Finite Group Databasebeta

← #37575964

Subgroups — #37575964

Order 24448.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259300C211yesmaximal, characteristic, complemented
2C2×#20259299C211yesmaximal, characteristic, complemented
3C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic
4C2×C4⋊C4×D191C211yesmaximal, characteristic, complemented
5#27132596C211yesmaximal, characteristic, complemented
6#27132593C211yesmaximal, characteristic, complemented
7#27132586C211yesmaximal, characteristic, complemented
8#27132567C211yesmaximal, characteristic, complemented
9#27132558C211yesmaximal, characteristic, complemented
10#27132539C211yesmaximal, characteristic
11#27132539C211yesmaximal, characteristic
12#27132522C211yesmaximal, characteristic, complemented
13#27132521C211yesmaximal, characteristic, complemented
14#27132520C211yesmaximal, characteristic, complemented
15#396×C191C211yesmaximal, characteristic, complemented
16C22×Dic382C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C23×Dic191C2211yescharacteristic
19C22×C4×D191C2211yescharacteristic
20C22×C4×D191C2211yescharacteristic
21C22×C4×D191C2211yescharacteristic
22#20259300—24no—
23#20259299—24no—
24(C22⋊D191)⋊C4—24no—
25C2×C4⋊Dic191C2211yescharacteristic
26C2×C4⋊Dic191C2211yescharacteristic
27C2×Dic191⋊C4C2211yescharacteristic
28C2×Dic191⋊C4C2211yescharacteristic
29C2×Dic191⋊C4C2211yescharacteristic
30C2×Dic191⋊C4C2211yescharacteristic
31C2×Dic191⋊C4C2211yescharacteristic
32C2×Dic191⋊C4C2211yescharacteristic
33C2×Dic191⋊C4C2211yescharacteristic
34C4⋊C4×D191—24no—
35C2×C22⋊Dic191C2211yescharacteristic
36C2×C22⋊Dic191C2211yescharacteristic
37C2×(C2×D191)⋊C4C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×C4×Dic191C2211yescharacteristic
41C191⋊#95C2211yescharacteristic
42C191⋊#95C2211yescharacteristic
43C191⋊#95C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C191⋊#95C2211yescharacteristic
47C191⋊#95C2211yescharacteristic
48C2×C4⋊C4×C191C2211yescharacteristic
49C2×C4⋊C4×C191C2211yescharacteristic
50C2×C4⋊C4×C191C2211yescharacteristic
51C2×C22⋊C4×C191C2211yescharacteristic
52C2×C22⋊C4×C191C2211yescharacteristic
53C2×C22⋊C4×C191C2211yescharacteristic
54#95×C191C2211yescharacteristic
55C23×D191C2311yescharacteristic
56C2×Dic382—22no—
57C2×Dic382—22no—
58C2×Dic382—22no—
59C2×C22⋊D191—22no—
60C2×C22⋊D191—22no—
61C2×C22⋊D191Q814yescomplemented
62C22×Dic191—22no—
63C22×Dic191—22no—
64C22×Dic191—22no—
65C22×Dic191C2311yescharacteristic
66C22×Dic191C2311yescharacteristic
67C22×Dic191C2311yescharacteristic
68C22×Dic191C2311yescharacteristic
69C22×Dic191C2311yescharacteristic
70C22×Dic191C2311yescharacteristic
71C22×Dic191C2311yescharacteristic
72C2×C4×D191—22no—
73C2×C4×D191—22no—
74C2×C4×D191—22no—
75C2×C4×D191—22no—
76C2×C4×D191—22no—
77C2×C4×D191—22no—
78C2×C4×D191—41no—
79C2×C4×D191—41no—
80C2×C4×D191D414yes—
81C4⋊Dic191—22no—
82C4⋊Dic191—22no—
83Dic191⋊C4—22no—
84Dic191⋊C4—22no—
85Dic191⋊C4—22no—
86Dic191⋊C4—22no—
87Dic191⋊C4—22no—
88Dic191⋊C4—41no—
89Dic191⋊C4D414yescomplemented
90C22⋊Dic191—22no—
91C22⋊Dic191—41no—
92(C2×D191)⋊C4—22no—
93(C2×D191)⋊C4—22no—
94(C2×D191)⋊C4—22no—
95C4×Dic191—22no—
96C24×C191C2311yescharacteristic
97C22×C4×C191C2311yescharacteristic
98C22×C4×C191C2311yescharacteristic
99C22×C4×C191C2311yescharacteristic
100C22×C4×C191C2311yescharacteristic
101C22×C4×C191C2311yescharacteristic
102C22×C4×C191C2311yescharacteristic
103C4⋊C4×C191—22no—
