Finite Group Databasebeta

← #36356540

Subgroups — #36356540

Order 22144.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149647C211yesmaximal, characteristic
2C2×C4×C22⋊D173C211yesmaximal, characteristic, complemented
3C22×#14583803C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D173C211yesmaximal, characteristic, complemented
5#26011391C211yesmaximal, characteristic, complemented
6#26011382C211yesmaximal, characteristic, complemented
7#26011381C211yesmaximal, characteristic, complemented
8#26011371C211yesmaximal, characteristic, complemented
9#26011370C211yesmaximal, characteristic, complemented
10#26011367C211yesmaximal, characteristic, complemented
11#26011364C211yesmaximal, characteristic, complemented
12#26011362C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
14C4×#14583803C211yesmaximal, characteristic, complemented
15C2×C4×D4×C173C211yesmaximal, characteristic, complemented
16C24×D173C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22#19149647C418yescomplemented
23C4×C22⋊D173—24no—
24C2×#14583805C2211yescharacteristic, complemented
25C2×#14583805C2211yescharacteristic, complemented
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic, complemented
29C2×C22⋊Dic173C2211yescharacteristic, complemented
30C2×#14583803—24no—
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C2×#14583803C2211yescharacteristic
39C22⋊C4×D173—24no—
40C2×C4×Dic173C2211yescharacteristic, complemented
41C2×C4×Dic173C2211yescharacteristic, complemented
42#19149586C2211yescharacteristic, complemented
43#19149586C2211yescharacteristic, complemented
44#19149586C2211yescharacteristic, complemented
45#19149586C2211yescharacteristic, complemented
46#19149586C2211yescharacteristic, complemented
47#19149586C2211yescharacteristic, complemented
48C22×D4×C173C2211yescharacteristic
49C23×C4×C173C2211yescharacteristic
50C23×C4×C173C2211yescharacteristic, complemented
51C4×D4×C173—24no—
52C2×C4⋊C4×C173C2211yescharacteristic, complemented
53C2×C22⋊C4×C173C2211yescharacteristic
54C2×C22⋊C4×C173C2211yescharacteristic, complemented
55C2×C42×C173C2211yescharacteristic, complemented
56C23×D173—22no—
57C23×D173—22no—
58C23×D173—22no—
59C23×D173—22no—
60C23×D173—41no—
61C23×D173—41no—
62C23×D173C2×C414yescomplemented
63C23×D173C2311yescharacteristic
64C23×D173C2311yescharacteristic
65C2×C22⋊D173—22no—
66C2×C22⋊D173—22no—
67C2×C22⋊D173—41no—
68C2×C22⋊D173—41no—
69C2×C22⋊D173C2×C414yescomplemented
70C2×C22⋊D173C2×C414yescomplemented
71C22×Dic173—22no—
72C22×Dic173—22no—
73C22×Dic173—41no—
74C22×Dic173C2311yescharacteristic
75C22×Dic173C2311yescharacteristic
76C22×Dic173C2311yescharacteristic
77C22×Dic173C2311yescharacteristic
78C22×Dic173C2311yescharacteristic
79C22×Dic173C2311yescharacteristic
80C2×C4×D173—22no—
81C2×C4×D173—22no—
82C2×C4×D173—22no—
83C2×C4×D173—22no—
84C2×C4×D173—41no—
85C2×C4×D173—41no—
86#14583805—22no—
87#14583805—41no—
88C22⋊Dic173—22no—
89C22⋊Dic173—22no—
90C22⋊Dic173—41no—
91C22⋊Dic173C2×C414yescomplemented
92#14583803—22no—
93#14583803—22no—
94#14583803—22no—
95#14583803—22no—
96#14583803—22no—
97#14583803C2×C414yescomplemented
98#14583803C2×C414yescomplemented
99#14583803D414yescomplemented
100C4×Dic173—22no—
101C4×Dic173—41no—
