Finite Group Databasebeta

← #37576202

Subgroups — #37576202

Order 24448.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259299C211yesmaximal, characteristic
2C2×#20259278C211yesmaximal, characteristic
3#27132604C211yesmaximal, characteristic, complemented
4#27132597C211yesmaximal, characteristic, complemented
5#27132594C211yesmaximal, characteristic, complemented
6#27132583C211yesmaximal, characteristic, complemented
7#27132567C211yesmaximal, characteristic, complemented
8#27132558C211yesmaximal, characteristic, complemented
9#27132549C211yesmaximal, characteristic, complemented
10#27132545C211yesmaximal, characteristic, complemented
11#27132536C211yesmaximal, characteristic
12#27132535C211yesmaximal, characteristic
13#27132524C211yesmaximal, characteristic
14C2×C191⋊#95C211yesmaximal, characteristic
15#380×C191C211yesmaximal, characteristic
16C22×Dic382C2211yescharacteristic
17C23×Dic191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19#20259299—24no—
20#20259278—24no—
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4C2211yescharacteristic
23C2×Dic191⋊C4C2211yescharacteristic
24C2×Dic191⋊C4C2211yescharacteristic
25C2×Dic191⋊C4C2211yescharacteristic
26C2×Dic191⋊C4C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×Dic191⋊C4C2211yescharacteristic
30C2×C22⋊Dic191C2211yescharacteristic
31C2×C22⋊Dic191C2211yescharacteristic
32C2×C4×Dic191C2211yescharacteristic
33C2×C4×Dic191C2211yescharacteristic
34C191⋊#95C2211yescharacteristic
35C191⋊#95C2211yescharacteristic
36C191⋊#95C2211yescharacteristic
37C191⋊#95C2211yescharacteristic
38C191⋊#95C2211yescharacteristic
39C191⋊#95C2211yescharacteristic
40C191⋊#95C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C23×C4×C191C2211yescharacteristic
47C2×C4⋊C4×C191C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C22⋊C4×C191C2211yescharacteristic
50C2×C22⋊C4×C191C2211yescharacteristic
51#95×C191C2211yescharacteristic
52#95×C191C2211yescharacteristic
53C2×Dic382—22no—
54C2×Dic382—22no—
55C2×Dic382—41no—
56C22×Dic191—22no—
57C22×Dic191—22no—
58C22×Dic191—22no—
59C22×Dic191—22no—
60C22×Dic191—41no—
61C22×Dic191Q814yescomplemented
62C22×Dic191C2311yescharacteristic
63C22×Dic191C2311yescharacteristic
64C22×Dic191C2311yescharacteristic
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C4⋊Dic191—22no—
71Dic191⋊C4—22no—
72Dic191⋊C4—22no—
73Dic191⋊C4—22no—
74Dic191⋊C4—22no—
75Dic191⋊C4—22no—
76Dic191⋊C4—22no—
77Dic191⋊C4—41no—
78Dic191⋊C4—41no—
79C22⋊Dic191—22no—
80C22⋊Dic191D414yes—
81C4×Dic191—22no—
82C4×Dic191—22no—
83C24×C191C2311yescharacteristic
84C22×C4×C191—22no—
85C22×C4×C191—22no—
86C22×C4×C191—41no—
87C22×C4×C191C2311yescharacteristic
88C22×C4×C191C2311yescharacteristic
89C22×C4×C191C2311yescharacteristic
90C22×C4×C191C2311yescharacteristic
91C22×C4×C191C2311yescharacteristic
92C22×C4×C191C2311yescharacteristic
93C4⋊C4×C191—22no—
94C4⋊C4×C191—41no—
95C22⋊C4×C191—22no—
96C22⋊C4×C191—22no—
97Dic382—44no—
98C2×Dic191—28no—
