Finite Group Databasebeta

← #37575864

Subgroups — #37575864

Order 24448.

908 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15690287C211yesmaximal, characteristic, complemented
2C2×#20259300C211yesmaximal, characteristic, complemented
3C2×#20259295C211yesmaximal, characteristic, complemented
4C2×#20259284C211yesmaximal, characteristic, complemented
5#27132580C211yesmaximal, characteristic, complemented
6#27132578C211yesmaximal, characteristic, complemented
7#27132573C211yesmaximal, characteristic, complemented
8#27132572C211yesmaximal, characteristic, complemented
9#27132567C211yesmaximal, characteristic, complemented
10#27132563C211yesmaximal, characteristic, complemented
11#27132542C211yesmaximal, characteristic, complemented
12#27132541C211yesmaximal, characteristic, complemented
13#27132534C211yesmaximal, characteristic, complemented
14#27132522C211yesmaximal, characteristic, complemented
15#385×C191C211yesmaximal, characteristic, complemented
16C2×#15690287—24no—
17C2×#15690287—24no—
18C2×#15690287—24no—
19C22×Dic382C2211yescharacteristic
20C22×D764C2211yescharacteristic
21C22×D764C2211yescharacteristic
22C22×D764C2211yescharacteristic
23C22×C22⋊D191C2211yescharacteristic
24C22×C22⋊D191C2211yescharacteristic
25C22×C22⋊D191C2211yescharacteristic
26C22×C4×D191C2211yescharacteristic
27C22×C4×D191C2211yescharacteristic
28C22×C4×D191C2211yescharacteristic
29#20259300—24no—
30#20259295—24no—
31#20259284—24no—
32C2×Dic191⋊C4C2211yescharacteristic
33C2×Dic191⋊C4C2211yescharacteristic
34C2×Dic191⋊C4C2211yescharacteristic
35C2×C22⋊Dic191C2211yescharacteristic
36C2×(C2×D191)⋊C4C2211yescharacteristic
37C2×(C2×D191)⋊C4C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×(C2×D191)⋊C4C2211yescharacteristic
41C2×(C2×D191)⋊C4C2211yescharacteristic
42C2×(C2×D191)⋊C4C2211yescharacteristic
43C2×(C2×D191)⋊C4C2211yescharacteristic
44C2×(C2×D191)⋊C4C2211yescharacteristic
45C2×C4×Dic191C2211yescharacteristic
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
59C23×D191C2311yescharacteristic
60#15690287—232no—
61C2×Dic382—22no—
62C2×Dic382—22no—
63C2×Dic382D414yescomplemented
64C2×D764—22no—
65C2×D764—22no—
66C2×D764—22no—
67C2×D764—22no—
68C2×D764—22no—
69C2×D764—22no—
70C2×D764—22no—
71C2×D764—22no—
72C2×D764—41no—
73C2×C22⋊D191—22no—
74C2×C22⋊D191—22no—
75C2×C22⋊D191—22no—
76C2×C22⋊D191—22no—
77C2×C22⋊D191—22no—
78C2×C22⋊D191—22no—
79C2×C22⋊D191—22no—
80C2×C22⋊D191—41no—
81C2×C22⋊D191D414yescomplemented
82C22×Dic191C2311yescharacteristic
83C22×Dic191C2311yescharacteristic
84C22×Dic191C2311yescharacteristic
85C22×Dic191C2311yescharacteristic
86C22×Dic191C2311yescharacteristic
87C2×C4×D191—22no—
88C2×C4×D191—22no—
89C2×C4×D191—22no—
90C2×C4×D191—22no—
91C2×C4×D191—22no—
92C2×C4×D191—22no—
93C2×C4×D191—22no—
94C2×C4×D191—22no—
95C2×C4×D191D414yescomplemented
96Dic191⋊C4—22no—
97Dic191⋊C4—22no—
98Dic191⋊C4—22no—
99C22⋊Dic191—41no—
100(C2×D191)⋊C4—22no—
