Finite Group Databasebeta

← #16830998

Subgroups — #16830998

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×#11745313C211yesmaximal, characteristic, complemented
2C22×#90783C211yesmaximal, characteristic, complemented
3D4×Dic58C211yesmaximal, characteristic, complemented
4#14271558C211yesmaximal, characteristic, complemented
5#14271407C211yesmaximal, characteristic, complemented
6#14271402C211yesmaximal, characteristic, complemented
7#14271382C211yesmaximal, characteristic, complemented
8#14271368C211yesmaximal, characteristic, complemented
9#14271325C211yesmaximal, characteristic, complemented
10#14271318C211yesmaximal, characteristic, complemented
11#14271314C211yesmaximal, characteristic, complemented
12#14271301C211yesmaximal, characteristic, complemented
13#14271263C211yesmaximal, characteristic, complemented
14#14271235C211yesmaximal, characteristic, complemented
15C29×#546C211yesmaximal, characteristic
16C2×#41×C29C2211yescharacteristic
17C22×D4×C29C2211yescharacteristic
18C22×Dic58C2211yescharacteristic
19C22×Dic58C2211yescharacteristic, complemented
20C29×#124C2211yescharacteristic
21C29×#123—21no—
22C29×C22⋊Q8C2211yescharacteristic
23C29×#121—21no—
24C29×#121C2211yescharacteristic
25C29×#120—21no—
26C4×D4×C29C2211yescharacteristic
27C4×D4×C29C2211yescharacteristic
28C2×C22⋊C4×C29—21no—
29#11745313—24no—
30C2×#90785C2211yescharacteristic, complemented
31C2×#90785C2211yescharacteristic, complemented
32C2×#90785C2211yescharacteristic, complemented
33D4×Dic29—21no—
34C2×#90783—24no—
35C2×#90783C2211yescharacteristic
36C2×#90783C2211yescharacteristic
37C2×#90783C2211yescharacteristic
38C2×#90783C2211yescharacteristic
39C2×#90783C2211yescharacteristic
40C2×#90783C2211yescharacteristic
41#11745286C2211yescharacteristic
42#11745286C2211yescharacteristic, complemented
43C2×#90777C2211yescharacteristic, complemented
44C22×#6597C2211yescharacteristic
45#11745250—21no—
46#11745240—21no—
47#11745240—21no—
48#11745231C2211yescharacteristic, complemented
49C4×Dic58C2211yescharacteristic, complemented
50#11745183C2211yescharacteristic, complemented
51#11745180C2211yescharacteristic
52#11745180C2211yescharacteristic
53C22⋊#6597C2211yescharacteristic
54C22⋊#6597C2211yescharacteristic, complemented
55#11745158C2211yescharacteristic, complemented
56#11745158C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745155C2211yescharacteristic, complemented
59C4⋊#6597C2211yescharacteristic, complemented
60#11745151C2211yescharacteristic, complemented
61C24×C29—21no—
62#41×C29—22no—
63#41×C29—41no—
64C2×Q8×C29C2311yescharacteristic
65C2×D4×C29—22no—
66C2×D4×C29—21no—
67C2×D4×C29—21no—
68C2×D4×C29—21no—
69C2×D4×C29—41no—
70C2×D4×C29D411yescharacteristic
71C2×D4×C29C2311yescharacteristic
72C2×D4×C29C2311yescharacteristic
73C22×C4×C29—21no—
74C22×C4×C29—21no—
75C22×C4×C29C2311yescharacteristic
76C22×C4×C29C2311yescharacteristic
77C22×Dic29—21no—
78C22×Dic29—21no—
79C2×Dic58—22no—
80C2×Dic58—22no—
81C2×Dic58—21no—
82C2×Dic58—21no—
83C2×Dic58—41no—
84C2×Dic58—41no—
85C2×Dic58C2311yescharacteristic
86C2×Dic58C2311yescharacteristic
87C2×Dic58C2311yescharacteristic
88C4⋊C4×C29—21no—
89C4⋊C4×C29D411yescharacteristic
90C4⋊C4×C29C2311yescharacteristic
91C22⋊C4×C29—21no—
92C22⋊C4×C29—21no—
93C22⋊C4×C29—21no—
94C22⋊C4×C29—21no—
95C22⋊C4×C29—21no—
96C22⋊C4×C29D411yescharacteristic
97C22⋊C4×C29D411yescharacteristic
98C42×C29C2311yescharacteristic
99C22⋊Dic29—21no—
100C22⋊Dic29—21no—
101#90785—22no—
102#90785—22no—
103#90785—41no—
104#90783—22no—
105#90783—22no—
106#90783—22no—
107#90783—22no—
108#90783—41no—
109#90783D414yescomplemented
110C4⋊Dic29—21no—
111C4⋊Dic29—21no—
112C4⋊Dic29C2311yescharacteristic
113#90779—21no—
114#90779—21no—
115#90779—21no—
116C4×Dic29—21no—
117#90777—22no—
118C2×#6597—22no—
119C2×#6597C2311yescharacteristic
120C2×#6597C2311yescharacteristic
121C2×#6597C2311yescharacteristic
122C2×#6597C2311yescharacteristic
123C23×C29—21no—
124C23×C29—21no—
125C23×C29—42no—
126C23×C29—42no—
127C23×C29—81no—
128C23×C29C2×D411yescharacteristic
129C23×C29C2×D411yescharacteristic
130Q8×C29—21no—
131Q8×C29—21no—
132D4×C29—22no—
133D4×C29—21no—
134D4×C29—21no—
135D4×C29—21no—
136D4×C29—42no—
137D4×C29—42no—
138D4×C29—81no—
139D4×C29C2×D412yes—
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—42no—
147C2×C4×C29—81no—
148C2×C4×C29C2×D411yescharacteristic
149C2×C4×C29C2×D411yescharacteristic
150C2×C4×C29C2×D411yescharacteristic
151C2×C4×C29C2×D411yescharacteristic
152C2×C4×C29C2411yescharacteristic