102C24×C173C2311yescharacteristic
103C24×C173C2311yescharacteristic
104C2×D4×C173—22no—
105C2×D4×C173—22no—
106C2×D4×C173C2×C414yescomplemented
107C22×C4×C173—22no—
108C22×C4×C173—22no—
109C22×C4×C173—22no—
110C22×C4×C173—22no—
111C22×C4×C173—41no—
112C22×C4×C173—41no—
113C22×C4×C173C2311yescharacteristic
114C22×C4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116C22×C4×C173C2311yescharacteristic
117C22×C4×C173C2311yescharacteristic
118C4⋊C4×C173—22no—
119C22⋊C4×C173—22no—
120C22⋊C4×C173D414yescomplemented
121C42×C173—22no—
122C22×D173—21no—
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—21no—
126C22×D173—28no—
127C22×D173—21no—
128C22×D173—21no—
129C22×D173—21no—
130C22×D173—21no—
131C22×D173—21no—
132C22×D173—21no—
133C22×D173—28no—
134C22×D173—44no—
135C22×D173—44no—
136C22×D173—44no—
137C22×D173—44no—
138C22×D173—44no—
139C22×D173C22×C412yes—
140C22×D173C22×C412yes—
141C22×D173C2×D412yes—
142C22×D173#4112yes—
143C22⋊D173—28no—
144C22⋊D173—44no—
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—21no—
155C2×Dic173—21no—
156C2×Dic173—21no—
157C2×Dic173—21no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173—44no—
161C2×Dic173C22×C412yes—
162C2×Dic173C22×C412yes—
163C2×Dic173C2×D412yes—
164C4×D173—44no—
165C4×D173—44no—
166C23×C173—21no—
167C23×C173—21no—
168C23×C173—21no—
169C23×C173—21no—
170C23×C173—21no—
171C23×C173—21no—
172C23×C173—21no—
173C23×C173—21no—
174C23×C173—21no—
175C23×C173—21no—
176C23×C173C22×C412yes—
177C23×C173C22×C412yes—
178C23×C173C2×D412yes—
179C23×C173#4112yes—
180C23×C173C2411yescharacteristic
181D4×C173—28no—
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×C173—21no—
190C2×C4×C173—21no—
191C2×C4×C173—21no—
192C2×C4×C173—21no—
193C2×C4×C173—44no—
194C2×C4×C173—44no—
195C2×C4×C173C22×C412yes—
196C2×C4×C173C2×D412yes—
197C2×C4×C173C2×D412yes—
198C2×C4×C173C2×D412yes—
199C22×C173—22no—
200C22×C173—22no—
201C22×C173—22no—
202C22×C173—22no—
203C22×C173—22no—
204C22×C173—22no—
205C22×C173—22no—
206C22×C173—22no—
207C22×C173—22no—
208C22×C173—41no—
209C22×C173—41no—
210C22×C173—41no—
211C22×C173—41no—
212C22×C173C4×D414yescomplemented
213C22×C173C23×C411yescharacteristic
214C22×C173C22×D411yescharacteristic
215C22×C173C22×D411yescharacteristic
216C22×C173C2×#4111yescharacteristic
217C22×C173C2×#4111yescharacteristic
218C22×C173H(5,𝔽2)11yescharacteristic
219C22×C173H(5,𝔽2)11yescharacteristic
220C2×D173—22no—
221C2×D173—22no—
222C2×D173—22no—
223C2×D173—22no—
224C2×D173—22no—
225C2×D173—22no—
226C2×D173—22no—
227C2×D173—22no—
228C2×D173—22no—
229C2×D173—416no—
230C2×D173—41no—
231C2×D173—41no—
232C2×D173—41no—
233C2×D173—41no—
234C2×D173C4×D414yescomplemented
235C4×C173—22no—
236C4×C173—22no—
237C4×C173—22no—
238C4×C173—22no—
239C4×C173—41no—
240Dic173—22no—
241Dic173—22no—
242Dic173—22no—
243Dic173—41no—
244Dic173—41no—
245Dic173—41no—
246C2×C173—24no—
247C2×C173—42no—
248C2×C173C2×C4×D411yescharacteristic
249C2×C173C2×C4×D411yescharacteristic
250C2×C173#51811yescharacteristic
251C2×C173D4211yescharacteristic
252C2×C173#54611yescharacteristic
253C2×C173#54611yescharacteristic
254C2×C173#55111yescharacteristic
255D173—24no—
256D173—42no—
257C173#235811yescharacteristic, complemented, Sylow
