Finite Group Databasebeta

← #36356222

Subgroups — #36356222

Order 22144.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149647C211yesmaximal, characteristic
2C2×#19149639C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D173C211yesmaximal, characteristic, complemented
4#26011391C211yesmaximal, characteristic, complemented
5#26011385C211yesmaximal, characteristic, complemented
6#26011385C211yesmaximal, characteristic, complemented
7#26011382C211yesmaximal, characteristic, complemented
8#26011381C211yesmaximal, characteristic, complemented
9#26011371C211yesmaximal, characteristic, complemented
10#26011368C211yesmaximal, characteristic, complemented
11#26011367C211yesmaximal, characteristic, complemented
12#26011362C211yesmaximal, characteristic, complemented
13#26011346C211yesmaximal, characteristic, complemented
14C4×#14583803C211yesmaximal, characteristic, complemented
15#390×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic, complemented
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22#19149647C418yescomplemented
23#19149639—24no—
24C2×#14583805C2211yescharacteristic, complemented
25C2×#14583805C2211yescharacteristic, complemented
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic, complemented
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C22⋊C4×D173—24no—
39C2×C4×Dic173C2211yescharacteristic, complemented
40C2×C4×Dic173C2211yescharacteristic, complemented
41#19149586C2211yescharacteristic, complemented
42#19149586C2211yescharacteristic, complemented
43#19149586C2211yescharacteristic, complemented
44#19149586C2211yescharacteristic, complemented
45#19149586C2211yescharacteristic, complemented
46#19149586C2211yescharacteristic, complemented
47C22×D4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic, complemented
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic, complemented
52C2×C22⋊C4×C173C2211yescharacteristic, complemented
53C2×C42×C173C2211yescharacteristic, complemented
54C23×D173—22no—
55C23×D173—22no—
56C23×D173—41no—
57C23×D173—41no—
58C23×D173—41no—
59C23×D173—41no—
60C23×D173C2×C414yescomplemented
61C23×D173C2311yescharacteristic
62C23×D173C2311yescharacteristic
63C2×C22⋊D173—22no—
64C2×C22⋊D173—22no—
65C2×C22⋊D173—41no—
66C2×C22⋊D173—41no—
67C2×C22⋊D173C2×C414yes—
68C2×C22⋊D173C2×C414yescomplemented
69C22×Dic173—22no—
70C22×Dic173—41no—
71C22×Dic173—41no—
72C22×Dic173C2311yescharacteristic
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C2×C4×D173—22no—
79C2×C4×D173—41no—
80C2×C4×D173—41no—
81C2×C4×D173—41no—
82C2×C4×D173—41no—
83C2×C4×D173D414yescomplemented
84#14583805—22no—
85#14583805—41no—
86C22⋊Dic173—22no—
87C22⋊Dic173—41no—
88C22⋊Dic173—41no—
89C22⋊Dic173C2×C414yescomplemented
90#14583803—22no—
91#14583803—22no—
92#14583803—22no—
93#14583803—22no—
94#14583803—41no—
95#14583803—41no—
96#14583803C2×C414yescomplemented
97#14583803C2×C414yescomplemented
98C4×Dic173—22no—
99C4×Dic173—41no—
100C24×C173C2311yescharacteristic
101C24×C173C2311yescharacteristic
102C2×D4×C173—41no—
103C2×D4×C173—41no—
104C2×D4×C173C2×C414yescomplemented
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110C4⋊C4×C173—41no—
111C22⋊C4×C173—22no—
112C22⋊C4×C173—22no—
113C22⋊C4×C173—41no—
114C22⋊C4×C173—41no—
115C42×C173—41no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—21no—
122C22×D173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—28no—
127C22×D173—44no—
128C22×D173—44no—
129C22×D173—44no—
130C22×D173—44no—
131C22×D173—44no—
132C22×D173—44no—
133C22×D173C22×C412yes—
134C22×D173C22×C412yes—
135C22×D173C2×D412yes—
136C22×D173#4112yes—
137C22⋊D173—28no—
138C22⋊D173—44no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173—21no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—21no—
150C2×Dic173—21no—
151C2×Dic173—21no—
152C2×Dic173—21no—
153C2×Dic173—21no—
154C2×Dic173—44no—
155C2×Dic173C22×C412yes—
156C2×Dic173C22×C412yes—
157C2×Dic173C2×D412yes—
158C4×D173—44no—
159C4×D173—44no—