101(C2×D191)⋊C4—22no—
102(C2×D191)⋊C4—22no—
103(C2×D191)⋊C4—22no—
104(C2×D191)⋊C4—22no—
105(C2×D191)⋊C4—22no—
106(C2×D191)⋊C4—41no—
107(C2×D191)⋊C4—41no—
108(C2×D191)⋊C4—41no—
109C4×Dic191—22no—
110C24×C191C2311yescharacteristic
111C22×C4×C191—22no—
112C22×C4×C191—22no—
113C22×C4×C191—22no—
114C22×C4×C191C2311yescharacteristic
115C22×C4×C191C2311yescharacteristic
116C22×C4×C191C2311yescharacteristic
117C22×C4×C191C2311yescharacteristic
118C22×C4×C191C2311yescharacteristic
119C22×C4×C191C2311yescharacteristic
120C4⋊C4×C191—22no—
121C4⋊C4×C191—22no—
122C22⋊C4×C191—22no—
123C22⋊C4×C191—22no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—21no—
128C22×D191—21no—
129C22×D191—21no—
130C22×D191—21no—
131C22×D191—21no—
132C22×D191—21no—
133C22×D191—21no—
134C22×D191—21no—
135C22×D191—21no—
136C22×D191—21no—
137C22×D191—21no—
138C22×D191—21no—
139C22×D191—21no—
140C22×D191—21no—
141C22×D191—21no—
142C22×D191C2×D412yes—
143C22×D191C2×D412yes—
144C22×D191#4112yescomplemented
145Dic382—28no—
146D764—28no—
147D764—28no—
148D764—44no—
149C22⋊D191—28no—
150C22⋊D191—28no—
151C22⋊D191—44no—
152C2×Dic191—21no—
153C2×Dic191—21no—
154C2×Dic191—21no—
155C2×Dic191—21no—
156C2×Dic191—21no—
157C2×Dic191—21no—
158C2×Dic191—21no—
159C2×Dic191—21no—
160C2×Dic191—21no—
161C2×Dic191—21no—
162C2×Dic191—21no—
163C2×Dic191C2×D412yes—
164C2×Dic191C2×D412yes—
165C2×Dic191C2×D412yes—
166C2×Dic191C2×D412yes—
167C4×D191—28no—
168C4×D191—28no—
169C4×D191—44no—
170C23×C191—21no—
171C23×C191—21no—
172C23×C191—21no—
173C23×C191—21no—
174C23×C191—21no—
175C23×C191—21no—
176C23×C191C2×D412yes—
177C23×C191C2411yescharacteristic
178C2×C4×C191—28no—
179C2×C4×C191—21no—
180C2×C4×C191—21no—
181C2×C4×C191—21no—
182C2×C4×C191—21no—
183C2×C4×C191—21no—
184C2×C4×C191—21no—
185C2×C4×C191—21no—
186C2×C4×C191—21no—
187C2×C4×C191—21no—
188C2×C4×C191—21no—
189C2×C4×C191—21no—
190C2×C4×C191—21no—
191C2×C4×C191—21no—
192C2×C4×C191—21no—
193C2×C4×C191—21no—
194C2×C4×C191C2×D412yes—
195C2×C4×C191C2×D412yes—
196C2×C4×C191#4112yescomplemented
197C22×C191—22no—
198C22×C191—22no—
199C22×C191—22no—
200C22×C191—41no—
201C22×C191—41no—
202C22×C191—41no—
203C22×C191—41no—
204C22×C191C22×D411yescharacteristic
205C22×C191C22×D411yescharacteristic
206C22×C191C22×D411yescharacteristic
207C22×C191C2×#4111yescharacteristic
208C22×C191C2×#4111yescharacteristic
209C22×C191H(5,𝔽2)11yescharacteristic
210C22×C1912−1+411yescharacteristic
211C2×D191—22no—
212C2×D191—22no—
213C2×D191—22no—
214C2×D191—22no—
215C2×D191—22no—
216C2×D191—22no—
217C2×D191—22no—
218C2×D191—22no—
219C2×D191—22no—
220C2×D191—41no—
221C2×D191—41no—
222C2×D191—41no—
223C2×D191—41no—
224C2×D191—41no—
225C2×D191—41no—
226C2×D191—41no—
227C2×D191—41no—
228C2×D191—41no—
229C2×D191—41no—
230C2×D191—41no—
231C2×D191—41no—
232C4×C191—22no—
233C4×C191—22no—
234C4×C191—22no—
235C4×C191—41no—
236C4×C191—41no—
237C4×C191—41no—
238Dic191—22no—
239Dic191—41no—
