Finite Group Databasebeta

← #36365054

Subgroups — #36365054

Order 22144.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149705C211yesmaximal, characteristic, complemented
2C22×#14583814C211yesmaximal, characteristic, complemented
3#26012150C211yesmaximal, characteristic, complemented
4#26011878C211yesmaximal, characteristic, complemented
5#26011735C211yesmaximal, characteristic, complemented
6#26011725C211yesmaximal, characteristic, complemented
7#26011706C211yesmaximal, characteristic, complemented
8#26011688C211yesmaximal, characteristic, complemented
9#26011642C211yesmaximal, characteristic, complemented
10#26011634C211yesmaximal, characteristic, complemented
11#26011629C211yesmaximal, characteristic, complemented
12#26011625C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011563C211yesmaximal, characteristic, complemented
15D42×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C22×Dic346C2211yescharacteristic, complemented
18#19149705—24no—
19C2×#14583814—24no—
20C2×#14583814C2211yescharacteristic
21C2×#14583814C2211yescharacteristic
22C2×#14583814C2211yescharacteristic, complemented
23C2×#14583814C2211yescharacteristic, complemented
24C2×#14583814C2211yescharacteristic, complemented
25C2×#14583814C2211yescharacteristic, complemented
26C2×#14583811C2211yescharacteristic
27C2×#14583811C2211yescharacteristic, complemented
28C2×#14583811C2211yescharacteristic, complemented
29C2×#14583810C2211yescharacteristic, complemented
30C22×#13063436C2211yescharacteristic, complemented
31#19149667C2211yescharacteristic, complemented
32#19149665—21no—
33#19149658C2211yescharacteristic, complemented
34#19149657—21no—
35#19149653C2211yescharacteristic, complemented
36#19149651—21no—
37C4×Dic346C2211yescharacteristic, complemented
38#19149639—21no—
39#19149612C2211yescharacteristic, complemented
40C4⋊#13063436C2211yescharacteristic, complemented
41#19149602C2211yescharacteristic, complemented
42#19149602C2211yescharacteristic, complemented
43#19149600C2211yescharacteristic
44#19149600C2211yescharacteristic, complemented
45#19149600C2211yescharacteristic, complemented
46#19149599C2211yescharacteristic, complemented
47C22⋊#13063436C2211yescharacteristic, complemented
48C22⋊#13063436C2211yescharacteristic, complemented
49#19149590C2211yescharacteristic, complemented
50C22×D4×C173—21no—
51C22×D4×C173C2211yescharacteristic
52C22×D4×C173C2211yescharacteristic
53#127×C173C2211yescharacteristic
54#121×C173—21no—
55#121×C173C2211yescharacteristic
56#121×C173C2211yescharacteristic
57#120×C173—21no—
58#120×C173—21no—
59C4×D4×C173C2211yescharacteristic
60C4×D4×C173C2211yescharacteristic
61#14583826—22no—
62#14583826—41no—
63C2×Dic346—22no—
64C2×Dic346—21no—
65C2×Dic346—41no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C2×C4×D173—21no—
73#14583814—22no—
74#14583814—22no—
75#14583814—22no—
76#14583814—22no—
77#14583814—41no—
78#14583814D414yescomplemented
79#14583811—22no—
80#14583811—22no—
81#14583811—41no—
82#14583810—22no—
83C2×#13063436—22no—
84C2×#13063436C2311yescharacteristic
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95C4×Dic173—21no—
96C24×C173—21no—
97C24×C173—21no—
98C2×D4×C173—22no—
99C2×D4×C173—22no—
100C2×D4×C173—21no—
101C2×D4×C173—21no—
102C2×D4×C173—21no—
103C2×D4×C173—21no—
104C2×D4×C173—21no—
105C2×D4×C173—21no—
106C2×D4×C173—41no—
107C2×D4×C173—41no—
108C2×D4×C173—44no—
109C2×D4×C173D411yescharacteristic
110C2×D4×C173C2311yescharacteristic
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173C2311yescharacteristic
115C22×C4×C173C2311yescharacteristic
116C4⋊C4×C173D411yescharacteristic
117C4⋊C4×C173C2311yescharacteristic