104C4⋊C4×C191—22no—
105C4⋊C4×C191—41no—
106C22⋊C4×C191—22no—
107C22⋊C4×C191—22no—
108C22⋊C4×C191—41no—
109C22×D191—21no—
110C22×D191—21no—
111C22×D191—21no—
112C22×D191—21no—
113C22×D191C2×D412yes—
114C22×D191C2×Q812yes—
115C22×D191#4112yes—
116Dic382—44no—
117C22⋊D191—28no—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—21no—
121C2×Dic191—21no—
122C2×Dic191—21no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—21no—
126C2×Dic191—21no—
127C2×Dic191—21no—
128C2×Dic191—21no—
129C2×Dic191—21no—
130C2×Dic191—21no—
131C2×Dic191—21no—
132C2×Dic191—21no—
133C2×Dic191—21no—
134C2×Dic191—44no—
135C2×Dic191C2×D412yes—
136C2×Dic191C2×D412yes—
137C2×Dic191C2×D412yes—
138C2×Dic191C2×Q812yes—
139C2×Dic191#4112yes—
140C4×D191—28no—
141C4×D191—44no—
142C4×D191—44no—
143C23×C191—21no—
144C23×C191—21no—
145C23×C191—21no—
146C23×C191—21no—
147C23×C191—21no—
148C23×C191—21no—
149C23×C191C2×Q812yes—
150C23×C191C2411yescharacteristic
151C2×C4×C191—21no—
152C2×C4×C191—21no—
153C2×C4×C191—21no—
154C2×C4×C191—21no—
155C2×C4×C191—21no—
156C2×C4×C191—21no—
157C2×C4×C191—21no—
158C2×C4×C191—21no—
159C2×C4×C191—21no—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191C2×D412yes—
168C2×C4×C191C2×D412yes—
169C22×C191—22no—
170C22×C191—22no—
171C22×C191—22no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191C22×D411yescharacteristic
177C22×C191C22×D411yescharacteristic
178C22×C191C22×Q811yescharacteristic
179C22×C191C2×#4111yescharacteristic
180C22×C191C2×#4111yescharacteristic
181C22×C191H(5,𝔽2)11yescharacteristic
182C22×C1912−1+411yescharacteristic
183C2×D191—22no—
184C2×D191—22no—
185C2×D191—22no—
186C2×D191—22no—
187C2×D191—22no—
188C2×D191—22no—
189C2×D191C22⋊Q814yescomplemented
190C4×C191—22no—
191C4×C191—22no—
192C4×C191—41no—
193C4×C191—41no—
194C4×C191—41no—
195C4×C191—41no—
196Dic191—22no—
197Dic191—22no—
198Dic191—41no—
199Dic191—41no—
200Dic191—41no—
201Dic191—41no—
202Dic191#12114yes—
203C2×C191—42no—
204C2×C191C2×#12111yescharacteristic
205C2×C191C2×C22⋊Q811yescharacteristic
206C2×C191#53711yescharacteristic
207C2×C191#54611yescharacteristic
208C2×C191D4×Q811yescharacteristic
209C2×C191#55211yescharacteristic
210C2×C191#55411yescharacteristic
211D191—24no—
212C191#241711yescharacteristic, complemented, Sylow
213#2417—1911nomaximal, Sylow
214C2×C22⋊Q8—1911no—
215C2×C22⋊Q8—1911no—
216C2×C4×D4—1911no—
217C22×C4⋊C4—1911no—
218#398—1911no—
219#398—1911no—
220#396—1911no—
221#396D19111yescharacteristic, complemented
222#391—1911no—
223#389—1911no—
224#386—1911no—
225#386—1911no—
226#380—1911no—
227#380—1911no—
228#380—1911no—
229C22×Q8—1911no—
230C22×D4—1911no—
231C23×C4—1911no—
232C23×C4—1911no—
233C23×C4—1911no—
234C23×C4—1911no—
235C22⋊Q8—3824no—
236C22⋊Q8—3824no—
237C4×D4—3824no—
238C2×C4⋊C4—3824no—
239C2×C4⋊C4—1911no—
240C2×C4⋊C4—1911no—
241C2×C4⋊C4—1911no—
242C2×C4⋊C4—1911no—
243C2×C4⋊C4—1911no—
244C2×C4⋊C4—1911no—
245C2×C4⋊C4—1911no—
246C2×C4⋊C4—1911no—
247C2×C4⋊C4—1911no—
248C2×C4⋊C4C2×D19111yescharacteristic
249C2×C4⋊C4C2×D19111yescharacteristic
250C2×C4⋊C4C2×D19111yescharacteristic
251C2×C22⋊C4—1911no—
252C2×C22⋊C4—1911no—
253C2×C22⋊C4—1911no—
254C2×C22⋊C4—1911no—
255C2×C22⋊C4—1911no—
256C2×C22⋊C4C2×D19111yescharacteristic
257C2×C22⋊C4C2×D19111yescharacteristic
258C2×C22⋊C4C2×D19111yescharacteristic
259C2×C42—1911no—
260#95—1911no—
261#95—1911no—
262#95—1911no—
263#95—1911no—
264#95—1911no—
265#95—1911no—
266#95—1911no—