240Dic191—41no—
241Dic191—41no—
242Dic191#12014yes—
243C2×C191—42no—
244C2×C191C2×#12011yescharacteristic
245C2×C191#52511yescharacteristic
246C2×C191#53711yescharacteristic
247C2×C191#54611yescharacteristic
248C2×C191#54711yescharacteristic
249C2×C191#54811yescharacteristic
250C2×C191#55011yescharacteristic
251D191—42no—
252D191—42no—
253D191—42no—
254C191#240711yescharacteristic, complemented, Sylow
255#2407—1911nomaximal, Sylow
256C22×#41—1911no—
257C2×C22⋊Q8—1911no—
258C2×#121—1911no—
259C2×C4×D4—1911no—
260#394—1911no—
261#394—1911no—
262#392—1911no—
263#392—1911no—
264#391—1911no—
265#390—1911no—
266#386—1911no—
267#386—1911no—
268#385—1911no—
269#385D19111yescharacteristic, complemented
270#380—1911no—
271C2×#41—3824no—
272C2×#41—3824no—
273C2×#41—3824no—
274C22×Q8—1911no—
275C22×D4—1911no—
276C22×D4—1911no—
277C22×D4—1911no—
278C22×D4—1911no—
279C22×D4—1911no—
280C22×D4—1911no—
281C23×C4—1911no—
282C23×C4—1911no—
283C23×C4—1911no—
284C23×C4C2×D19111yescharacteristic
285C22⋊Q8—3824no—
286#121—3824no—
287C4×D4—3824no—
288C2×C4⋊C4—1911no—
289C2×C4⋊C4—1911no—
290C2×C4⋊C4—1911no—
291C2×C4⋊C4C2×D19111yescharacteristic
292C2×C4⋊C4C2×D19111yescharacteristic
293C2×C22⋊C4—1911no—
294C2×C22⋊C4—1911no—
295C2×C22⋊C4—1911no—
296C2×C22⋊C4—1911no—
297C2×C22⋊C4—1911no—
298C2×C22⋊C4—1911no—
299C2×C22⋊C4—1911no—
300C2×C22⋊C4—1911no—
301C2×C22⋊C4—1911no—
302C2×C22⋊C4—1911no—
303C2×C22⋊C4C2×D19111yescharacteristic
304C2×C22⋊C4C2×D19111yescharacteristic
305C2×C42—1911no—
306#95—1911no—
307#95—1911no—
308#95—1911no—
309#95—1911no—
310#95C2×D19111yescharacteristic
311#95C2×D19111yescharacteristic
312C24—1911no—
313C24—1911no—
314C24—1911no—
315C24C22×D19111yescharacteristic
316#41—38232no—
317C2×Q8—3822no—
318C2×Q8—3822no—
319C2×Q8—1914no—
320C2×D4—3822no—
321C2×D4—3822no—
322C2×D4—3822no—
323C2×D4—3822no—
324C2×D4—3822no—
325C2×D4—3822no—
326C2×D4—3822no—
327C2×D4—3822no—
328C2×D4—3822no—
329C2×D4—3822no—
330C2×D4—3822no—
331C2×D4—3822no—
332C2×D4—3822no—
333C2×D4—3822no—
334C2×D4—3822no—
335C2×D4—1914no—
336C2×D4—7641no—
337C2×D4—7641no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—3822no—
342C22×C4—3822no—
343C22×C4—3822no—
344C22×C4—3822no—
345C22×C4—3822no—
346C22×C4—3822no—
347C22×C4—3822no—
348C22×C4—3822no—
349C22×C4—1911no—
350C22×C4—1911no—
351C22×C4—1911no—
352C22×C4—1911no—
353C22×C4—1914no—
354C22×C4—1911no—
355C22×C4C22×D19111yescharacteristic
356C22×C4C22×D19111yescharacteristic
357C22×C4C22×D19111yescharacteristic
358C22×C4C22×D19111yescharacteristic
359C22×C4C22×D19111yescharacteristic
360C22×C4C22×D19111yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—3822no—
364C4⋊C4—3822no—
365C4⋊C4—3822no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—3822no—
369C22⋊C4—3822no—
370C22⋊C4—3822no—
371C22⋊C4—3822no—
372C22⋊C4—3822no—
373C22⋊C4—3822no—
374C22⋊C4—7641no—
375C22⋊C4—7641no—
376C22⋊C4—7641no—
377C22⋊C4—7641no—
