Finite Group Databasebeta

← #18011085

Subgroups — #18011085

Order 4736.

696 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#3866C212yesmaximal, complemented
2#14558483C212yesmaximal, complemented
3#14558482C212yesmaximal, complemented
4#14558481C212yesmaximal, complemented
5#14558481C212yesmaximal, complemented
6#14558480C212yesmaximal, complemented
7#14558475C211yesmaximal, characteristic, complemented
8C2×(C2×#3866)⋊C4C211yesmaximal, characteristic, complemented
9#121×D37C211yesmaximal, characteristic
10C23×#3866C2211yescharacteristic
11C23×#3866C2212yes—
12C23×#3866C2211yescharacteristic
13D4×#3866—28no—
14C2×C4⋊#3866C2212yes—
15C2×C4⋊#3866C2211yescharacteristic
16C2×C4⋊#3866C2211yescharacteristic
17C2×C22⋊#3866C2212yes—
18C2×C22⋊#3866C2212yes—
19C2×C22⋊#3866C2212yes—
20C2×C22⋊#3866C2212yes—
21C2×C4×#3866C2212yes—
22(C2×#3866)⋊C4C2212yes—
23(C2×#3866)⋊C4C2212yes—
24(C2×#3866)⋊C4C2212yes—
25(C2×#3866)⋊C4C2212yes—
26(C2×#3866)⋊C4C2212yes—
27C2×D4×D37C2212yes—
28C2×D4×D37C2211yescharacteristic
29C22×C4×D37C2211yescharacteristic
30#11924511C411yescharacteristic, complemented
31#11924510C411yescharacteristic, complemented
32#11924509C412yescomplemented
33#11924508C412yescomplemented
34#11924507C411yescharacteristic, complemented
35C4⋊C4×D37C2211yescharacteristic
36C22⋊C4×D37C2212yes—
37#121×C37C411yescharacteristic, complemented
38C23×D37C2312yes—
39C23×D37C2311yescharacteristic
40C22×#3866—22no—
41C22×#3866—24no—
42C22×#3866—41no—
43C22×#3866—41no—
44C22×#3866—42no—
45C22×#3866—42no—
46C22×#3866—41no—
47C22×#3866—41no—
48C22×#3866D414yescomplemented
49C22×#3866C2312yes—
50C22×#3866C2312yes—
51C22×#3866C2312yes—
52C22×#3866C2312yes—
53C2×D4×C37C2×C412yescomplemented
54C2×D4×C37C2×C411yescharacteristic, complemented
55C22×C4×C37C2×C411yescharacteristic, complemented
56C2×C22⋊D37C2×C412yescomplemented
57C2×C22⋊D37C2×C412yescomplemented
58C2×C22⋊D37C2×C412yescomplemented
59C22×Dic37C2×C411yescharacteristic, complemented
60D4×D37—24no—
61D4×D37—24no—
62D4×D37—41no—
63D4×D37—41no—
64C2×D148C2×C412yescomplemented
65C2×D148C2×C411yescharacteristic, complemented
66C2×C4×D37—41no—
67C2×C4×D37—41no—
68C2×C4×D37C2×C411yescharacteristic, complemented
69C2×C4×D37C2×C411yescharacteristic, complemented
70C2×C4×D37C2311yescharacteristic
71C2×C4×D37C2312yes—
72C2×C4×D37C2311yescharacteristic
73C22⋊#3866—24no—
74C22⋊#3866—24no—
75C22⋊#3866—42no—
76C22⋊#3866—42no—
77C4⋊#3866—24no—
78C4⋊#3866—41no—
79C4⋊#3866—41no—
80C4×#3866—24no—
81C4⋊C4×C37C2×C411yescharacteristic, complemented
82C22⋊C4×C37C2×C412yescomplemented
83C22⋊Dic37C2×C412yescomplemented
84#10596938C2×C412yescomplemented
85#10596938C2×C412yescomplemented
86C4⋊Dic37C2×C411yescharacteristic, complemented
87#10596936C2×C412yescomplemented
88C23×C37C22×C412yes—
89C23×C37C22×C411yescharacteristic
90C22×D37—21no—
91C22×D37—21no—
92C22×D37—21no—
93C22×D37—21no—
94C22×D37—22no—
95C22×D37—22no—
96C22×D37—22no—
97C22×D37—22no—
98C22×D37—22no—
99C22×D37—22no—
100C22×D37C22×C412yes—
101C22×D37C22×C411yescharacteristic
102C22×D37C2×D412yes—
103C22×D37#4112yes—
104C22×D37C2411yescharacteristic
105C2×#3866—22no—
106C2×#3866—22no—
107C2×#3866—22no—
108C2×#3866—22no—
109C2×#3866—22no—
110C2×#3866—22no—
111C2×#3866—22no—
