Finite Group Databasebeta

← #37576176

Subgroups — #37576176

Order 24448.

852 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic382C211yesmaximal, characteristic, complemented
2C2×#20259308C212yesmaximal, complemented
3C2×#20259294C211yesmaximal, characteristic
4#27132576C212yesmaximal, complemented
5#27132575C212yesmaximal, complemented
6#27132567C212yesmaximal, complemented
7#27132543C211yesmaximal, characteristic, complemented
8#27132542C211yesmaximal, characteristic, complemented
9#27132540C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic191C211yesmaximal, characteristic, complemented
11#386×C191C211yesmaximal, characteristic, complemented
12C22×Dic382—28no—
13C22×Dic382—24no—
14C22×Dic382C2212yes—
15C22×Dic382C2212yes—
16C22×C22⋊D191C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C23×Dic191C2211yescharacteristic
20#20259308—28no—
21#20259294—24no—
22C2×Dic191⋊C4C2211yescharacteristic
23C2×C22⋊Dic191C2211yescharacteristic
24C2×C22⋊Dic191C2212yes—
25C2×C22⋊Dic191C2211yescharacteristic
26C2×(C2×D191)⋊C4C2212yes—
27C2×(C2×D191)⋊C4C2212yes—
28C2×(C2×D191)⋊C4C2211yescharacteristic
29C2×C4×Dic191C2212yes—
30C191⋊#95C2212yes—
31C191⋊#95C2212yes—
32C191⋊#95C2212yes—
33C191⋊#95C2212yes—
34C22×D4×C191C2211yescharacteristic
35C23×C4×C191C2211yescharacteristic
36C2×C22⋊C4×C191C2211yescharacteristic
37C2×C22⋊C4×C191C2212yes—
38#95×C191C2212yes—
39C23×D191C2311yescharacteristic
40C2×Dic382—232no—
41C2×Dic382—24no—
42C2×Dic382—24no—
43C2×Dic382—24no—
44C2×Dic382—24no—
45C2×Dic382—24no—
46C2×Dic382D418yescomplemented
47C2×C22⋊D191—22no—
48C2×C22⋊D191—24no—
49C2×C22⋊D191—22no—
50C2×C22⋊D191—22no—
51C2×C22⋊D191—22no—
52C22×Dic191—24no—
53C22×Dic191—24no—
54C22×Dic191—22no—
55C22×Dic191D414yes—
56C22×Dic191C2311yescharacteristic
57C22×Dic191C2312yes—
58C22×Dic191C2312yes—
59C22×Dic191C2311yescharacteristic
60C22×Dic191C2311yescharacteristic
61Dic191⋊C4—22no—
62C22⋊Dic191—22no—
63C22⋊Dic191—24no—
64C22⋊Dic191—22no—
65(C2×D191)⋊C4—22no—
66(C2×D191)⋊C4—24no—
67(C2×D191)⋊C4—24no—
68C4×Dic191—24no—
69C24×C191C2311yescharacteristic
70C24×C191C2311yescharacteristic
71C2×D4×C191—22no—
72C2×D4×C191—22no—
73C2×D4×C191—22no—
74C22×C4×C191—22no—
75C22×C4×C191—24no—
76C22×C4×C191C2311yescharacteristic
77C22×C4×C191C2312yes—
78C22×C4×C191C2312yes—
79C22⋊C4×C191—22no—
80C22⋊C4×C191—24no—
81C22×D191—22no—
82C22×D191—21no—
83C22×D191—21no—
84C22×D191—21no—
85C22×D191—22no—
86Dic382—216no—
87Dic382—216no—
88C22⋊D191—44no—
89C22⋊D191—44no—
90C2×Dic191—28no—
91C2×Dic191—22no—
92C2×Dic191—22no—
93C2×Dic191—22no—
94C2×Dic191—28no—
95C2×Dic191—22no—
96C2×Dic191—21no—
97C2×Dic191—22no—
98C2×Dic191—21no—
99C2×Dic191—21no—
100C2×Dic191—22no—
101C2×Dic191C2×D414yes—
102C2×Dic191C2×D412yes—
103C2×Dic191C2×D414yes—
104C2×Dic191C2×D412yes—
105C23×C191—22no—
106C23×C191—22no—
107C23×C191—22no—
108C23×C191—21no—
109C23×C191—21no—
110C23×C191—21no—
111C23×C191—22no—
112C23×C191C2×D412yes—
113C23×C191#4112yes—
114C23×C191#4112yes—
115C23×C191C2411yescharacteristic
116D4×C191—44no—
117C2×C4×C191—28no—
118C2×C4×C191—22no—
119C2×C4×C191—22no—
120C2×C4×C191—22no—
121C2×C4×C191—21no—
122C2×C4×C191—22no—
123C2×C4×C191—21no—
124C2×C4×C191—21no—
125C2×C4×C191—22no—
126C2×C4×C191C2×D414yes—
127C22×C191—22no—
128C22×C191—22no—
129C22×C191—24no—
130C22×C191—24no—
131C22×C191—41no—
132C22×C191—41no—
133C22×C191—41no—
134C22×C191—42no—
135C22×C191—42no—
136C22×C191#12414yescomplemented
137C22×C191C22×D411yescharacteristic
138C22×C191C22×D412yes—
139C22×C191C2×#4111yescharacteristic
140C22×C191C2×#4111yescharacteristic
141C22×C1912−1+412yes—
142C2×D191—41no—
143C2×D191—41no—
144C2×D191—41no—
145C2×D191—42no—
146C2×D191—42no—
147C4×C191—24no—
148C4×C191—41no—
149C4×C191—42no—
150Dic191—24no—
151Dic191—22no—
152Dic191—41no—
153Dic191—42no—
154Dic191#12014yes—
155C2×C191—24no—
156C2×C191—81no—
157C2×C191C2×#12011yescharacteristic