160C23×C173—21no—
161C23×C173—21no—
162C23×C173—21no—
163C23×C173—21no—
164C23×C173—21no—
165C23×C173—21no—
166C23×C173—21no—
167C23×C173—21no—
168C23×C173—21no—
169C23×C173—21no—
170C23×C173—21no—
171C23×C173—21no—
172C23×C173C22×C412yes—
173C23×C173C22×C412yes—
174C23×C173C2411yescharacteristic
175D4×C173—44no—
176C2×C4×C173—21no—
177C2×C4×C173—21no—
178C2×C4×C173—21no—
179C2×C4×C173—21no—
180C2×C4×C173—21no—
181C2×C4×C173—21no—
182C2×C4×C173—21no—
183C2×C4×C173—21no—
184C2×C4×C173—21no—
185C2×C4×C173—21no—
186C2×C4×C173—21no—
187C2×C4×C173—21no—
188C2×C4×C173—21no—
189C2×C4×C173C22×C412yes—
190C2×C4×C173C2×D412yes—
191C22×C173—22no—
192C22×C173—22no—
193C22×C173—22no—
194C22×C173—22no—
195C22×C173—22no—
196C22×C173—22no—
197C22×C173—41no—
198C22×C173—41no—
199C22×C173—41no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173—41no—
203C22×C173—41no—
204C22×C173—41no—
205C22×C173C23×C411yescharacteristic
206C22×C173C22×D411yescharacteristic
207C22×C173C2×#4111yescharacteristic
208C22×C173H(5,𝔽2)11yescharacteristic
209C22×C173H(5,𝔽2)11yescharacteristic
210C22×C173H(5,𝔽2)11yescharacteristic
211C22×C173H(5,𝔽2)11yescharacteristic
212C2×D173—22no—
213C2×D173—22no—
214C2×D173—22no—
215C2×D173—22no—
216C2×D173—22no—
217C2×D173—22no—
218C2×D173—22no—
219C2×D173—22no—
220C2×D173—22no—
221C2×D173—416no—
222C2×D173—41no—
223C2×D173—41no—
224C2×D173—41no—
225C2×D173—41no—
226C2×D173C4×D414yes—
227C4×C173—22no—
228C4×C173—22no—
229C4×C173—41no—
230C4×C173—41no—
231C4×C173—41no—
232Dic173—22no—
233Dic173—22no—
234Dic173—22no—
235Dic173—41no—
236Dic173—41no—
237Dic173—41no—
238C2×C173—42no—
239C2×C173—42no—
240C2×C173C2×C4×D411yescharacteristic
241C2×C173#51811yescharacteristic
242C2×C173#51811yescharacteristic
243C2×C173#53411yescharacteristic
244C2×C173#53511yescharacteristic
245C2×C173#53811yescharacteristic
246C2×C173#53811yescharacteristic
247D173—24no—
248D173—42no—
249C173#232111yescharacteristic, complemented, Sylow
250#2321—1731nomaximal, Sylow
251C2×#120—1731no—
252C2×C4×D4—1731no—
253C22×C22⋊C4—1731no—
254#390—1731no—
255#390D17311yescharacteristic, complemented
256#388—1731no—
257#388—1731no—
258#388—1731no—
259#388—1731no—
260#386—1731no—
261#386—1731no—
262#386—1731no—
263#385—1731no—
264#379—1731no—
265C4×C22⋊C4—1731no—
266C25—1731no—
267C22×D4—1731no—
268C22×D4—1731no—
269C22×D4C2×D17311yescharacteristic
270C23×C4—1731no—
271C23×C4—1731no—
272C23×C4—1731no—
273#120—1738no—
274C4×D4—3464no—
275C2×C4⋊C4—1731no—
276C2×C4⋊C4—1731no—
277C2×C4⋊C4C2×D17311yescharacteristic, complemented
278C2×C22⋊C4—3464no—
279C2×C22⋊C4—1731no—
280C2×C22⋊C4—1731no—
281C2×C22⋊C4—1731no—
282C2×C22⋊C4—1731no—
283C2×C22⋊C4—1731no—
284C2×C22⋊C4—1731no—
285C2×C22⋊C4—1731no—
286C2×C22⋊C4—1731no—
287C2×C22⋊C4—1731no—
288C2×C22⋊C4—1731no—
289C2×C22⋊C4—1731no—
290C2×C22⋊C4—1731no—
291C2×C22⋊C4C2×D17311yescharacteristic
292C2×C22⋊C4C2×D17311yescharacteristic
293C2×C22⋊C4C2×D17311yescharacteristic, complemented
294C2×C22⋊C4C2×D17311yescharacteristic, complemented
295C2×C42—1731no—
296C2×C42—1731no—
297C2×C42C2×D17311yescharacteristic, complemented
298#95—1731no—
299#95—1731no—
300#95—1731no—
301#95—1731no—
302#95—1731no—
303#95—1731no—
304C24—3462no—
305C24—3462no—
306C24—1731no—
307C24—1731no—
308C24—1734no—
309C24—6921no—
310C24—6921no—
311C24—6921no—
312C24—6921no—
313C24C22×D17311yescharacteristic
314C24C22×D17311yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—3462no—
318C2×D4—3462no—
319C2×D4—1734no—
320C2×D4—1734no—
321C2×D4—6921no—
322C2×D4—6921no—
323C2×D4C4×D17314yescomplemented
324C22×C4—3462no—
325C22×C4—3462no—
326C22×C4—1731no—
327C22×C4—1731no—
328C22×C4—1731no—
329C22×C4—1731no—
330C22×C4—1731no—
331C22×C4—1731no—
332C22×C4—1734no—
333C22×C4—6921no—
334C22×C4—6921no—
335C22×C4—6921no—
336C22×C4—6921no—
337C22×C4—6921no—
338C22×C4—6921no—
339C22×C4C22×D17311yescharacteristic
340C22×C4C22×D17311yescharacteristic
341C22×C4C22×D17311yescharacteristic