153C2×Dic29—21no—
154C2×Dic29—21no—
155C2×Dic29—21no—
156C2×Dic29—21no—
157C2×Dic29—42no—
158C2×Dic29—42no—
159Dic58—21no—
160Dic58—21no—
161Dic58—21no—
162Dic58—21no—
163Dic58—21no—
164Dic58—42no—
165Dic58—42no—
166Dic58C2×D412yes—
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—22no—
176C22×C29—21no—
177C22×C29—21no—
178C22×C29—41no—
179C22×C29—41no—
180C22×C29—41no—
181C22×C29—41no—
182C22×C29—41no—
183C22×C29—84no—
184C22×C29#13712yescomplemented
185C22×C29C22×D411yescharacteristic
186C4×C29—21no—
187C4×C29—21no—
188C4×C29—21no—
189C4×C29—21no—
190C4×C29—41no—
191C4×C29—41no—
192C4×C29C22×D411yescharacteristic
193C4×C29H(5,𝔽2)11yescharacteristic
194Dic29—22no—
195Dic29—41no—
196Dic29—41no—
197Dic29—41no—
198C2×#137—291no—
199C22×SD16—291no—
200D4×Q8—291no—
201#546D2911yescharacteristic
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×C29—42no—
216C2×C29—81no—
217C2×C29D4211yescharacteristic
218C2×C29C2×#13711yescharacteristic
219C2×C29#57711yescharacteristic
220C2×#41C2×D2911yescharacteristic
221C22×Q8—291no—
222C22×Q8—291no—
223C22×D4C2×D2911yescharacteristic
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—291no—
238C4⋊Q8—581no—
239#124C2×D2911yescharacteristic
240#123—21no—
241C22⋊Q8—291no—
242C22⋊Q8—291no—
243C22⋊Q8—581no—
244C22⋊Q8—581no—
245C22⋊Q8C2×D2911yescharacteristic
246#121—21no—
247#121C2×D2911yescharacteristic
248#120—21no—
249C4×Q8—291no—
250C4×D4—581no—
251C4×D4C2×D2911yescharacteristic
252C4×D4C2×D2911yescharacteristic
253C2×C22⋊C4—21no—
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—21no—
267#41—22no—
268#41—41no—
269C2×Q8—291no—
270C2×Q8—291no—
271C2×Q8—291no—
272C2×Q8—582no—
273C2×Q8—582no—
274C2×Q8—581no—
275C2×Q8—581no—
276C2×Q8—1161no—
277C2×Q8—1161no—
278C2×Q8C22×D2911yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D2911yescharacteristic
285C2×D4C22×D2911yescharacteristic
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—291no—
310C4⋊C4—581no—
311C4⋊C4—581no—
312C4⋊C4—581no—
313C4⋊C4—581no—
314C4⋊C4—581no—
315C4⋊C4C22⋊D2911yescharacteristic
316C4⋊C4C22×D2911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4C22⋊D2911yescharacteristic
325C22⋊C4C22⋊D2911yescharacteristic
326C42—581no—
327C42C22×D2911yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D2911yescharacteristic
334C23C2×C22⋊D2911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—292no—
338Q8—581no—
339Q8—581no—
340Q8—581no—
341Q8—581no—
342Q8—581no—
343Q8—1162no—
344Q8—1162no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D2912yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—581no—
362C2×C4—581no—
363C2×C4—581no—
364C2×C4—581no—
365C2×C4—1162no—
366C2×C4—1162no—
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—22no—
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#1174531312yescomplemented
389C22C22×C22⋊D2911yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—582no—
397C4—1161no—
398C4—1161no—
399C4—1161no—
400C4C2×D4×D2911yescharacteristic
401C4#1174535911yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D2911yescharacteristic
407C2C2×#1174531311yescharacteristic
408C2#1427197411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
Fitting, prime radicalC29×#546C2
socleC22×C29C22×D4
hypercenter#546D29
hypocenterC29#3266
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#11745313C2semidirect
2. C22×#90783C2semidirect
3. D4×Dic58C2semidirect
4. #14271558C2semidirect
5. #14271407C2semidirect
6. #14271402C2semidirect
7. #14271382C2semidirect
8. #14271368C2semidirect
9. #14271325C2semidirect
10. #14271318C2semidirect
11. #14271314C2semidirect
12. #14271301C2semidirect
13. #14271263C2semidirect
14. #14271235C2semidirect
19. C22×Dic58C22semidirect
30. C2×#90785C22semidirect
31. C2×#90785C22semidirect
32. C2×#90785C22semidirect
42. #11745286C22semidirect
43. C2×#90777C22semidirect
48. #11745231C22semidirect
49. C4×Dic58C22semidirect
50. #11745183C22semidirect
54. C22⋊#6597C22semidirect
55. #11745158C22semidirect
56. #11745158C22semidirect
57. #11745158C22semidirect
58. #11745155C22semidirect
59. C4⋊#6597C22semidirect
60. #11745151C22semidirect
109. #90783D4semidirect
184. C22×C29#137semidirect
265. C29#3266semidirect
388. C22#11745313semidirect

Maximal subgroups

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

subgroupindex
C2×#117453132
C22×#907832
D4×Dic582
#142715582
#142714072
#142714022
#142713822
#142713682
#142713252
#142713182
#142713142
#142713012
#142712632
#142712352
C29×#5462
#326629