Finite Group Databasebeta

← #37575975

Subgroups — #37575975

Order 24448.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259303C211yesmaximal, characteristic, complemented
2C2×#20259290C211yesmaximal, characteristic
3C2×(C22⋊D191)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D191C211yesmaximal, characteristic, complemented
5#27132598C211yesmaximal, characteristic, complemented
6#27132591C211yesmaximal, characteristic, complemented
7#27132578C211yesmaximal, characteristic, complemented
8#27132557C211yesmaximal, characteristic, complemented
9#27132552C211yesmaximal, characteristic, complemented
10#27132540C211yesmaximal, characteristic, complemented
11#27132539C211yesmaximal, characteristic, complemented
12#27132534C211yesmaximal, characteristic, complemented
13#27132522C211yesmaximal, characteristic, complemented
14#27132518C211yesmaximal, characteristic, complemented
15#394×C191C211yesmaximal, characteristic, complemented
16C24×D191C2211yescharacteristic
17C22×C22⋊D191C2211yescharacteristic
18C22×C22⋊D191C2211yescharacteristic
19C23×Dic191C2211yescharacteristic, complemented
20C22×C4×D191C2211yescharacteristic
21C22×C4×D191C2211yescharacteristic
22#20259303—24no—
23#20259290—24no—
24(C22⋊D191)⋊C4—24no—
25C2×Dic191⋊C4C2211yescharacteristic, complemented
26C2×Dic191⋊C4C2211yescharacteristic, complemented
27C2×Dic191⋊C4C2211yescharacteristic, complemented
28C2×Dic191⋊C4C2211yescharacteristic, complemented
29C2×C22⋊Dic191C2211yescharacteristic
30C2×C22⋊Dic191C2211yescharacteristic
31C2×C22⋊Dic191C2211yescharacteristic
32C2×C22⋊Dic191C2211yescharacteristic, complemented
33C2×(C2×D191)⋊C4C2211yescharacteristic
34C2×(C2×D191)⋊C4C2211yescharacteristic
35C2×(C2×D191)⋊C4C2211yescharacteristic
36C2×(C2×D191)⋊C4C2211yescharacteristic
37C2×(C2×D191)⋊C4C2211yescharacteristic
38C2×(C2×D191)⋊C4C2211yescharacteristic
39C2×(C2×D191)⋊C4C2211yescharacteristic
40C2×(C2×D191)⋊C4C2211yescharacteristic
41C22⋊C4×D191—24no—
42C2×C4×Dic191C2211yescharacteristic, complemented
43C191⋊#95C2211yescharacteristic, complemented
44C191⋊#95C2211yescharacteristic, complemented
45C191⋊#95C2211yescharacteristic, complemented
46C191⋊#95C2211yescharacteristic, complemented
47C191⋊#95C2211yescharacteristic, complemented
48C22×D4×C191C2211yescharacteristic
49C2×C4⋊C4×C191C2211yescharacteristic, complemented
50C2×C22⋊C4×C191C2211yescharacteristic
51C2×C22⋊C4×C191C2211yescharacteristic
52C2×C22⋊C4×C191C2211yescharacteristic, complemented
53C2×C22⋊C4×C191C2211yescharacteristic, complemented
54#95×C191C2211yescharacteristic, complemented
55C23×D191—22no—
56C23×D191—22no—
57C23×D191—22no—
58C23×D191—22no—
59C23×D191—22no—
60C23×D191—41no—
61C23×D191—41no—
62C23×D191C2311yescharacteristic
63C23×D191C2311yescharacteristic
64C2×C22⋊D191—22no—
65C2×C22⋊D191—22no—
66C2×C22⋊D191—22no—
67C2×C22⋊D191—22no—
68C2×C22⋊D191—41no—
69C2×C22⋊D191D414yes—
70C22×Dic191—22no—
71C22×Dic191—22no—
72C22×Dic191—41no—
73C22×Dic191C2311yescharacteristic
74C22×Dic191C2311yescharacteristic
75C22×Dic191C2311yescharacteristic
