Finite Group Databasebeta

← #37575853

Subgroups — #37575853

Order 24448.

864 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690287C211yesmaximal, characteristic
2C2×#20259307C211yesmaximal, characteristic, complemented
3C2×#20259304C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×D191C211yesmaximal, characteristic, complemented
5#27132584C211yesmaximal, characteristic, complemented
6#27132583C211yesmaximal, characteristic, complemented
7#27132579C211yesmaximal, characteristic, complemented
8#27132578C211yesmaximal, characteristic, complemented
9#27132541C211yesmaximal, characteristic, complemented
10#27132539C211yesmaximal, characteristic, complemented
11#27132537C211yesmaximal, characteristic, complemented
12#27132536C211yesmaximal, characteristic, complemented
13#27132534C211yesmaximal, characteristic, complemented
14#27132525C211yesmaximal, characteristic, complemented
15#385×C191C211yesmaximal, characteristic, complemented
16C2×#15690287—24no—
17C2×#15690287—24no—
18C2×#15690287—24no—
19C22×Dic382C2211yescharacteristic
20C22×D764C2211yescharacteristic
21C22×C22⋊D191C2211yescharacteristic
22C22×C22⋊D191C2211yescharacteristic
23C22×C4×D191C2211yescharacteristic
24C22×C4×D191C2211yescharacteristic
25C22×C4×D191C2211yescharacteristic
26C22×C4×D191C2211yescharacteristic
27#20259307—24no—
28#20259304—24no—
29C2×C4⋊Dic191C2211yescharacteristic
30C2×C4⋊Dic191C2211yescharacteristic
31C2×Dic191⋊C4C2211yescharacteristic
32C2×Dic191⋊C4C2211yescharacteristic
33C2×Dic191⋊C4C2211yescharacteristic
34C2×Dic191⋊C4C2211yescharacteristic
35C4⋊C4×D191—24no—
36C2×C22⋊Dic191C2211yescharacteristic
37C2×C22⋊Dic191C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×(C2×D191)⋊C4C2211yescharacteristic
41C2×(C2×D191)⋊C4C2211yescharacteristic
42C2×(C2×D191)⋊C4C2211yescharacteristic
43C2×(C2×D191)⋊C4C2211yescharacteristic
44C191⋊#95C2211yescharacteristic
45C191⋊#95C2211yescharacteristic
46C191⋊#95C2211yescharacteristic
47C191⋊#95C2211yescharacteristic
48C191⋊#95C2211yescharacteristic
49C191⋊#95C2211yescharacteristic
50C23×C4×C191C2211yescharacteristic
51C2×C4⋊C4×C191C2211yescharacteristic
52C2×C4⋊C4×C191C2211yescharacteristic
53C2×C22⋊C4×C191C2211yescharacteristic
54C2×C22⋊C4×C191C2211yescharacteristic
55#95×C191C2211yescharacteristic
56#95×C191C2211yescharacteristic
57C23×D191C2311yescharacteristic
58C23×D191C2311yescharacteristic
59#15690287—232no—
60C2×Dic382—22no—
61C2×Dic382—22no—
62C2×Dic382—22no—
63C2×D764—22no—
64C2×D764—22no—
65C2×D764D414yes—
66C2×C22⋊D191—22no—
67C2×C22⋊D191—22no—
68C2×C22⋊D191—22no—
69C2×C22⋊D191—22no—
70C2×C22⋊D191—22no—
71C2×C22⋊D191D414yes—
72C22×Dic191C2311yescharacteristic
73C22×Dic191C2311yescharacteristic
74C22×Dic191C2311yescharacteristic
75C22×Dic191C2311yescharacteristic
76C22×Dic191C2311yescharacteristic