342C22×C4C22×D17311yescharacteristic
343C22×C4C22×D17311yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—3462no—
346C4⋊C4—6921no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—3462no—
352C22⋊C4—3462no—
353C22⋊C4—3462no—
354C22⋊C4—3462no—
355C22⋊C4—3462no—
356C22⋊C4—1734no—
357C22⋊C4—1734no—
358C22⋊C4—1734no—
359C22⋊C4—6921no—
360C22⋊C4—6921no—
361C22⋊C4—6921no—
362C22⋊C4—6921no—
363C42—41no—
364C42—3462no—
365C42—6921no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—3468no—
379C23—3461no—
380C23—3461no—
381C23—3461no—
382C23—3461no—
383C23—3461no—
384C23—3461no—
385C23—3461no—
386C23—3461no—
387C23—3461no—
388C23—3461no—
389C23—1732no—
390C23—1732no—
391C23—1732no—
392C23—1732no—
393C23—6924no—
394C23—6924no—
395C23—6924no—
396C23—6924no—
397C23—6924no—
398C23—6924no—
399C23C2×C4×D17312yes—
400C23C2×C4×D17312yes—
401C23C23×D17311yescharacteristic
402D4—44no—
403D4—3468no—
404D4—6924no—
405C2×C4—21no—
406C2×C4—21no—
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—3461no—
419C2×C4—3461no—
420C2×C4—3461no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—3461no—
425C2×C4—3461no—
426C2×C4—3461no—
427C2×C4—3461no—
428C2×C4—3461no—
429C2×C4—3461no—
430C2×C4—3461no—
431C2×C4—3461no—
432C2×C4—3461no—
433C2×C4—1732no—
434C2×C4—1732no—
435C2×C4—1732no—
436C2×C4—6924no—
437C2×C4—6924no—
438C2×C4—6924no—
439C2×C4C2×C4×D17312yes—
440C2×C4D4×D17312yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—3462no—
456C22—3462no—
457C22—3462no—
458C22—3462no—
459C22—3462no—
460C22—3462no—
461C22—3462no—
462C22—3462no—
463C22—3462no—
464C22—1734no—
465C22—69216no—
466C22—6921no—
467C22—6921no—
468C22—6921no—
469C22—6921no—
470C22C22×C4×D17311yescharacteristic
471C22C2×D4×D17311yescharacteristic
472C22#41×D17311yescharacteristic
473C22#1914972411yescharacteristic
474C22#1914972411yescharacteristic
475C22#1914972411yescharacteristic
476C22#1914972511yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—3462no—
483C4—3462no—
484C4—3462no—
485C4—6921no—
486C4—6921no—
487C4—6921no—
488C2—42no—
489C2—42no—
490C2—3464no—
491C2—6922no—
492C2C4×D4×D17311yescharacteristic
493C2#2601196311yescharacteristic
494C2#2601196411yescharacteristic
495C2#2601204911yescharacteristic
496C2#2601205411yescharacteristic
497C2#2601207311yescharacteristic
498C2#2601207511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#390×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149647C2
hypercenter#390D173
hypocenterC173#2321
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#19149639C2semidirect
3. C2×C22⋊C4×D173C2semidirect
4. #26011391C2semidirect
5. #26011385C2semidirect
6. #26011385C2semidirect
7. #26011382C2semidirect
8. #26011381C2semidirect
9. #26011371C2semidirect
10. #26011368C2semidirect
11. #26011367C2semidirect
12. #26011362C2semidirect
13. #26011346C2semidirect
14. C4×#14583803C2semidirect
15. #390×C173C2semidirect
19. C23×Dic173C22semidirect
22. #19149647C4semidirect
24. C2×#14583805C22semidirect
25. C2×#14583805C22semidirect
29. C2×C22⋊Dic173C22semidirect
39. C2×C4×Dic173C22semidirect
40. C2×C4×Dic173C22semidirect
41. #19149586C22semidirect
42. #19149586C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
48. C2×C4⋊C4×C173C22semidirect
51. C2×C22⋊C4×C173C22semidirect
52. C2×C22⋊C4×C173C22semidirect
53. C2×C42×C173C22semidirect
60. C23×D173C2×C4semidirect
68. C2×C22⋊D173C2×C4semidirect
83. C2×C4×D173D4semidirect
89. C22⋊Dic173C2×C4semidirect
96. #14583803C2×C4semidirect
97. #14583803C2×C4semidirect
104. C2×D4×C173C2×C4semidirect
249. C173#2321semidirect
255. #390D173semidirect
277. C2×C4⋊C4C2×D173semidirect
293. C2×C22⋊C4C2×D173semidirect
294. C2×C22⋊C4C2×D173semidirect
297. C2×C42C2×D173semidirect
323. C2×D4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496472
C2×#191496392
C2×C22⋊C4×D1732
#260113912
#260113852
#260113822
#260113812
#260113712
#260113682
#260113672
#260113622
#260113462
C4×#145838032
#390×C1732
#2321173