Finite Group Databasebeta

← #36356216

Subgroups — #36356216

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×#19149649C211yesmaximal, characteristic
2C2×#19149641C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D173C211yesmaximal, characteristic, complemented
4#26011393C211yesmaximal, characteristic, complemented
5#26011391C211yesmaximal, characteristic, complemented
6#26011384C211yesmaximal, characteristic, complemented
7#26011381C211yesmaximal, characteristic, complemented
8#26011380C211yesmaximal, characteristic, complemented
9#26011369C211yesmaximal, characteristic, complemented
10#26011369C211yesmaximal, characteristic, complemented
11#26011367C211yesmaximal, characteristic, complemented
12#26011365C211yesmaximal, characteristic, complemented
13#26011346C211yesmaximal, characteristic, complemented
14C4×#14583803C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×D692C2211yescharacteristic
18C22×D692C2211yescharacteristic
19C22×C22⋊D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22C22×C4×D173C2211yescharacteristic, complemented
23#19149649C418yescomplemented
24#19149641—24no—
25C2×C4⋊Dic173C2211yescharacteristic, complemented
26C2×#14583805C2211yescharacteristic, complemented
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic, complemented
38C2×#14583803C2211yescharacteristic, complemented
39C2×#14583803C2211yescharacteristic, complemented
40C22⋊C4×D173—24no—
41C2×C4×Dic173C2211yescharacteristic, complemented
42C2×C4×Dic173C2211yescharacteristic, complemented
43#19149586C2211yescharacteristic, complemented
44#19149586C2211yescharacteristic, complemented
45#19149586C2211yescharacteristic, complemented
46#19149586C2211yescharacteristic, complemented
47C2×C4⋊C4×C173C2211yescharacteristic, complemented
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C42×C173C2211yescharacteristic, complemented
52#95×C173C2211yescharacteristic, complemented
53#95×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
63C23×D173C2311yescharacteristic
64C2×D692—22no—
65C2×D692—22no—
66C2×D692—41no—
67C2×D692—41no—
68C2×D692C2×C414yes—
69C2×D692C2×C414yescomplemented
70C2×C22⋊D173—41no—
71C2×C22⋊D173—41no—
72C2×C22⋊D173C2×C414yescomplemented
73C22×Dic173C2311yescharacteristic
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C2×C4×D173—22no—
79C2×C4×D173—22no—
80C2×C4×D173—41no—
81C2×C4×D173—41no—
82C2×C4×D173—41no—
83C2×C4×D173—41no—
84C2×C4×D173—41no—
85C2×C4×D173—41no—
86C2×C4×D173D414yescomplemented
87C4⋊Dic173—41no—
88#14583805—41no—
89C22⋊Dic173—22no—
90C22⋊Dic173—41no—
91#14583803—22no—
92#14583803—22no—
93#14583803—22no—
94#14583803—22no—
95#14583803—22no—
96#14583803—41no—
97#14583803—41no—
98#14583803—41no—
99#14583803—41no—
100#14583803C2×C414yescomplemented
101#14583803C2×C414yescomplemented
102C4×Dic173—22no—
103C4×Dic173—41no—
104C24×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108C22×C4×C173C2311yescharacteristic
109C22×C4×C173C2311yescharacteristic
110C22×C4×C173C2311yescharacteristic
111C4⋊C4×C173—22no—
112C22⋊C4×C173—22no—
113C22⋊C4×C173—41no—
114C22⋊C4×C173C2×C414yescomplemented
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—21no—
127C22×D173—21no—
128C22×D173—21no—
129C22×D173—21no—
130C22×D173—21no—
131C22×D173—21no—
132C22×D173—28no—
133C22×D173—44no—
134C22×D173—44no—
135C22×D173—44no—
136C22×D173—44no—
137C22×D173—44no—
138C22×D173—44no—
139C22×D173C22×C412yes—
140C22×D173C22×C412yes—
141C22×D173C22×C412yes—
142C22×D173C2×D412yes—
143C22×D173#4112yes—
144D692—28no—
145D692—44no—