258#2358—1731nomaximal, Sylow
259C2×#120—1731no—
260C2×C4×D4—1731no—
261C2×C4×D4D17311yescharacteristic, complemented
262C22×C22⋊C4—1731no—
263C22×C22⋊C4—1731no—
264#390—1731no—
265#388—1731no—
266#388—1731no—
267#386—1731no—
268#386—1731no—
269#386—1731no—
270#385—1731no—
271#385—1731no—
272C4×C22⋊C4—1731no—
273C4×C22⋊C4—1731no—
274C25—1731no—
275C22×D4—1731no—
276C22×D4—1731no—
277C22×D4C2×D17311yescharacteristic
278C23×C4—1731no—
279C23×C4—1731no—
280C23×C4—1731no—
281C23×C4C2×D17311yescharacteristic
282C23×C4C2×D17311yescharacteristic, complemented
283#120—1738no—
284C4×D4—24no—
285C4×D4—3464no—
286C2×C4⋊C4—1731no—
287C2×C4⋊C4—1731no—
288C2×C4⋊C4C2×D17311yescharacteristic, complemented
289C2×C22⋊C4—3464no—
290C2×C22⋊C4—3464no—
291C2×C22⋊C4—1731no—
292C2×C22⋊C4—1731no—
293C2×C22⋊C4—1731no—
294C2×C22⋊C4—1731no—
295C2×C22⋊C4—1731no—
296C2×C22⋊C4—1731no—
297C2×C22⋊C4—1731no—
298C2×C22⋊C4—1731no—
299C2×C22⋊C4—1731no—
300C2×C22⋊C4—1731no—
301C2×C22⋊C4—1731no—
302C2×C22⋊C4—1731no—
303C2×C22⋊C4C2×D17311yescharacteristic
304C2×C22⋊C4C2×D17311yescharacteristic, complemented
305C2×C42—1731no—
306C2×C42—1731no—
307C2×C42C2×D17311yescharacteristic, complemented
308#95—1731no—
309#95—1731no—
310#95—1731no—
311#95—1731no—
312#95—1731no—
313#95—1731no—
314C24—3462no—
315C24—3462no—
316C24—3462no—
317C24—3462no—
318C24—1731no—
319C24—1731no—
320C24—1734no—
321C24—6921no—
322C24—6921no—
323C24C22×D17311yescharacteristic
324C24C22×D17311yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—3462no—
328C2×D4—3462no—
329C2×D4—1734no—
330C2×D4—1734no—
331C2×D4—6921no—
332C2×D4—6921no—
333C2×D4C4×D17314yescomplemented
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—3462no—
341C22×C4—3462no—
342C22×C4—3462no—
343C22×C4—3462no—
344C22×C4—3462no—
345C22×C4—3462no—
346C22×C4—1731no—
347C22×C4—1731no—
348C22×C4—1731no—
349C22×C4—1731no—
350C22×C4—1731no—
351C22×C4—1731no—
352C22×C4—6921no—
353C22×C4—6921no—
354C22×C4—6921no—
355C22×C4C22×D17311yescharacteristic
356C22×C4C22×D17311yescharacteristic
357C22×C4C22×D17311yescharacteristic
358C22×C4C22×D17311yescharacteristic
359C22×C4C22×D17311yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—3462no—
362C4⋊C4—6921no—
363C22⋊C4—22no—
364C22⋊C4—3462no—
365C22⋊C4—3462no—
366C22⋊C4—3462no—
367C22⋊C4—3462no—
368C22⋊C4—3462no—
369C22⋊C4—3462no—
370C22⋊C4—3462no—
371C22⋊C4—1734no—
372C22⋊C4—1734no—
373C22⋊C4—1734no—
374C22⋊C4—1734no—
375C22⋊C4—6921no—
376C22⋊C4C22⋊D17314yescomplemented
377C42—22no—
378C42—3462no—
379C42—6921no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—3468no—
391C23—3461no—
392C23—3461no—
393C23—3461no—
394C23—3461no—
395C23—3468no—
396C23—3461no—
397C23—3461no—
398C23—3461no—
399C23—3461no—
400C23—3461no—
401C23—3461no—
402C23—1732no—
403C23—1732no—
404C23—1732no—
405C23—1732no—
406C23—6924no—
407C23—6924no—
408C23—6924no—
409C23—6924no—
410C23—6924no—
411C23C2×C4×D17312yes—
412C23C2×C4×D17312yes—
413C23C2×C22⋊D17312yes—
414C23#1458382612yes—
415C23C23×D17311yescharacteristic
416D4—28no—
417D4—3468no—
418D4—6924no—
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—44no—
431C2×C4—44no—
432C2×C4—3461no—
433C2×C4—3461no—
434C2×C4—3461no—
435C2×C4—3461no—
436C2×C4—3461no—
437C2×C4—3461no—