76C22×Dic191C2311yescharacteristic
77C22×Dic191C2311yescharacteristic
78C22×Dic191C2311yescharacteristic
79C2×C4×D191—22no—
80C2×C4×D191—22no—
81C2×C4×D191—22no—
82C2×C4×D191—22no—
83C2×C4×D191—41no—
84C2×C4×D191—41no—
85Dic191⋊C4—22no—
86Dic191⋊C4—22no—
87Dic191⋊C4—22no—
88Dic191⋊C4—41no—
89C22⋊Dic191—22no—
90C22⋊Dic191—22no—
91C22⋊Dic191—22no—
92C22⋊Dic191D414yescomplemented
93(C2×D191)⋊C4—22no—
94(C2×D191)⋊C4—22no—
95(C2×D191)⋊C4—22no—
96(C2×D191)⋊C4—22no—
97(C2×D191)⋊C4—22no—
98(C2×D191)⋊C4—22no—
99(C2×D191)⋊C4—22no—
100(C2×D191)⋊C4—41no—
101C4×Dic191—22no—
102C24×C191C2311yescharacteristic
103C24×C191C2311yescharacteristic
104C2×D4×C191—22no—
105C2×D4×C191—22no—
106C2×D4×C191—41no—
107C22×C4×C191C2311yescharacteristic
108C22×C4×C191C2311yescharacteristic
109C22×C4×C191C2311yescharacteristic
110C22×C4×C191C2311yescharacteristic
111C22×C4×C191C2311yescharacteristic
112C4⋊C4×C191—41no—
113C22⋊C4×C191—22no—
114C22⋊C4×C191—22no—
115C22⋊C4×C191—41no—
116C22⋊C4×C191—41no—
117C22×D191—21no—
118C22×D191—21no—
119C22×D191—21no—
120C22×D191—21no—
121C22×D191—28no—
122C22×D191—21no—
123C22×D191—21no—
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—21no—
128C22×D191—44no—
129C22×D191—44no—
130C22×D191—44no—
131C22×D191—44no—
132C22×D191—44no—
133C22×D191—44no—
134C22×D191C2×D412yes—
135C22×D191#4112yes—
136C22×D191#4112yes—
137C22×D191#4112yes—
138C22⋊D191—28no—
139C22⋊D191—44no—
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×Dic191—21no—
152C2×Dic191—21no—
153C2×Dic191—21no—
154C2×Dic191—21no—
155C2×Dic191—44no—
156C2×Dic191C2×D412yes—
157C2×Dic191C2×D412yes—
158C2×Dic191C2×D412yes—
159C4×D191—44no—
160C4×D191—44no—
161C23×C191—21no—
162C23×C191—21no—
163C23×C191—21no—
164C23×C191—21no—
165C23×C191—21no—
166C23×C191—21no—
167C23×C191—21no—
168C23×C191—21no—
169C23×C191—21no—
170C23×C191—21no—
171C23×C191—21no—
172C23×C191—21no—
173C23×C191C2×D412yes—
174C23×C191C2×D412yes—
175C23×C191C2411yescharacteristic
176D4×C191—44no—
177C2×C4×C191—21no—
178C2×C4×C191—21no—
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—
192C22×C191—22no—
193C22×C191—22no—
194C22×C191—22no—
195C22×C191—22no—
196C22×C191—22no—
197C22×C191—22no—
198C22×C191—41no—
199C22×C191—41no—
200C22×C191—41no—
201C22×C191—41no—
202C22×C191—41no—
203C22×C191—41no—
204C22×C191—41no—
205C22×C191—41no—
206C22×C191C22×D411yescharacteristic
207C22×C191C22×D411yescharacteristic
208C22×C191C2×#4111yescharacteristic
209C22×C191C2×#4111yescharacteristic
210C22×C191C2×#4111yescharacteristic
211C22×C191H(5,𝔽2)11yescharacteristic
212C22×C191H(5,𝔽2)11yescharacteristic
213C2×D191—22no—
214C2×D191—22no—
215C2×D191—22no—
216C2×D191—22no—
217C2×D191—22no—
218C2×D191—22no—
219C2×D191—22no—
220C2×D191—22no—
221C2×D191—22no—
222C2×D191—416no—
223C2×D191—41no—
224C2×D191—41no—
225C2×D191—41no—
226C2×D191—41no—
227C2×D191#12314yescomplemented
228C4×C191—41no—
229C4×C191—41no—
230C4×C191—41no—
231C4×C191—41no—
232C4×C191—41no—
233Dic191—22no—
234Dic191—22no—
235Dic191—22no—
236Dic191—41no—
237Dic191—41no—
238Dic191—41no—
239C2×C191—42no—
