4×Q8×C29C2211yescharacteristic
27C4×D4×C29—21no—
28C4×D4×C29C2211yescharacteristic
29#11745312—24no—
30C2×#90785C2211yescharacteristic
31C2×#90785C2211yescharacteristic
32#11745304—24no—
33C2×#90784C2211yescharacteristic
34C2×#90784C2211yescharacteristic
35C2×#90784C2211yescharacteristic
36C2×#90784C2211yescharacteristic
37#11745302—21no—
38C2×#90783C2211yescharacteristic
39C2×#90782C2211yescharacteristic
40C2×#90782C2211yescharacteristic
41#11745292C2211yescharacteristic
42#11745292C2211yescharacteristic
43C2×#90777C2211yescharacteristic
44C22×#6597C2211yescharacteristic
45#11745255—21no—
46#11745246—21no—
47#11745246—21no—
48#11745236C2211yescharacteristic
49C4×Dic58C2211yescharacteristic
50#11745183C2211yescharacteristic
51#11745183C2211yescharacteristic
52#11745180C2211yescharacteristic
53C22⋊#6597C2211yescharacteristic
54C22⋊#6597C2211yescharacteristic
55#11745158C2211yescharacteristic
56#11745157C2211yescharacteristic
57#11745157C2211yescharacteristic
58#11745156C2211yescharacteristic
59C4⋊#6597C2211yescharacteristic
60#11745151C2211yescharacteristic
61#41×C29—22no—
62#41×C29—41no—
63C2×Q8×C29—21no—
64C2×Q8×C29—22no—
65C2×Q8×C29—41no—
66C2×Q8×C29C2311yescharacteristic
67C2×Q8×C29C2311yescharacteristic
68C2×D4×C29—21no—
69C2×D4×C29—21no—
70C2×D4×C29D411yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C22×C4×C29—21no—
73C22×C4×C29—21no—
74C22×C4×C29C2311yescharacteristic
75C22×C4×C29C2311yescharacteristic
76C2×C22⋊D29—21no—
77C2×C22⋊D29—21no—
78C2×C22⋊D29—21no—
79C2×C22⋊D29—21no—
80#90805—22no—
81#90805—22no—
82#90805—41no—
83#90805—41no—
84C2×D116—21no—
85C2×D116—21no—
86C2×D116C2311yescharacteristic
87C2×D116C2311yescharacteristic
88C2×C4×D29—21no—
89C2×C4×D29—21no—
90C2×Dic58C2311yescharacteristic
91C4⋊C4×C29—21no—
92C4⋊C4×C29—21no—
93C4⋊C4×C29D411yescharacteristic
94C4⋊C4×C29C2311yescharacteristic
95C22⋊C4×C29—21no—
96C22⋊C4×C29—21no—
97C22⋊C4×C29—21no—
98C22⋊C4×C29—21no—
99C22⋊C4×C29D411yescharacteristic
100C22⋊C4×C29D411yescharacteristic
101C42×C29—21no—
102C42×C29C2311yescharacteristic
103#90785—22no—
104#90785D414yescomplemented
105#90784—22no—
106#90784—22no—
107#90784—22no—
108#90784—41no—
109#90783—22no—
110#90782—22no—
111#90782—41no—
112#90781—21no—
113#90781—21no—
114C4⋊Dic29C2311yescharacteristic
115#90779—21no—
116C4×Dic29—21no—
117#90777—22no—
118C2×#6597—22no—
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122C2×#6597C2311yescharacteristic
123C23×C29—21no—
124C23×C29C2×D411yescharacteristic
125C23×C29C2×D411yescharacteristic
126C22×D29—21no—
127C22×D29—21no—
128Q8×C29—21no—
129Q8×C29—21no—
130Q8×C29—21no—
131Q8×C29—42no—
132Q8×C29C2×D412yes—
133D4×C29—21no—
134D4×C29—21no—
135D4×C29—41no—
136D4×C29—42no—
137D4×C29—42no—
138C2×C4×C29—21no—
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—42no—
146C2×C4×C29—42no—
147C2×C4×C29C2×D411yescharacteristic
148C2×C4×C29C2×D411yescharacteristic
149C2×C4×C29C2×D411yescharacteristic
150C2×C4×C29C2×D411yescharacteristic
151C2×C4×C29C2411yescharacteristic
152C22⋊D29—42no—
153C22⋊D29—42no—
154C22⋊D29—81no—
155C22⋊D29—81no—
156C2×Dic29—21no—
157C2×Dic29—21no—
158D116—21no—
159D116—21no—
160D116—21no—
161D116—21no—
162D116—42no—
163D116—81no—
164C4×D29—42no—
165C4×D29—42no—
166Dic58—21no—
167Dic58C2×D412yes—
168#6597—21no—
169#6597—21no—
170#6597—21no—
171#6597C2×D412yes—
172#3271—291nomaximal, Sylow
173C22×C29—21no—
174C22×C29—21no—
175C22×C29—21no—
176C22×C29—21no—
177C22×C29—21no—
178C22×C29—21no—
179C22×C29—41no—
180C22×C29—41no—
181C22×C29—41no—
182C22×C29C22×D411yescharacteristic
183C2×D29—41no—
184C2×D29—41no—
185C2×D29—41no—
186C2×D29—41no—
187C2×D29—41no—
188C2×D29—41no—
189C4×C29—21no—
190C4×C29—21no—
191C4×C29—22no—
192C4×C29—21no—
193C4×C29—21no—
194C4×C29—41no—
195C4×C29—41no—
196C4×C29C22×D411yescharacteristic
197C4×C29H(5,𝔽2)11yescharacteristic
198Dic29—22no—
199Dic29—41no—
200C2×#137—291no—
201C2×#135—291no—
202#550—291no—
203#548D2911yescharacteristic, complemented
204#496—291no—
205#469—291no—
206#465—291no—
207#464—291no—
208#460—291no—
209#452—291no—
210#449—291no—
211#449—291no—
212#448—291no—
213#441—291no—
214#435—291no—
215C2×C29—41no—
216C2×C29—41no—
217C2×C29—81no—
218C2×C29D4211yescharacteristic