99C2×Dic191—21no—
100C2×Dic191—21no—
101C2×Dic191—21no—
102C2×Dic191—21no—
103C2×Dic191—21no—
104C2×Dic191—21no—
105C2×Dic191—21no—
106C2×Dic191—21no—
107C2×Dic191—21no—
108C2×Dic191—21no—
109C2×Dic191—21no—
110C2×Dic191—21no—
111C2×Dic191—21no—
112C2×Dic191—21no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—21no—
118C2×Dic191—44no—
119C2×Dic191C2×D412yes—
120C2×Dic191C2×D412yes—
121C2×Dic191C2×Q812yes—
122C2×Dic191C2×Q812yes—
123C2×Dic191#4112yes—
124C23×C191—21no—
125C23×C191—21no—
126C23×C191—21no—
127C23×C191—21no—
128C23×C191C2×D412yes—
129C23×C191C2×Q812yes—
130C23×C191#4112yes—
131C23×C191C2411yescharacteristic
132C2×C4×C191—21no—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191—21no—
141C2×C4×C191—21no—
142C2×C4×C191—21no—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—21no—
147C2×C4×C191—21no—
148C2×C4×C191—21no—
149C2×C4×C191—44no—
150C2×C4×C191#4112yes—
151C22×C191—22no—
152C22×C191—22no—
153C22×C191—22no—
154C22×C191—22no—
155C22×C191—22no—
156C22×C191—22no—
157C22×C191C22⋊Q814yes—
158C22×C191C22×D411yescharacteristic
159C22×C191C22×Q811yescharacteristic
160C22×C191C2×#4111yescharacteristic
161C22×C191C2×#4111yescharacteristic
162C22×C191C2×#4111yescharacteristic
163C22×C191H(5,𝔽2)11yescharacteristic
164C22×C1912−1+411yescharacteristic
165C4×C191—22no—
166C4×C191—41no—
167C4×C191—41no—
168C4×C191—41no—
169C4×C191—41no—
170C4×C191—41no—
171Dic191—22no—
172Dic191—22no—
173Dic191—22no—
174Dic191—22no—
175Dic191—22no—
176Dic191—41no—
177Dic191—41no—
178Dic191—41no—
179C2×C191—24no—
180C2×C191C2×C22⋊Q811yescharacteristic
181C2×C191#53311yescharacteristic
182C2×C191#54411yescharacteristic
183C2×C191#54611yescharacteristic
184C2×C191#54711yescharacteristic
185C2×C191#55111yescharacteristic
186C2×C191#55211yescharacteristic
187C191#243811yescharacteristic, complemented, Sylow
188#2438—1911nomaximal, Sylow
189C2×C22⋊Q8—1911no—
190C2×#117—1911no—
191#400—1911no—
192#398—1911no—
193#398—1911no—
194#395—1911no—
195#391—1911no—
196#389—1911no—
197#387—1911no—
198#387—1911no—
199#385—1911no—
200#385—1911no—
201#381—1911no—
202#380D19111yescharacteristic
203C2×#95—1911no—
204C22×Q8—1911no—
205C23×C4—1911no—
206C23×C4—1911no—
207C23×C4C2×D19111yescharacteristic
208C22⋊Q8—3824no—
209#117—3824no—
210C2×C4⋊C4—1911no—
211C2×C4⋊C4—1911no—
212C2×C4⋊C4—1911no—
213C2×C4⋊C4—1911no—
214C2×C4⋊C4—1911no—
215C2×C4⋊C4—1911no—
216C2×C4⋊C4—1911no—
217C2×C4⋊C4—1911no—
218C2×C4⋊C4—1911no—
219C2×C4⋊C4C2×D19111yescharacteristic
220C2×C4⋊C4C2×D19111yescharacteristic
221C2×C22⋊C4—1911no—
222C2×C22⋊C4—1911no—
223C2×C22⋊C4C2×D19111yescharacteristic
224C2×C22⋊C4C2×D19111yescharacteristic
225C2×C42—1911no—
226C2×C42—1911no—
227#95—1911no—
228#95—1911no—
229#95—1911no—
230#95—1911no—
231#95—1911no—
232#95—1911no—
233#95—1911no—
234#95—1911no—
235#95—1911no—
236#95—1911no—
237#95—1911no—
238#95—1911no—
239#95C2×D19111yescharacteristic
240#95C2×D19111yescharacteristic
241C24C22×D19111yescharacteristic