438C2×C4—3461no—
439C2×C4—3461no—
440C2×C4—3461no—
441C2×C4—3461no—
442C2×C4—3461no—
443C2×C4—3461no—
444C2×C4—3461no—
445C2×C4—3461no—
446C2×C4—3461no—
447C2×C4—1732no—
448C2×C4—1732no—
449C2×C4—1732no—
450C2×C4—6924no—
451C2×C4—6924no—
452C2×C4—6924no—
453C2×C4C2×C4×D17312yes—
454C2×C4D4×D17312yes—
455C2×C4C2×C22⋊D17312yes—
456C2×C4C2×C22⋊D17312yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—3462no—
471C22—3462no—
472C22—3462no—
473C22—3462no—
474C22—3462no—
475C22—3462no—
476C22—3462no—
477C22—3462no—
478C22—3462no—
479C22—1734no—
480C22—69216no—
481C22—6921no—
482C22—6921no—
483C22—6921no—
484C22—6921no—
485C22C4×C22⋊D17314yescomplemented
486C22C22×C4×D17311yescharacteristic
487C22C2×D4×D17311yescharacteristic
488C22C22×C22⋊D17311yescharacteristic
489C22#41×D17311yescharacteristic
490C22C2×#1458382611yescharacteristic
491C22#1914972411yescharacteristic
492C22#1914972511yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—3462no—
499C4—3462no—
500C4—3462no—
501C4—6921no—
502C4—6921no—
503C4—6921no—
504C2—24no—
505C2—42no—
506C2—3464no—
507C2—6922no—
508C2C4×D4×D17311yescharacteristic
509C2C2×C4×C22⋊D17311yescharacteristic
510C2#2601196311yescharacteristic
511C2D4×C22⋊D17311yescharacteristic
512C2#2601212911yescharacteristic
513C2#2601213711yescharacteristic
514C2#2601217011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
FittingC2×C4×D4×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149647C2
hypercenterC2×C4×D4D173
hypocenterC173#2358
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D173C2semidirect
3. C22×#14583803C2semidirect
4. C2×C22⋊C4×D173C2semidirect
5. #26011391C2semidirect
6. #26011382C2semidirect
7. #26011381C2semidirect
8. #26011371C2semidirect
9. #26011370C2semidirect
10. #26011367C2semidirect
11. #26011364C2semidirect
12. #26011362C2semidirect
13. C22⋊C4×Dic173C2semidirect
14. C4×#14583803C2semidirect
15. C2×C4×D4×C173C2semidirect
22. #19149647C4semidirect
24. C2×#14583805C22semidirect
25. C2×#14583805C22semidirect
28. C2×C22⋊Dic173C22semidirect
29. C2×C22⋊Dic173C22semidirect
40. C2×C4×Dic173C22semidirect
41. C2×C4×Dic173C22semidirect
42. #19149586C22semidirect
43. #19149586C22semidirect
44. #19149586C22semidirect
45. #19149586C22semidirect
46. #19149586C22semidirect
47. #19149586C22semidirect
50. C23×C4×C173C22semidirect
52. C2×C4⋊C4×C173C22semidirect
54. C2×C22⋊C4×C173C22semidirect
55. C2×C42×C173C22semidirect
62. C23×D173C2×C4semidirect
69. C2×C22⋊D173C2×C4semidirect
70. C2×C22⋊D173C2×C4semidirect
91. C22⋊Dic173C2×C4semidirect
97. #14583803C2×C4semidirect
98. #14583803C2×C4semidirect
99. #14583803D4semidirect
106. C2×D4×C173C2×C4semidirect
120. C22⋊C4×C173D4semidirect
212. C22×C173C4×D4semidirect
234. C2×D173C4×D4semidirect
257. C173#2358semidirect
261. C2×C4×D4D173semidirect
282. C23×C4C2×D173semidirect
288. C2×C4⋊C4C2×D173semidirect
304. C2×C22⋊C4C2×D173semidirect
307. C2×C42C2×D173semidirect
333. C2×D4C4×D173semidirect
376. C22⋊C4C22⋊D173semidirect
485. C22C4×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496472
C2×C4×C22⋊D1732
C22×#145838032
C2×C22⋊C4×D1732
#260113912
#260113822
#260113812
#260113712
#260113702
#260113672
#260113642
#260113622
C22⋊C4×Dic1732
C4×#145838032
C2×C4×D4×C1732
#2358173