219C2×C29#57511yescharacteristic
220C2×C29#57711yescharacteristic
221D29—81no—
222D29—81no—
223C2×#41—291no—
224C2×#41—291no—
225C2×#41C2×D2911yescharacteristic
226C22×Q8C2×D2911yescharacteristic
227#137—584no—
228#135—584no—
229C2×Q16—291no—
230C2×Q16—291no—
231C2×SD16—291no—
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—291no—
235C2×SD16—291no—
236C2×D8—291no—
237C2×D8—291no—
238C2×M16—291no—
239C22×C8—291no—
240#127—291no—
241#127—581no—
242#124—21no—
243#124C2×D2911yescharacteristic
244C22⋊Q8—21no—
245C22⋊Q8C2×D2911yescharacteristic
246#121—21no—
247#121—291no—
248#121—291no—
249#121—581no—
250#121—581no—
251#121C2×D2911yescharacteristic
252C4×Q8—291no—
253C4×Q8C2×D2911yescharacteristic
254C4×D4—21no—
255C4×D4—581no—
256C4×D4C2×D2911yescharacteristic
257#106—291no—
258C4⋊C8—291no—
259#103—291no—
260#103—291no—
261#103—291no—
262#102—291no—
263#102—291no—
264#102—291no—
265C22⋊C8—291no—
266C22⋊C8—291no—
267#97—291no—
268C29#327111yescharacteristic, complemented, Sylow
269#41—22no—
270#41—41no—
271#41—582no—
272#41—582no—
273#41—1161no—
274#41—1161no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—291no—
279C2×Q8C22×D2911yescharacteristic
280C2×Q8C22×D2911yescharacteristic
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—291no—
284C2×D4—291no—
285C2×D4—581no—
286C2×D4—581no—
287C2×D4—581no—
288C2×D4—581no—
289C2×D4—581no—
290C2×D4—581no—
291C2×D4C22⋊D2911yescharacteristic
292C2×D4C22×D2911yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—581no—
296C22×C4—581no—
297C22×C4C22×D2911yescharacteristic
298C22×C4C22×D2911yescharacteristic
299Q16—294no—
300Q16—582no—
301SD16—582no—
302SD16—582no—
303SD16—582no—
304SD16—582no—
305SD16—1161no—
306D8—582no—
307D8—1161no—
308M16—582no—
309C2×C8—291no—
310C2×C8—291no—
311C2×C8—291no—
312C2×C8—291no—
313C2×C8—582no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—291no—
317C4⋊C4—581no—
318C4⋊C4C22⋊D2911yescharacteristic
319C4⋊C4C22×D2911yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C22⋊C4C22⋊D2911yescharacteristic
327C22⋊C4C22⋊D2911yescharacteristic
328C42—21no—
329C42—581no—
330C42C22×D2911yescharacteristic
331C23—21no—
332C23—581no—
333C23—581no—
334C23C2×C22⋊D2911yescharacteristic
335C23C2×C22⋊D2911yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—42no—
340Q8—292no—
341Q8—581no—
342Q8D4×D2912yes—
343D4—21no—
344D4—21no—
345D4—41no—
346D4—42no—
347D4—42no—
348D4—581no—
349D4—581no—
350D4—581no—
351D4—581no—
352D4—1162no—
353D4—1162no—
354D4—1162no—
355D4—2321no—
356D4—2321no—
357D4—2321no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—581no—
368C2×C4—581no—
369C2×C4—1162no—
370C2×C4—1162no—
371C2×C4C2×C22⋊D2911yescharacteristic
372C2×C4C2×C22⋊D2911yescharacteristic
373C2×C4C2×C22⋊D2911yescharacteristic
374C2×C4C2×C22⋊D2911yescharacteristic
375C2×C4C23×D2911yescharacteristic
376C8—292no—
377C8—581no—
378C8—581no—
379C8—581no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—1161no—
390C22—1161no—
391C22—1161no—
392C22—1161no—
393C22—1161no—
394C22—1161no—
395C22C22×C22⋊D2911yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—582no—
404C4—1161no—
405C4C2×D4×D2911yescharacteristic
406C4#1174535911yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—2321no—
411C2—2321no—
412C2D4×C22⋊D2911yescharacteristic
413C2#1427196111yescharacteristic
414C2#1427197111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#548C2
socleC22×C29C22×D4
hypercenter#548D29
hypocenterC29#3271
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#11745304C2semidirect
2. C2×#11745312C2semidirect
3. #14271834C2semidirect
4. #14271553C2semidirect
5. #14271414C2semidirect
6. #14271394C2semidirect
7. #14271382C2semidirect
8. #14271363C2semidirect
9. #14271323C2semidirect
10. #14271308C2semidirect
11. #14271306C2semidirect
12. #14271302C2semidirect
13. #14271268C2semidirect
14. #14271235C2semidirect
15. C29×#548C2semidirect
104. #90785D4semidirect
203. #548D29semidirect
268. C29#3271semidirect

Maximal subgroups

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

subgroupindex
C2×#117453042
C2×#117453122
#142718342
#142715532
#142714142
#142713942
#142713822
#142713632
#142713232
#142713082
#142713062
#142713022
#142712682
#142712352
C29×#5482
#327129