242C2×Q8—3822no—
243C2×Q8—3822no—
244C2×Q8—7641no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—3822no—
249C22×C4—3822no—
250C22×C4—3822no—
251C22×C4—3822no—
252C22×C4—1911no—
253C22×C4—1911no—
254C22×C4—1911no—
255C22×C4—1914no—
256C22×C4—1911no—
257C22×C4—1911no—
258C22×C4—1911no—
259C22×C4—1911no—
260C22×C4—1911no—
261C22×C4—7641no—
262C22×C4C22×D19111yescharacteristic
263C22×C4C22×D19111yescharacteristic
264C22×C4C22×D19111yescharacteristic
265C22×C4C22×D19111yescharacteristic
266C22×C4C22×D19111yescharacteristic
267C22×C4C22×D19111yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—3822no—
271C4⋊C4—3822no—
272C4⋊C4—3822no—
273C4⋊C4—3822no—
274C4⋊C4—3822no—
275C4⋊C4—3822no—
276C4⋊C4—3822no—
277C4⋊C4—7641no—
278C4⋊C4—7641no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—3822no—
282C22⋊C4—1914no—
283C42—3822no—
284C42—3822no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23D4×D19112yes—
290C23Q8×D19112yes—
291C23#1569028712yes—
292C23C23×D19111yescharacteristic
293Q8—7644no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—3821no—
313C2×C4—3821no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—3821no—
317C2×C4—3821no—
318C2×C4—3821no—
319C2×C4—3821no—
320C2×C4—3821no—
321C2×C4—3821no—
322C2×C4—3828no—
323C2×C4—3821no—
324C2×C4—3821no—
325C2×C4—3821no—
326C2×C4—3821no—
327C2×C4—3821no—
328C2×C4—3821no—
329C2×C4—3821no—
330C2×C4—3821no—
331C2×C4—3821no—
332C2×C4—1912no—
333C2×C4—1912no—
334C2×C4—1912no—
335C2×C4—1912no—
336C2×C4—1912no—
337C2×C4—7644no—
338C2×C4#1569028612yes—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22#2025930014yes—
346C22C2×D4×D19111yescharacteristic
347C22C2×Q8×D19111yescharacteristic
348C22#41×D19111yescharacteristic
349C22C2×#1569028611yescharacteristic
350C22C2×#1569028711yescharacteristic
351C22#2025936711yescharacteristic
352C22#2025936911yescharacteristic
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—3822no—
360C4—3822no—
361C4—3822no—
362C4—3822no—
363C4—3822no—
364C4—7641no—
365C4—7641no—
366C4—7641no—
367C2—24no—
368C2C2×#2025930011yescharacteristic
369C2#2713321511yescharacteristic
370C2#2713329211yescharacteristic
371C2#2713330211yescharacteristic
372C2#2713331311yescharacteristic
373C2#2713333911yescharacteristic
374C2#2713334611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#380×C191C2
prime radicalC24×C191C23
hypercenter#380D191
hypocenterC191#2438
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #27132604C2semidirect
4. #27132597C2semidirect
5. #27132594C2semidirect
6. #27132583C2semidirect
7. #27132567C2semidirect
8. #27132558C2semidirect
9. #27132549C2semidirect
10. #27132545C2semidirect
61. C22×Dic191Q8semidirect
187. C191#2438semidirect

Maximal subgroups

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

subgroupindex
C2×#202592992
C2×#202592782
#271326042
#271325972
#271325942
#271325832
#271325672
#271325582
#271325492
#271325452
#271325362
#271325352
#271325242
C2×C191⋊#952
#380×C1912
#2438191