158C2×C191C2×#12411yescharacteristic
159C2×C191#53611yescharacteristic
160C2×C191#54812yes—
161C2×C191#54812yes—
162D191—81no—
163C191#243611yescharacteristic, complemented, Sylow
164#2436—1911nomaximal, Sylow
165C23×Q8—1911no—
166C2×#124—1912no—
167C2×#121—1911no—
168#393—1912no—
169#393—1912no—
170#391—1912no—
171#386—1911no—
172#386—1911no—
173#386—1911no—
174#386D19111yescharacteristic, complemented
175C4×C22⋊C4—1911no—
176C22×Q8—3828no—
177C22×Q8—3824no—
178C22×Q8—1912no—
179C22×Q8—1912no—
180C22×D4—1911no—
181C22×D4—1911no—
182C22×D4C2×D19111yescharacteristic
183C23×C4—1911no—
184C23×C4—1911no—
185C23×C4C2×D19111yescharacteristic
186#124—3828no—
187#121—3824no—
188C2×C4⋊C4—1911no—
189C2×C22⋊C4—1911no—
190C2×C22⋊C4—1912no—
191C2×C22⋊C4—1911no—
192C2×C22⋊C4—1911no—
193C2×C22⋊C4—1912no—
194C2×C22⋊C4—1912no—
195C2×C22⋊C4C2×D19112yes—
196C2×C22⋊C4C2×D19111yescharacteristic
197C2×C42—1912no—
198#95—1912no—
199#95—1912no—
200#95—1912no—
201#95—1912no—
202#95C2×D19112yes—
203C24—1911no—
204C24C22×D19111yescharacteristic
205C24C22×D19111yescharacteristic
206C2×Q8—38232no—
207C2×Q8—3824no—
208C2×Q8—3824no—
209C2×Q8—3824no—
210C2×Q8—3824no—
211C2×Q8—3824no—
212C2×Q8—1918no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—3822no—
217C2×D4—3822no—
218C2×D4—3822no—
219C2×D4—3822no—
220C2×D4—3824no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—3822no—
224C22×C4—3824no—
225C22×C4—3824no—
226C22×C4—1911no—
227C22×C4—1912no—
228C22×C4—1914no—
229C22×C4—1912no—
230C22×C4—1911no—
231C22×C4—1911no—
232C22×C4C22×D19111yescharacteristic
233C22×C4C22×D19112yes—
234C22×C4C22×D19112yes—
235C4⋊C4—3822no—
236C22⋊C4—22no—
237C22⋊C4—24no—
238C22⋊C4—3822no—
239C22⋊C4—3824no—
240C22⋊C4—3822no—
241C22⋊C4—3822no—
242C22⋊C4—3824no—
243C22⋊C4—3824no—
244C42—3824no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—3822no—
253C23—3821no—
254C23—3821no—
255C23—3821no—
256C23—3822no—
257C23D4×D19112yes—
258C23#1569028612yes—
259C23#1569028712yes—
260C23C23×D19111yescharacteristic
261Q8—38216no—
262Q8—38216no—
263D4—44no—
264D4—7644no—
265D4—7644no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—3828no—
276C2×C4—3822no—
277C2×C4—3822no—
278C2×C4—3822no—
279C2×C4—3828no—
280C2×C4—3822no—
281C2×C4—3821no—
282C2×C4—3821no—
283C2×C4—3822no—
284C2×C4—3821no—
285C2×C4—3822no—
286C2×C4—1914no—
287C2×C4—1912no—
288C2×C4—1914no—
289C2×C4—1912no—
290C2×C4D4×D19114yes—
291C22—22no—
292C22—24no—
293C22—24no—
294C22—22no—
295C22—41no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—7641no—
301C22—7642no—
302C22—7642no—
303C22—7641no—
304C22—7641no—
305C22#2025930814yescomplemented
306C22C2×D4×D19111yescharacteristic
307C22C2×D4×D19112yes—
308C22C2×#1569028611yescharacteristic
309C22C2×#1569028711yescharacteristic
310C22#2025936912yes—
311C4—24no—
312C4—42no—
313C4—41no—
314C4—3824no—
315C4—3822no—
316C4—1914no—
317C4—7642no—
318C4—7641no—
319C2—24no—
320C2—81no—
321C2—15281no—
322C2#120×D19111yescharacteristic
323C2C2×#2025930811yescharacteristic
324C2#2713323511yescharacteristic
325C2#2713331912yes—
326C2#2713332212yes—
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#386×C191C2
prime radicalC2×#20259294C2
hypercenter#386D191
hypocenterC191#2436
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C23×Dic382C2semidirect
2. C2×#20259308C2semidirect
4. #27132576C2semidirect
5. #27132575C2semidirect
6. #27132567C2semidirect
7. #27132543C2semidirect
8. #27132542C2semidirect
9. #27132540C2semidirect
10. C22⋊C4×Dic191C2semidirect
11. #386×C191C2semidirect
46. C2×Dic382D4semidirect
136. C22×C191#124semidirect
163. C191#2436semidirect
174. #386D191semidirect
305. C22#20259308semidirect

Maximal subgroups

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

subgroupindex
C23×Dic3822
C2×#202593082
C2×#202592942
#271325762
#271325752
#271325672
#271325432
#271325422
#271325402
C22⋊C4×Dic1912
#386×C1912
#2436191