240C2×C191—42no—
241C2×C191C2×#12311yescharacteristic
242C2×C191#52511yescharacteristic
243C2×C191#53811yescharacteristic
244C2×C191D4211yescharacteristic
245C2×C191#54611yescharacteristic
246C2×C191#54611yescharacteristic
247C2×C191#55111yescharacteristic
248D191—24no—
249D191—42no—
250C191#241811yescharacteristic, complemented, Sylow
251#2418—1911nomaximal, Sylow
252C2×#123—1911no—
253C2×#120—1911no—
254C2×C4×D4—1911no—
255C22×C22⋊C4—1911no—
256#399—1911no—
257#397—1911no—
258#394—1911no—
259#394D19111yescharacteristic, complemented
260#388—1911no—
261#388—1911no—
262#386—1911no—
263#386—1911no—
264#385—1911no—
265#380—1911no—
266#379—1911no—
267C25—1911no—
268C22×D4—1911no—
269C22×D4—1911no—
270C22×D4C2×D19111yescharacteristic
271C23×C4—1911no—
272C23×C4—1911no—
273C23×C4—1911no—
274#123—3824no—
275#120—3824no—
276C4×D4—3824no—
277C2×C4⋊C4—1911no—
278C2×C4⋊C4—1911no—
279C2×C4⋊C4—1911no—
280C2×C4⋊C4—1911no—
281C2×C4⋊C4C2×D19111yescharacteristic, complemented
282C2×C22⋊C4—3824no—
283C2×C22⋊C4—1911no—
284C2×C22⋊C4—1911no—
285C2×C22⋊C4—1911no—
286C2×C22⋊C4—1911no—
287C2×C22⋊C4—1911no—
288C2×C22⋊C4—1911no—
289C2×C22⋊C4—1911no—
290C2×C22⋊C4—1911no—
291C2×C22⋊C4—1911no—
292C2×C22⋊C4—1911no—
293C2×C22⋊C4—1911no—
294C2×C22⋊C4—1911no—
295C2×C22⋊C4C2×D19111yescharacteristic
296C2×C22⋊C4C2×D19111yescharacteristic
297C2×C22⋊C4C2×D19111yescharacteristic, complemented
298C2×C22⋊C4C2×D19111yescharacteristic, complemented
299C2×C42—1911no—
300#95—1911no—
301#95—1911no—
302#95—1911no—
303#95—1911no—
304#95—1911no—
305#95C2×D19111yescharacteristic, complemented
306C24—3822no—
307C24—3822no—
308C24—3822no—
309C24—3822no—
310C24—3822no—
311C24—1911no—
312C24—1911no—
313C24—7641no—
314C24—7641no—
315C24C22×D19111yescharacteristic
316C24C22×D19111yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—3822no—
321C2×D4—3822no—
322C2×D4—3822no—
323C2×D4—3822no—
324C2×D4—1914no—
325C2×D4—7641no—
326C22×C4—3822no—
327C22×C4—3822no—
328C22×C4—3822no—
329C22×C4—3822no—
330C22×C4—3822no—
331C22×C4—3822no—
332C22×C4—1911no—
333C22×C4—1911no—
334C22×C4—1911no—
335C22×C4—1911no—
336C22×C4—1911no—
337C22×C4—1911no—
338C22×C4—7641no—
339C22×C4—7641no—
340C22×C4—7641no—
341C22×C4C22×D19111yescharacteristic
342C22×C4C22×D19111yescharacteristic
343C22×C4C22×D19111yescharacteristic
344C22×C4C22×D19111yescharacteristic
345C22×C4C22×D19111yescharacteristic
346C4⋊C4—41no—
347C4⋊C4—3822no—
348C4⋊C4—3822no—
349C4⋊C4—3822no—
350C4⋊C4—7641no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—3822no—
356C22⋊C4—3822no—
357C22⋊C4—3822no—
358C22⋊C4—3822no—
359C22⋊C4—3822no—
360C22⋊C4—3822no—
361C22⋊C4—3822no—
362C22⋊C4—3822no—
363C22⋊C4—3822no—
364C22⋊C4—3822no—
365C22⋊C4—1914no—
366C22⋊C4—7641no—
367C42—3822no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—3828no—
381C23—3821no—
382C23—3821no—
383C23—3821no—
384C23—3821no—
385C23—3821no—
386C23—3821no—
387C23—3821no—
388C23—3821no—
389C23—3821no—
390C23—3821no—
391C23—1912no—
392C23—1912no—
393C23—1912no—
394C23—1912no—
395C23—7644no—
396C23—7644no—