267#95C2×D19111yescharacteristic
268C24—1911no—
269C24C22×D19111yescharacteristic
270C2×Q8—3822no—
271C2×Q8—3822no—
272C2×Q8—3822no—
273C2×D4—3822no—
274C2×D4—3822no—
275C2×D4—1914no—
276C22×C4—3822no—
277C22×C4—3822no—
278C22×C4—3822no—
279C22×C4—3822no—
280C22×C4—3822no—
281C22×C4—3822no—
282C22×C4—3822no—
283C22×C4—3822no—
284C22×C4—3822no—
285C22×C4—1911no—
286C22×C4—1911no—
287C22×C4—1911no—
288C22×C4—1911no—
289C22×C4—1911no—
290C22×C4—1911no—
291C22×C4—1914no—
292C22×C4—1911no—
293C22×C4—7641no—
294C22×C4—7641no—
295C22×C4C22×D19111yescharacteristic
296C22×C4C22×D19111yescharacteristic
297C22×C4C22×D19111yescharacteristic
298C22×C4C22×D19111yescharacteristic
299C22×C4C22×D19111yescharacteristic
300C22×C4C22×D19111yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—3822no—
305C4⋊C4—3822no—
306C4⋊C4—3822no—
307C4⋊C4—3822no—
308C4⋊C4—3822no—
309C4⋊C4—3822no—
310C4⋊C4—3822no—
311C4⋊C4—1914no—
312C4⋊C4—7641no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—3822no—
317C22⋊C4—3822no—
318C22⋊C4—3822no—
319C22⋊C4—3822no—
320C22⋊C4—7641no—
321C42—3822no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—3821no—
329C23—3821no—
330C23—3821no—
331C23—3821no—
332C23—1912no—
333C23—1912no—
334C23—1912no—
335C23Q8×D19112yes—
336C23C23×D19111yescharacteristic
337Q8—7644no—
338D4—3828no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—3828no—
356C2×C4—3821no—
357C2×C4—3821no—
358C2×C4—3821no—
359C2×C4—3821no—
360C2×C4—3821no—
361C2×C4—3821no—
362C2×C4—3821no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—3821no—
369C2×C4—3821no—
370C2×C4—3821no—
371C2×C4—3821no—
372C2×C4—1912no—
373C2×C4—1912no—
374C2×C4—1912no—
375C2×C4—1912no—
376C2×C4—1912no—
377C2×C4—7644no—
378C2×C4—7644no—
379C2×C4—7644no—
380C2×C4D4×D19112yes—
381C2×C4D4×D19112yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3822no—
390C22—3822no—
391C22—3822no—
392C22—3822no—
393C22—3822no—
394C22—3822no—
395C22—1914no—
396C22C2×D4×D19111yescharacteristic
397C22C2×D4×D19111yescharacteristic
398C22C2×Q8×D19111yescharacteristic
399C22#41×D19111yescharacteristic
400C22#41×D19111yescharacteristic
401C22#2025936711yescharacteristic
402C22#2025936911yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—3822no—
410C4—3822no—
411C4—1914no—
412C4—7641no—
413C4—7641no—
414C4—7641no—
415C4—7641no—
416C2—42no—
417C2—3824no—
418C2#121×D19111yescharacteristic
419C2C22⋊Q8×D19111yescharacteristic
420C2#2713324211yescharacteristic
421C2#2713330211yescharacteristic
422C2#2713332611yescharacteristic
423C2#2713334911yescharacteristic
424C2#2713336711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#396×C191C2
prime radicalC22×C22⋊D191C22
hypercenter#396D191
hypocenterC191#2417
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#20259300C2semidirect
2. C2×#20259299C2semidirect
4. C2×C4⋊C4×D191C2semidirect
5. #27132596C2semidirect
6. #27132593C2semidirect
7. #27132586C2semidirect
8. #27132567C2semidirect
9. #27132558C2semidirect
12. #27132522C2semidirect
13. #27132521C2semidirect
14. #27132520C2semidirect
15. #396×C191C2semidirect
61. C2×C22⋊D191Q8semidirect
89. Dic191⋊C4D4semidirect
189. C2×D191C22⋊Q8semidirect
212. C191#2417semidirect
221. #396D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593002
C2×#202592992
C2×(C22⋊D191)⋊C42
C2×C4⋊C4×D1912
#271325962
#271325932
#271325862
#271325672
#271325582
#271325392
#271325222
#271325212
#271325202
#396×C1912
#2417191