77C22×Dic191C2311yescharacteristic
78C2×C4×D191—22no—
79C2×C4×D191—22no—
80C2×C4×D191—22no—
81C2×C4×D191—22no—
82C2×C4×D191—22no—
83C2×C4×D191—22no—
84C2×C4×D191—22no—
85C2×C4×D191—22no—
86C2×C4×D191—22no—
87C2×C4×D191—22no—
88C2×C4×D191—22no—
89C2×C4×D191D414yes—
90C4⋊Dic191—22no—
91C4⋊Dic191—22no—
92Dic191⋊C4—22no—
93Dic191⋊C4—22no—
94Dic191⋊C4—22no—
95Dic191⋊C4—22no—
96C22⋊Dic191—22no—
97C22⋊Dic191—22no—
98(C2×D191)⋊C4—22no—
99(C2×D191)⋊C4—22no—
100(C2×D191)⋊C4—22no—
101(C2×D191)⋊C4—22no—
102(C2×D191)⋊C4—22no—
103(C2×D191)⋊C4—22no—
104C24×C191C2311yescharacteristic
105C22×C4×C191—22no—
106C22×C4×C191—22no—
107C22×C4×C191—22no—
108C22×C4×C191C2311yescharacteristic
109C22×C4×C191C2311yescharacteristic
110C22×C4×C191C2311yescharacteristic
111C22×C4×C191C2311yescharacteristic
112C22×C4×C191C2311yescharacteristic
113C22×C4×C191C2311yescharacteristic
114C4⋊C4×C191—22no—
115C4⋊C4×C191—22no—
116C22⋊C4×C191—22no—
117C22⋊C4×C191—22no—
118C22×D191—21no—
119C22×D191—21no—
120C22×D191—21no—
121C22×D191—21no—
122C22×D191—21no—
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—21no—
128C22×D191C2×D412yes—
129C22×D191C2×D412yes—
130C22×D191C2×D412yes—
131C22×D191C2×D412yes—
132Dic382—28no—
133D764—28no—
134C22⋊D191—28no—
135C22⋊D191—28no—
136C2×Dic191—21no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—21no—
144C2×Dic191—21no—
145C2×Dic191—21no—
146C2×Dic191—21no—
147C2×Dic191—21no—
148C2×Dic191—21no—
149C2×Dic191—21no—
150C2×Dic191—21no—
151C2×Dic191C2×D412yes—
152C2×Dic191C2×D412yes—
153C2×Dic191#4112yescomplemented
154C4×D191—28no—
155C4×D191—28no—
156C4×D191—28no—
157C4×D191—44no—
158C23×C191—21no—
159C23×C191—21no—
160C23×C191—21no—
161C23×C191—21no—
162C23×C191—21no—
163C23×C191—21no—
164C23×C191C2×D412yes—
165C23×C191C2411yescharacteristic
166C2×C4×C191—28no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—21no—
172C2×C4×C191—21no—
173C2×C4×C191—21no—
174C2×C4×C191—21no—
175C2×C4×C191—21no—
176C2×C4×C191—21no—
177C2×C4×C191—21no—
178C2×C4×C191—21no—
179C2×C4×C191—21no—
180C2×C4×C191—21no—
181C2×C4×C191—21no—
182C2×C4×C191C2×D412yes—
183C2×C4×C191C2×D412yes—
184C2×C4×C191#4112yescomplemented
185C22×C191—22no—
186C22×C191—22no—
187C22×C191—22no—
188C22×C191—41no—
189C22×C191—41no—
190C22×C191—41no—
191C22×C191—41no—
192C22×C191C22×D411yescharacteristic
193C22×C191C22×D411yescharacteristic
194C22×C191C22×D411yescharacteristic
195C22×C191C2×#4111yescharacteristic
196C22×C191C2×#4111yescharacteristic
197C22×C1912−1+411yescharacteristic
198C22×C1912−1+411yescharacteristic
199C2×D191—22no—
200C2×D191—22no—
201C2×D191—22no—
202C2×D191—22no—
203C2×D191—22no—
204C2×D191—22no—
205C2×D191—22no—
206C2×D191—22no—
207C2×D191—22no—
208C2×D191—41no—
209C2×D191—41no—
210C2×D191—41no—
211C2×D191—41no—
212C2×D191#12014yes—
213C4×C191—22no—
214C4×C191—22no—
215C4×C191—22no—