146C22⋊D173—44no—
147C2×Dic173—21no—
148C2×Dic173—21no—
149C2×Dic173—21no—
150C2×Dic173—21no—
151C2×Dic173—21no—
152C2×Dic173—21no—
153C2×Dic173—21no—
154C2×Dic173—21no—
155C2×Dic173—21no—
156C2×Dic173—21no—
157C2×Dic173—21no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173C22×C412yes—
161C2×Dic173C2×D412yes—
162C4×D173—44no—
163C4×D173—44no—
164C4×D173—44no—
165C23×C173—21no—
166C23×C173—21no—
167C23×C173—21no—
168C23×C173—21no—
169C23×C173—21no—
170C23×C173—21no—
171C23×C173C22×C412yes—
172C23×C173C2411yescharacteristic
173C2×C4×C173—21no—
174C2×C4×C173—21no—
175C2×C4×C173—21no—
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×C173C22×C412yes—
189C2×C4×C173C22×C412yes—
190C2×C4×C173C2×D412yes—
191C22×C173—22no—
192C22×C173—22no—
193C22×C173—22no—
194C22×C173—41no—
195C22×C173—41no—
196C22×C173—41no—
197C22×C173—41no—
198C22×C173C23×C411yescharacteristic
199C22×C173C22×D411yescharacteristic
200C22×C173C2×#4111yescharacteristic
201C22×C173H(5,𝔽2)11yescharacteristic
202C22×C173H(5,𝔽2)11yescharacteristic
203C22×C173H(5,𝔽2)11yescharacteristic
204C22×C173H(5,𝔽2)11yescharacteristic
205C2×D173—22no—
206C2×D173—22no—
207C2×D173—22no—
208C2×D173—22no—
209C2×D173—22no—
210C2×D173—22no—
211C2×D173—22no—
212C2×D173—22no—
213C2×D173—22no—
214C2×D173—22no—
215C2×D173—22no—
216C2×D173—22no—
217C2×D173—416no—
218C2×D173—41no—
219C2×D173—41no—
220C2×D173—41no—
221C2×D173—41no—
222C2×D173—41no—
223C2×D173—41no—
224C2×D173—41no—
225C2×D173—41no—
226C2×D173C4×D414yes—
227C4×C173—22no—
228C4×C173—22no—
229C4×C173—22no—
230C4×C173—41no—
231C4×C173—41no—
232C4×C173—41no—
233Dic173—22no—
234Dic173—22no—
235Dic173—41no—
236Dic173—41no—
237Dic173—41no—
238C2×C173—42no—
239C2×C173C2×C4×D411yescharacteristic
240C2×C173#51811yescharacteristic
241C2×C173#51811yescharacteristic
242C2×C173#53411yescharacteristic
243C2×C173#53511yescharacteristic
244C2×C173#53811yescharacteristic
245C2×C173#53811yescharacteristic
246D173—24no—
247D173—42no—
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#390—1731no—
256#388—1731no—
257#388—1731no—
258#388—1731no—
259#388D17311yescharacteristic, complemented
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×D4—1731no—
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⋊C4—1731no—
292C2×C22⋊C4C2×D17311yescharacteristic
293C2×C22⋊C4C2×D17311yescharacteristic
294C2×C22⋊C4C2×D17311yescharacteristic
295C2×C42—1731no—
296C2×C42—1731no—
297C2×C42C2×D17311yescharacteristic, complemented
298#95—1731no—
299#95—1731no—
300#95—1731no—
301#95—1731no—
302#95C2×D17311yescharacteristic, complemented
303#95C2×D17311yescharacteristic, complemented
304C24—3462no—
305C24—3462no—
306C24—1731no—
307C24—1731no—
308C24—1731no—
309C24—1734no—
310C24—6921no—
311C24—6921no—
312C24—6921no—
313C24—6921no—
314C24C22×D17311yescharacteristic
315C2×D4—3462no—
316C2×D4—3462no—
317C2×D4—1734no—
318C2×D4—1734no—
319C2×D4—1734no—
320C2×D4—6921no—
321C2×D4—6921no—
322C2×D4—6921no—
323C2×D4—6921no—
324C22×C4—3462no—
325C22×C4—3462no—
326C22×C4—1731no—
327C22×C4—1731no—
328C22×C4—1731no—
329C22×C4—1731no—
330C22×C4—1731no—
331C22×C4—1734no—
332C22×C4—6921no—
333C22×C4—6921no—
334C22×C4—6921no—
335C22×C4—6921no—
336C22×C4—6921no—
337C22×C4—6921no—
338C22×C4C22×D17311yescharacteristic
339C22×C4C22×D17311yescharacteristic
340C22×C4C22×D17311yescharacteristic
341C22×C4C22×D17311yescharacteristic
342C22×C4C22×D17311yescharacteristic
343C22×C4C22×D17311yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—6921no—