378C42—3822no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—3821no—
386C23—3821no—
387C23—3821no—
388C23—3821no—
389C23—3821no—
390C23—3821no—
391C23—3821no—
392C23—3821no—
393C23—3821no—
394C23—3821no—
395C23—3821no—
396C23—3821no—
397C23—3821no—
398C23—3821no—
399C23—3821no—
400C23—3821no—
401C23—3821no—
402C23—3821no—
403C23—1912no—
404C23—1912no—
405C23—1912no—
406C23D4×D19112yes—
407C23C23×D19111yescharacteristic
408Q8—3828no—
409D4—3828no—
410D4—3828no—
411D4—3828no—
412D4—3828no—
413D4—7644no—
414D4—7644no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—3828no—
432C2×C4—3828no—
433C2×C4—3821no—
434C2×C4—3821no—
435C2×C4—3821no—
436C2×C4—3821no—
437C2×C4—3821no—
438C2×C4—3821no—
439C2×C4—3821no—
440C2×C4—3821no—
441C2×C4—3821no—
442C2×C4—3821no—
443C2×C4—3821no—
444C2×C4—1912no—
445C2×C4—1912no—
446C2×C4—1912no—
447C2×C4—1912no—
448C2×C4—7644no—
449C2×C4D4×D19112yes—
450C2×C4D4×D19112yes—
451C2×C4#1569028712yescomplemented
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—3822no—
460C22—3822no—
461C22—3822no—
462C22—3822no—
463C22—3822no—
464C22—3822no—
465C22—3822no—
466C22—3822no—
467C22—3822no—
468C22—7641no—
469C22—7641no—
470C22—7641no—
471C22—7641no—
472C22—7641no—
473C22—7641no—
474C22—7641no—
475C22—7641no—
476C22—7641no—
477C22—7641no—
478C22—7641no—
479C22—7641no—
480C22C2×D4×D19111yescharacteristic
481C22C2×D4×D19111yescharacteristic
482C22C2×D4×D19111yescharacteristic
483C22#41×D19111yescharacteristic
484C22C2×#1569028711yescharacteristic
485C22#2025936811yescharacteristic
486C22#2025937011yescharacteristic
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—3822no—
494C4—1914no—
495C4—7641no—
496C4—7641no—
497C4—7641no—
498C2—42no—
499C2—7642no—
500C2—7642no—
501C2—7642no—
502C2#120×D19111yescharacteristic
503C2#2713317211yescharacteristic
504C2#2713324311yescharacteristic
505C2#2713330511yescharacteristic
506C2#2713331411yescharacteristic
507C2#2713332311yescharacteristic
508C2#2713333411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#385×C191C2
hypercenter#385D191
hypocenterC191#2407
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#15690287C2semidirect
2. C2×#20259300C2semidirect
3. C2×#20259295C2semidirect
4. C2×#20259284C2semidirect
5. #27132580C2semidirect
6. #27132578C2semidirect
7. #27132573C2semidirect
8. #27132572C2semidirect
9. #27132567C2semidirect
10. #27132563C2semidirect
11. #27132542C2semidirect
12. #27132541C2semidirect
13. #27132534C2semidirect
14. #27132522C2semidirect
15. #385×C191C2semidirect
63. C2×Dic382D4semidirect
81. C2×C22⋊D191D4semidirect
95. C2×C4×D191D4semidirect
144. C22×D191#41semidirect
196. C2×C4×C191#41semidirect
254. C191#2407semidirect
269. #385D191semidirect
451. 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×#202593002
C2×#202592952
C2×#202592842
#271325802
#271325782
#271325732
#271325722
#271325672
#271325632
#271325422
#271325412
#271325342
#271325222
#385×C1912
#2407191