112C2×#3866—22no—
113C2×#3866—22no—
114C2×#3866—22no—
115C2×#3866—22no—
116C2×#3866—44no—
117C2×#3866—48no—
118C2×#3866—44no—
119C2×#3866C2×D414yes—
120D4×C37—24no—
121D4×C37—41no—
122C2×C4×C37—41no—
123C2×C4×C37C22×C412yes—
124C2×C4×C37C22×C411yescharacteristic
125C2×C4×C37C22×C411yescharacteristic
126C22⋊D37—24no—
127C22⋊D37—24no—
128C22⋊D37—42no—
129C2×Dic37—41no—
130C2×Dic37C22×C411yescharacteristic
131C2×Dic37C22×C412yes—
132C2×Dic37C22×C411yescharacteristic
133D148—24no—
134D148—41no—
135C4×D37—22no—
136C4×D37—22no—
137C4×D37—21no—
138C4×D37—21no—
139C4×D37—21no—
140C4×D37—21no—
141C4×D37—41no—
142C4×D37—41no—
143C22×C37—21no—
144C22×C37—21no—
145C22×C37—22no—
146C22×C37—22no—
147C22×C37—22no—
148C22×C37C4×D412yes—
149C22×C37C23×C411yescharacteristic
150C2×D37—21no—
151C2×D37—21no—
152C2×D37—22no—
153C2×D37—22no—
154C2×D37—22no—
155C2×D37—22no—
156C2×D37—22no—
157C2×D37—22no—
158C2×D37—22no—
159C2×D37—42no—
160C2×D37—42no—
161C2×D37—42no—
162C2×D37—42no—
163C2×D37C4×D412yes—
164C2×D37C22×D411yescharacteristic
165C2×D37C2×#4111yescharacteristic
166C2×D37H(5,𝔽2)11yescharacteristic
167C2×D37H(5,𝔽2)11yescharacteristic
168C2×D37H(5,𝔽2)11yescharacteristic
169C2×D372−1+411yescharacteristic
170#3866—24no—
171#3866—42no—
172#3866—42no—
173#3866—42no—
174C4×C37—22no—
175C4×C37—21no—
176C4×C37—21no—
177Dic37—22no—
178Dic37—21no—
179Dic37—21no—
180#2325—371nomaximal, Sylow
181C2×C37—22no—
182C2×C37—42no—
183C2×C37C2×C4×D411yescharacteristic
184C2×C37#51811yescharacteristic
185C2×C37#52011yescharacteristic
186D37—22no—
187D37—42no—
188D37#53411yescharacteristic
189D37#53711yescharacteristic
190D37#53811yescharacteristic
191D37#53911yescharacteristic
192C2×#121—371no—
193C2×C4×D4—372no—
194#388—372no—
195#387—372no—
196#386—372no—
197#386—372no—
198#385—372no—
199#380—371no—
200C2×#95—371no—
201C37#232511yescharacteristic, complemented, Sylow
202C22×D4—372no—
203C22×D4—371no—
204C23×C4—371no—
205C23×C4—371no—
206C23×C4—372no—
207C23×C4—371no—
208#121—371no—
209#121—372no—
210#121—371no—
211#121—371no—
212#121—372no—
213#121#386611yescharacteristic, complemented
214C4×D4—748no—
215C2×C4⋊C4—371no—
216C2×C4⋊C4—372no—
217C2×C4⋊C4—371no—
218C2×C4⋊C4—371no—
219C2×C22⋊C4—372no—
220C2×C22⋊C4—372no—
221C2×C22⋊C4—372no—
222C2×C22⋊C4—372no—
223C2×C22⋊C4—372no—
224C2×C42—372no—
225#95—372no—
226#95—372no—
227#95—372no—
228#95—372no—
229#95—372no—
230C24—371no—
231C24—372no—
232C2×D4—372no—
233C2×D4—372no—
234C2×D4—372no—
235C2×D4—372no—
236C2×D4—371no—
237C2×D4—744no—
238C2×D4—744no—
239C2×D4—1481no—
240C2×D4—1481no—
241C2×D4C2×#386612yescomplemented
242C2×D4C2×#386611yescharacteristic, complemented
243C22×C4—372no—
244C22×C4—374no—
245C22×C4—371no—
246C22×C4—371no—
247C22×C4—372no—
248C22×C4—371no—
249C22×C4—371no—
250C22×C4—371no—
251C22×C4—372no—
252C22×C4—372no—
253C22×C4—372no—
254C22×C4—744no—
255C22×C4—742no—
256C22×C4—1481no—
257C22×C4—1481no—
258C22×C4—1481no—
259C22×C4—1481no—
260C22×C4—1482no—
261C22×C4—1482no—
262C22×C4—1481no—
263C22×C4—1481no—
264C22×C4C2×#386611yescharacteristic, complemented
265C4⋊C4—372no—
266C4⋊C4—371no—
267C4⋊C4—744no—
268C4⋊C4—1481no—
269C4⋊C4—1481no—
270C4⋊C4C2×#386611yescharacteristic, complemented
271C22⋊C4—372no—
272C22⋊C4—372no—
273C22⋊C4—372no—