118C22⋊C4×C173—21no—
119C22⋊C4×C173—21no—
120C22⋊C4×C173—21no—
121C22⋊C4×C173D411yescharacteristic
122C22⋊C4×C173D411yescharacteristic
123C42×C173C2311yescharacteristic
124C22×D173—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129Dic346C2×D412yescomplemented
130D692—21no—
131D692—21no—
132C22⋊D173—42no—
133C22⋊D173—42no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—42no—
138C4×D173—42no—
139#13063436—21no—
140#13063436—21no—
141#13063436—21no—
142#13063436C2×D412yescomplemented
143C23×C173—21no—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173—42no—
147C23×C173—42no—
148C23×C173—42no—
149C23×C173—42no—
150C23×C173—81no—
151C23×C173—81no—
152C23×C173C2×D411yescharacteristic
153C23×C173C2×D411yescharacteristic
154D4×C173—22no—
155D4×C173—21no—
156D4×C173—21no—
157D4×C173—21no—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—42no—
161D4×C173—42no—
162D4×C173—42no—
163D4×C173—42no—
164D4×C173—42no—
165D4×C173—81no—
166D4×C173C2×D412yes—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173—42no—
173C2×C4×C173C2×D411yescharacteristic
174C2×C4×C173C2×D411yescharacteristic
175C2×C4×C173C2×D411yescharacteristic
176C2×C4×C173C2×D411yescharacteristic
177C2×C4×C173C2411yescharacteristic
178C22×C173—21no—
179C22×C173—21no—
180C22×C173—21no—
181C22×C173—22no—
182C22×C173—21no—
183C22×C173—21no—
184C22×C173—41no—
185C22×C173—41no—
186C22×C173—41no—
187C22×C173—41no—
188C22×C173—41no—
189C22×C173—41no—
190C22×C173—41no—
191C22×C173—41no—
192C22×C173—84no—
193C22×C173—84no—
194C22×C173AGL(1,ℤ/8)12yescomplemented
195C22×C173C22×D411yescharacteristic
196C2×D173—41no—
197C2×D173—41no—
198C2×D173—41no—
199C4×C173—21no—
200C4×C173—21no—
201C4×C173—21no—
202C4×C173—21no—
203C4×C173—41no—
204C4×C173C22×D411yescharacteristic
205C4×C173H(5,𝔽2)11yescharacteristic
206Dic173—22no—
207Dic173—41no—
208Dic173—41no—
209C2×C173—22no—
210C2×C173—41no—
211C2×C173—42no—
212C2×C173—81no—
213C2×C173—81no—
214C2×C173D4211yescharacteristic
215C2×C173C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C173#57711yescharacteristic
217D173—81no—
218C173#326511yescharacteristic, complemented, Sylow
219#3265—1731nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1731no—
221C22×SD16—1731no—
222#548—1731no—
223D42D17311yescharacteristic, complemented
224#496—1731no—
225#468—1731no—
226#466—1731no—
227#463—1731no—
228#460—1731no—
229#450—1731no—
230#449—1731no—
231#448—1731no—
232#447—1731no—
233#440—1731no—
234#435—1731no—
235C2×#41—1731no—
236C22×Q8—1731no—
237C22×D4—21no—
238C22×D4C2×D17311yescharacteristic
239C22×D4C2×D17311yescharacteristic
240AGL(1,ℤ/8)—3464no—
241C2×SD16—3464no—
242C2×SD16—1731no—
243C2×SD16—1731no—
244C2×SD16—1731no—
245C2×SD16—1731no—
246C2×SD16—1731no—
247C2×SD16—1731no—
248C2×D8—1731no—
249C2×D8—1731no—
250C2×D8—1731no—
251C2×M16—1731no—
252C22×C8—1731no—
253#127C2×D17311yescharacteristic
254#124—3461no—
255#124—1731no—
256C22⋊Q8—3461no—
257C22⋊Q8—1731no—
258#121—21no—
259#121—3461no—
260#121—1731no—
261#121C2×D17311yescharacteristic
262#121C2×D17311yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1731no—
266C4×D4—3461no—
267C4×D4C2×D17311yescharacteristic
268C4×D4C2×D17311yescharacteristic
269#107—1731no—
270C4⋊C8—1731no—
271#103—1731no—
272#103—1731no—
273#102—1731no—
274#102—1731no—
275#102—1731no—
276#102—1731no—
277C22⋊C8—1731no—
278C22⋊C8—1731no—
279#97—1731no—
280C24—21no—
281C24—21no—
282#41—3462no—
283#41—6921no—
284C2×Q8—3462no—
285C2×Q8—3461no—
286C2×Q8—1731no—
287C2×Q8—1731no—
288C2×Q8—6921no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—3461no—
301C2×D4—3461no—
302C2×D4—1731no—
303C2×D4C22⋊D17311yescharacteristic