216C4×C191—41no—
217C4×C191—41no—
218C4×C191—41no—
219Dic191—22no—
220Dic191—22no—
221Dic191—22no—
222Dic191—41no—
223Dic191—41no—
224Dic191—41no—
225C2×C191—42no—
226C2×C191C2×#12011yescharacteristic
227C2×C191#52511yescharacteristic
228C2×C191#53611yescharacteristic
229C2×C191#54711yescharacteristic
230C2×C191#54711yescharacteristic
231C2×C191#54711yescharacteristic
232C2×C191#54711yescharacteristic
233D191—24no—
234D191—42no—
235C191#240611yescharacteristic, complemented, Sylow
236#2406—1911nomaximal, Sylow
237C22×#41—1911no—
238C2×#123—1911no—
239C2×#123—1911no—
240C22×C4⋊C4—1911no—
241#395—1911no—
242#395—1911no—
243#394—1911no—
244#394—1911no—
245#386—1911no—
246#386—1911no—
247#385—1911no—
248#385—1911no—
249#385—1911no—
250#385D19111yescharacteristic, complemented
251#381—1911no—
252C2×#41—3824no—
253C2×#41—3824no—
254C2×#41—3824no—
255C22×Q8—1911no—
256C22×D4—1911no—
257C22×D4—1911no—
258C22×D4—1911no—
259C23×C4—1911no—
260C23×C4—1911no—
261C23×C4—1911no—
262C23×C4—1911no—
263C23×C4C2×D19111yescharacteristic
264#123—3824no—
265#123—3824no—
266C2×C4⋊C4—3824no—
267C2×C4⋊C4—1911no—
268C2×C4⋊C4—1911no—
269C2×C4⋊C4—1911no—
270C2×C4⋊C4—1911no—
271C2×C4⋊C4—1911no—
272C2×C4⋊C4—1911no—
273C2×C4⋊C4C2×D19111yescharacteristic
274C2×C4⋊C4C2×D19111yescharacteristic
275C2×C22⋊C4—1911no—
276C2×C22⋊C4—1911no—
277C2×C22⋊C4—1911no—
278C2×C22⋊C4—1911no—
279C2×C22⋊C4—1911no—
280C2×C22⋊C4—1911no—
281C2×C22⋊C4—1911no—
282C2×C22⋊C4—1911no—
283C2×C22⋊C4C2×D19111yescharacteristic
284C2×C22⋊C4C2×D19111yescharacteristic
285#95—1911no—
286#95—1911no—
287#95—1911no—
288#95—1911no—
289#95—1911no—
290#95—1911no—
291#95C2×D19111yescharacteristic
292#95C2×D19111yescharacteristic
293C24—1911no—
294C24—1911no—
295C24C22×D19111yescharacteristic
296#41—38232no—
297C2×Q8—3822no—
298C2×Q8—3822no—
299C2×Q8—3822no—
300C2×D4—3822no—
301C2×D4—3822no—
302C2×D4—3822no—
303C2×D4—3822no—
304C2×D4—3822no—
305C2×D4—3822no—
306C2×D4—3822no—
307C2×D4—1914no—
308C2×D4—1914no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—3822no—
313C22×C4—3822no—
314C22×C4—3822no—
315C22×C4—3822no—
316C22×C4—3822no—
317C22×C4—3822no—
318C22×C4—3822no—
319C22×C4—3822no—
320C22×C4—3822no—
321C22×C4—3822no—
322C22×C4—3822no—
323C22×C4—1911no—
324C22×C4—1911no—
325C22×C4—1911no—
326C22×C4—1911no—
327C22×C4—1911no—
328C22×C4—1911no—
329C22×C4—1914no—
330C22×C4C22×D19111yescharacteristic
331C22×C4C22×D19111yescharacteristic
332C22×C4C22×D19111yescharacteristic
333C22×C4C22×D19111yescharacteristic
334C22×C4C22×D19111yescharacteristic
335C22×C4C22×D19111yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—3822no—
339C4⋊C4—3822no—
340C4⋊C4—3822no—
341C4⋊C4—3822no—
342C4⋊C4—3822no—
343C4⋊C4—3822no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—3822no—
347C22⋊C4—3822no—
348C22⋊C4—3822no—
349C22⋊C4—3822no—
350C22⋊C4—3822no—