274C22⋊C4—744no—
275C22⋊C4—744no—
276C22⋊C4—1482no—
277C22⋊C4—1482no—
278C22⋊C4C2×#386612yescomplemented
279C42—744no—
280C23—371no—
281C23—372no—
282C23—372no—
283C23—371no—
284C23—372no—
285C23—741no—
286C23—741no—
287C23—741no—
288C23—741no—
289C23—742no—
290C23—742no—
291C23—742no—
292C23—742no—
293C23—742no—
294C23—742no—
295C23C22×#386612yes—
296C23C22×#386611yescharacteristic
297D4—24no—
298D4—41no—
299D4—744no—
300D4—744no—
301D4—744no—
302D4—1481no—
303D4—1482no—
304C2×C4—41no—
305C2×C4—374no—
306C2×C4—371no—
307C2×C4—372no—
308C2×C4—371no—
309C2×C4—742no—
310C2×C4—741no—
311C2×C4—741no—
312C2×C4—741no—
313C2×C4—741no—
314C2×C4—742no—
315C2×C4—742no—
316C2×C4—742no—
317C2×C4—742no—
318C2×C4—742no—
319C2×C4—742no—
320C2×C4—742no—
321C2×C4—742no—
322C2×C4—742no—
323C2×C4—742no—
324C2×C4—742no—
325C2×C4—742no—
326C2×C4—1481no—
327C2×C4—1481no—
328C2×C4—1481no—
329C2×C4—1484no—
330C2×C4—1488no—
331C2×C4—1484no—
332C2×C4C22×#386611yescharacteristic
333C2×C4C22×#386612yes—
334C2×C4C22×#386611yescharacteristic
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—371no—
341C22—371no—
342C22—371no—
343C22—371no—
344C22—371no—
345C22—371no—
346C22—372no—
347C22—742no—
348C22—742no—
349C22—742no—
350C22—742no—
351C22—741no—
352C22—741no—
353C22—742no—
354C22—742no—
355C22—742no—
356C22—1482no—
357C22—1482no—
358C22—1482no—
359C22—1482no—
360C22D4×#386612yes—
361C22C23×#386611yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—744no—
366C4—742no—
367C4—741no—
368C4—741no—
369C4—1482no—
370C4—1482no—
371C4—1482no—
372C2—22no—
373C2—42no—
374C2—371no—
375C2—371no—
376C2—371no—
377C2—371no—
378C2—742no—
379C2—1482no—
380C2C2×D4×#386611yescharacteristic
381C2#1455860011yescharacteristic
382C2#1455860211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#3866
commutator, socleC22×C37C23×C4
Fitting#121×C37C4
prime radical#121×D37C2
cosocleC22×D37C24
hypercenter#121#3866
hypocenterC37#2325
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×#3866C2semidirect
2. #14558483C2semidirect
3. #14558482C2semidirect
4. #14558481C2semidirect
5. #14558481C2semidirect
6. #14558480C2semidirect
7. #14558475C2semidirect
8. C2×(C2×#3866)⋊C4C2semidirect
30. #11924511C4semidirect
31. #11924510C4semidirect
32. #11924509C4semidirect
33. #11924508C4semidirect
34. #11924507C4semidirect
37. #121×C37C4semidirect
48. C22×#3866D4semidirect
53. C2×D4×C37C2×C4semidirect
54. C2×D4×C37C2×C4semidirect
55. C22×C4×C37C2×C4semidirect
56. C2×C22⋊D37C2×C4semidirect
57. C2×C22⋊D37C2×C4semidirect
58. C2×C22⋊D37C2×C4semidirect
59. C22×Dic37C2×C4semidirect
64. C2×D148C2×C4semidirect
65. C2×D148C2×C4semidirect
68. C2×C4×D37C2×C4semidirect
69. C2×C4×D37C2×C4semidirect
81. C4⋊C4×C37C2×C4semidirect
82. C22⋊C4×C37C2×C4semidirect
83. C22⋊Dic37C2×C4semidirect
84. #10596938C2×C4semidirect
85. #10596938C2×C4semidirect
86. C4⋊Dic37C2×C4semidirect
87. #10596936C2×C4semidirect
201. C37#2325semidirect
213. #121#3866semidirect
241. C2×D4C2×#3866semidirect
242. C2×D4C2×#3866semidirect
264. C22×C4C2×#3866semidirect
270. C4⋊C4C2×#3866semidirect
278. C22⋊C4C2×#3866semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#38662
#145584832
#145584822
#145584812
#145584802
#145584752
C2×(C2×#3866)⋊C42
#121×D372
#232537