304C2×D4C22×D17311yescharacteristic
305C2×D4C22×D17311yescharacteristic
306C2×D4C22×D17311yescharacteristic
307C22×C4—21no—
308C22×C4—3461no—
309C22×C4—3461no—
310C22×C4C22×D17311yescharacteristic
311C22×C4C22×D17311yescharacteristic
312SD16—3462no—
313SD16—3462no—
314SD16—3462no—
315SD16—3462no—
316SD16—1734no—
317SD16—6921no—
318D8—3462no—
319D8—3462no—
320D8—6921no—
321M16—3462no—
322C2×C8—3462no—
323C2×C8—1731no—
324C2×C8—1731no—
325C2×C8—1731no—
326C2×C8—1731no—
327C4⋊C4—3461no—
328C4⋊C4—3461no—
329C4⋊C4—1731no—
330C4⋊C4C22⋊D17311yescharacteristic
331C4⋊C4C22×D17311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3461no—
336C22⋊C4—3461no—
337C22⋊C4—3461no—
338C22⋊C4—3461no—
339C22⋊C4C22⋊D17311yescharacteristic
340C22⋊C4C22⋊D17311yescharacteristic
341C42—3461no—
342C42C22×D17311yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—3461no—
353C23C2×C22⋊D17311yescharacteristic
354C23C2×C22⋊D17311yescharacteristic
355Q8—3461no—
356Q8—3461no—
357Q8—3461no—
358Q8—1732no—
359Q8—6922no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—3461no—
373D4—3461no—
374D4—6922no—
375D4—6922no—
376D4D4×D17312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3461no—
386C2×C4—6922no—
387C2×C4—6922no—
388C2×C4C2×C22⋊D17311yescharacteristic
389C2×C4C2×C22⋊D17311yescharacteristic
390C2×C4C2×C22⋊D17311yescharacteristic
391C2×C4C2×C22⋊D17311yescharacteristic
392C2×C4C23×D17311yescharacteristic
393C8—3461no—
394C8—3461no—
395C8—3461no—
396C8—1732no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—6921no—
414C22—6921no—
415C22—6921no—
416C22#1914970512yescomplemented
417C22C22×C22⋊D17311yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—3462no—
424C4—6921no—
425C4—6921no—
426C4C2×D4×D17311yescharacteristic
427C4#1914972411yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—13841no—
434C2D4×C22⋊D17311yescharacteristic
435C2C2×#1914970511yescharacteristic
436C2#2601230211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingD42×C173C2
socleC22×C173C22×D4
hypercenterD42D173
hypocenterC173#3265
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19149705C2semidirect
2. C22×#14583814C2semidirect
3. #26012150C2semidirect
4. #26011878C2semidirect
5. #26011735C2semidirect
6. #26011725C2semidirect
7. #26011706C2semidirect
8. #26011688C2semidirect
9. #26011642C2semidirect
10. #26011634C2semidirect
11. #26011629C2semidirect
12. #26011625C2semidirect
13. #26011591C2semidirect
14. #26011563C2semidirect
15. D42×C173C2semidirect
16. C2×#14583826C22semidirect
17. C22×Dic346C22semidirect
22. C2×#14583814C22semidirect
23. C2×#14583814C22semidirect
24. C2×#14583814C22semidirect
25. C2×#14583814C22semidirect
27. C2×#14583811C22semidirect
28. C2×#14583811C22semidirect
29. C2×#14583810C22semidirect
30. C22×#13063436C22semidirect
31. #19149667C22semidirect
33. #19149658C22semidirect
35. #19149653C22semidirect
37. C4×Dic346C22semidirect
39. #19149612C22semidirect
40. C4⋊#13063436C22semidirect
41. #19149602C22semidirect
42. #19149602C22semidirect
44. #19149600C22semidirect
45. #19149600C22semidirect
46. #19149599C22semidirect
47. C22⋊#13063436C22semidirect
48. C22⋊#13063436C22semidirect
49. #19149590C22semidirect
78. #14583814D4semidirect
129. Dic346C2×D4semidirect
142. #13063436C2×D4semidirect
194. C22×C173AGL(1,ℤ/8)semidirect
218. C173#3265semidirect
223. D42D173semidirect
416. C22#19149705semidirect

Maximal subgroups

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

subgroupindex
C2×#191497052
C22×#145838142
#260121502
#260118782
#260117352
#260117252
#260117062
#260116882
#260116422
#260116342
#260116292
#260116252
#260115912
#260115632
D42×C1732
#3265173