351C22⋊C4—3822no—
352C22⋊C4—3822no—
353C22⋊C4—3822no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—3821no—
361C23—3821no—
362C23—3821no—
363C23—3821no—
364C23—3821no—
365C23—3821no—
366C23—3821no—
367C23—3821no—
368C23—3821no—
369C23—3821no—
370C23—1912no—
371C23—1912no—
372C23—1912no—
373C23—1912no—
374C23D4×D19112yes—
375C23C23×D19111yescharacteristic
376Q8—3828no—
377D4—3828no—
378D4—3828no—
379D4—3828no—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—3828no—
397C2×C4—3828no—
398C2×C4—3828no—
399C2×C4—3821no—
400C2×C4—3821no—
401C2×C4—3821no—
402C2×C4—3821no—
403C2×C4—3821no—
404C2×C4—3821no—
405C2×C4—3821no—
406C2×C4—3821no—
407C2×C4—3821no—
408C2×C4—3821no—
409C2×C4—3821no—
410C2×C4—3821no—
411C2×C4—3821no—
412C2×C4—3821no—
413C2×C4—3821no—
414C2×C4—1912no—
415C2×C4—1912no—
416C2×C4—1912no—
417C2×C4—7644no—
418C2×C4D4×D19112yes—
419C2×C4D4×D19112yes—
420C2×C4#1569028712yescomplemented
421C22—22no—
422C22—22no—
423C22—22no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—3822no—
429C22—3822no—
430C22—3822no—
431C22—3822no—
432C22—3822no—
433C22—3822no—
434C22—3822no—
435C22—3822no—
436C22—3822no—
437C22—1914no—
438C22—7641no—
439C22—7641no—
440C22—7641no—
441C22—7641no—
442C22C2×D4×D19111yescharacteristic
443C22C2×D4×D19111yescharacteristic
444C22C2×D4×D19111yescharacteristic
445C22#41×D19111yescharacteristic
446C22C2×#1569028711yescharacteristic
447C22#2025936911yescharacteristic
448C22#2025937011yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—3822no—
456C4—3822no—
457C4—3822no—
458C4—7641no—
459C4—7641no—
460C4—7641no—
461C2—42no—
462C2—3824no—
463C2—7642no—
464C2#120×D19111yescharacteristic
465C2#2713317211yescharacteristic
466C2#2713323611yescharacteristic
467C2#2713331111yescharacteristic
468C2#2713331411yescharacteristic
469C2#2713331511yescharacteristic
470C2#2713331811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#385×C191C2
prime radicalC22×#15690287C2
hypercenter#385D191
hypocenterC191#2406
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#20259307C2semidirect
3. C2×#20259304C2semidirect
4. C2×C4⋊C4×D191C2semidirect
5. #27132584C2semidirect
6. #27132583C2semidirect
7. #27132579C2semidirect
8. #27132578C2semidirect
9. #27132541C2semidirect
10. #27132539C2semidirect
11. #27132537C2semidirect
12. #27132536C2semidirect
13. #27132534C2semidirect
14. #27132525C2semidirect
15. #385×C191C2semidirect
153. C2×Dic191#41semidirect
184. C2×C4×C191#41semidirect
235. C191#2406semidirect
250. #385D191semidirect
420. C2×C4#15690287semidirect

Maximal subgroups

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

subgroupindex
C22×#156902872
C2×#202593072
C2×#202593042
C2×C4⋊C4×D1912
#271325842
#271325832
#271325792
#271325782
#271325412
#271325392
#271325372
#271325362
#271325342
#271325252
#385×C1912
#2406191