397C23—7644no—
398C23—7644no—
399C23—7644no—
400C23—7644no—
401C23D4×D19112yes—
402C23D4×D19112yes—
403C23C23×D19111yescharacteristic
404D4—44no—
405D4—3828no—
406D4—7644no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—3821no—
423C2×C4—3821no—
424C2×C4—3821no—
425C2×C4—3821no—
426C2×C4—3821no—
427C2×C4—3821no—
428C2×C4—3821no—
429C2×C4—3821no—
430C2×C4—3821no—
431C2×C4—3821no—
432C2×C4—3821no—
433C2×C4—3821no—
434C2×C4—3821no—
435C2×C4—3821no—
436C2×C4—3821no—
437C2×C4—1912no—
438C2×C4—1912no—
439C2×C4—1912no—
440C2×C4—7644no—
441C2×C4—7644no—
442C2×C4—7644no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—3822no—
458C22—3822no—
459C22—3822no—
460C22—3822no—
461C22—3822no—
462C22—3822no—
463C22—3822no—
464C22—3822no—
465C22—3822no—
466C22—1914no—
467C22—76416no—
468C22—7641no—
469C22—7641no—
470C22—7641no—
471C22—7641no—
472C22C2×D4×D19111yescharacteristic
473C22C2×D4×D19111yescharacteristic
474C22#41×D19111yescharacteristic
475C22#41×D19111yescharacteristic
476C22#41×D19111yescharacteristic
477C22#2025936711yescharacteristic
478C22#2025936711yescharacteristic
479C4—41no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—3822no—
485C4—3822no—
486C4—3822no—
487C4—7641no—
488C4—7641no—
489C4—7641no—
490C2—42no—
491C2—42no—
492C2—3824no—
493C2—7642no—
494C2#123×D19111yescharacteristic
495C2#2713317111yescharacteristic
496C2#2713324511yescharacteristic
497C2#2713329511yescharacteristic
498C2#2713330211yescharacteristic
499C2#2713330211yescharacteristic
500C2#2713334311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D191
commutator, socle, cosocleC23×C191C24
Fitting#394×C191C2
prime radicalC2×#20259290C2
hypercenter#394D191
hypocenterC191#2418
LCSG ▷ C23×C191 ▷ C191
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#20259303C2semidirect
3. C2×(C22⋊D191)⋊C4C2semidirect
4. C2×C22⋊C4×D191C2semidirect
5. #27132598C2semidirect
6. #27132591C2semidirect
7. #27132578C2semidirect
8. #27132557C2semidirect
9. #27132552C2semidirect
10. #27132540C2semidirect
11. #27132539C2semidirect
12. #27132534C2semidirect
13. #27132522C2semidirect
14. #27132518C2semidirect
15. #394×C191C2semidirect
19. C23×Dic191C22semidirect
25. C2×Dic191⋊C4C22semidirect
26. C2×Dic191⋊C4C22semidirect
27. C2×Dic191⋊C4C22semidirect
28. C2×Dic191⋊C4C22semidirect
32. C2×C22⋊Dic191C22semidirect
42. C2×C4×Dic191C22semidirect
43. C191⋊#95C22semidirect
44. C191⋊#95C22semidirect
45. C191⋊#95C22semidirect
46. C191⋊#95C22semidirect
47. C191⋊#95C22semidirect
49. C2×C4⋊C4×C191C22semidirect
52. C2×C22⋊C4×C191C22semidirect
53. C2×C22⋊C4×C191C22semidirect
54. #95×C191C22semidirect
92. C22⋊Dic191D4semidirect
227. C2×D191#123semidirect
250. C191#2418semidirect
259. #394D191semidirect
281. C2×C4⋊C4C2×D191semidirect
297. C2×C22⋊C4C2×D191semidirect
298. C2×C22⋊C4C2×D191semidirect
305. #95C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593032
C2×#202592902
C2×(C22⋊D191)⋊C42
C2×C22⋊C4×D1912
#271325982
#271325912
#271325782
#271325572
#271325522
#271325402
#271325392
#271325342
#271325222
#271325182
#394×C1912
#2418191