346C4⋊C4—6921no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—3462no—
350C22⋊C4—3462no—
351C22⋊C4—3462no—
352C22⋊C4—3462no—
353C22⋊C4—3462no—
354C22⋊C4—3462no—
355C22⋊C4—1734no—
356C22⋊C4—1734no—
357C22⋊C4—6921no—
358C22⋊C4—6921no—
359C22⋊C4—6921no—
360C22⋊C4—6921no—
361C22⋊C4—6921no—
362C22⋊C4C4×D17314yescomplemented
363C42—41no—
364C42—3462no—
365C42—6921no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—3468no—
373C23—3461no—
374C23—3461no—
375C23—3461no—
376C23—3461no—
377C23—3461no—
378C23—3461no—
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—1732no—
394C23—6924no—
395C23—6924no—
396C23—6924no—
397C23—6924no—
398C23—6924no—
399C23—6924no—
400C23C2×C4×D17312yes—
401C23C23×D17311yescharacteristic
402D4—3468no—
403D4—6924no—
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—21no—
419C2×C4—21no—
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—6924no—
436C2×C4—6924no—
437C2×C4—6924no—
438C2×C4C2×C4×D17312yes—
439C2×C4C2×C4×D17312yes—
440C2×C4D4×D17312yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—3462no—
449C22—3462no—
450C22—3462no—
451C22—3462no—
452C22—3462no—
453C22—3462no—
454C22—3462no—
455C22—3462no—
456C22—3462no—
457C22—3462no—
458C22—3462no—
459C22—3462no—
460C22—1734no—
461C22—69216no—
462C22—6921no—
463C22—6921no—
464C22—6921no—
465C22—6921no—
466C22—6921no—
467C22—6921no—
468C22—6921no—
469C22—6921no—
470C22C22×C4×D17311yescharacteristic
471C22C2×D4×D17311yescharacteristic
472C22#41×D17311yescharacteristic
473C22#1914972411yescharacteristic
474C22#1914972511yescharacteristic
475C22#1914972511yescharacteristic
476C22#1914972511yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—3462no—
484C4—3462no—
485C4—6921no—
486C4—6921no—
487C4—6921no—
488C2—42no—
489C2—3464no—
490C2—6922no—
491C2—6922no—
492C2C4×D4×D17311yescharacteristic
493C2#2601196311yescharacteristic
494C2#2601196511yescharacteristic
495C2#2601204911yescharacteristic
496C2#2601205911yescharacteristic
497C2#2601207711yescharacteristic
498C2#2601208011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149649C2
hypercenter#388D173
hypocenterC173#2321
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#19149641C2semidirect
3. C2×C22⋊C4×D173C2semidirect
4. #26011393C2semidirect
5. #26011391C2semidirect
6. #26011384C2semidirect
7. #26011381C2semidirect
8. #26011380C2semidirect
9. #26011369C2semidirect
10. #26011369C2semidirect
11. #26011367C2semidirect
12. #26011365C2semidirect
13. #26011346C2semidirect
14. C4×#14583803C2semidirect
15. #388×C173C2semidirect
22. C22×C4×D173C22semidirect
23. #19149649C4semidirect
25. C2×C4⋊Dic173C22semidirect
26. C2×#14583805C22semidirect
37. C2×#14583803C22semidirect
38. C2×#14583803C22semidirect
39. C2×#14583803C22semidirect
41. C2×C4×Dic173C22semidirect
42. C2×C4×Dic173C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
47. C2×C4⋊C4×C173C22semidirect
51. C2×C42×C173C22semidirect
52. #95×C173C22semidirect
53. #95×C173C22semidirect
60. C23×D173C2×C4semidirect
69. C2×D692C2×C4semidirect
72. C2×C22⋊D173C2×C4semidirect
86. C2×C4×D173D4semidirect
100. #14583803C2×C4semidirect
101. #14583803C2×C4semidirect
114. C22⋊C4×C173C2×C4semidirect
249. C173#2321semidirect
259. #388D173semidirect
277. C2×C4⋊C4C2×D173semidirect
297. C2×C42C2×D173semidirect
302. #95C2×D173semidirect
303. #95C2×D173semidirect
362. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496492
C2×#191496412
C2×C22⋊C4×D1732
#260113932
#260113912
#260113842
#260113812
#260113802
#260113692
#260113672
#260113652
#260113462
C4×#145838032
#388×C1732
#2321173