Finite Group Databasebeta

← #36365102

Subgroups — #36365102

Order 22144.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149703C211yesmaximal, characteristic, complemented
2C2×#19149700C211yesmaximal, characteristic, complemented
3#26012161C211yesmaximal, characteristic, complemented
4#26012134C211yesmaximal, characteristic, complemented
5#26011734C211yesmaximal, characteristic, complemented
6#26011724C211yesmaximal, characteristic, complemented
7#26011710C211yesmaximal, characteristic, complemented
8#26011683C211yesmaximal, characteristic, complemented
9#26011650C211yesmaximal, characteristic, complemented
10#26011635C211yesmaximal, characteristic, complemented
11#26011633C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011593C211yesmaximal, characteristic, complemented
14#26011564C211yesmaximal, characteristic, complemented
15#496×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583825C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149703—24no—
21#19149700—24no—
22C2×#14583816C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583811C2211yescharacteristic
26C2×#14583811C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic, complemented
29#19149675—21no—
30#19149661—21no—
31#19149655C2211yescharacteristic
32#19149652—21no—
33#19149651—21no—
34#19149651—21no—
35#19149650C2211yescharacteristic
36#19149650C2211yescharacteristic
37#19149650C2211yescharacteristic, complemented
38#19149649—21no—
39C4×Dic346C2211yescharacteristic, complemented
40C4×D692C2211yescharacteristic
41C4×D692C2211yescharacteristic
42D4×Dic173—21no—
43C22⋊C4×D173—21no—
44#19149613C2211yescharacteristic, complemented
45C4⋊#13063436C2211yescharacteristic, complemented
46#19149604C2211yescharacteristic, complemented
47#19149603C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic
49#19149600C2211yescharacteristic
50#19149600C2211yescharacteristic, complemented
51#19149600C2211yescharacteristic, complemented
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic, complemented
54C2×SD16×C173C2211yescharacteristic, complemented
55C2×SD16×C173C2211yescharacteristic, complemented
56C2×D8×C173C2211yescharacteristic
57C2×D8×C173C2211yescharacteristic
58#127×C173C2211yescharacteristic, complemented
59#124×C173C2211yescharacteristic
60#97×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64#14583825—22no—
65#14583825—22no—
66#14583825—41no—
67#14583825—41no—
68C2×Dic346C2311yescharacteristic
69C2×D692—21no—
70C2×D692D411yescharacteristic
71C2×D692C2311yescharacteristic
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78D4×D173—22no—
79D4×D173—41no—
80C22×Dic173—21no—
81C22×Dic173—21no—
82C2×C4×D173—21no—
83C2×C4×D173—21no—
84C2×C4×D173C2311yescharacteristic
85C2×C4×D173C2311yescharacteristic
86#14583816—22no—
87#14583815D414yescomplemented
88#14583814—22no—
89#14583811—22no—
90#14583811—22no—
91#14583809—22no—
92C2×#13063436C2311yescharacteristic
93C2×#13063436C2311yescharacteristic
94C8×D173—22no—
95C4⋊Dic173D411yescharacteristic
96C4⋊Dic173C2311yescharacteristic
97C4⋊Dic173C2311yescharacteristic
98#14583805—21no—
99#14583805—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803D411yescharacteristic
106#14583803D411yescharacteristic
107C4×Dic173—21no—
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114SD16×C173—22no—
115SD16×C173—41no—
116D8×C173—22no—
117D8×C173—41no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C22⋊C4×C173—21no—
121C22⋊C4×C173—21no—
122C42×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—42no—
125C22×D173—42no—
126C22×D173—81no—
127C22×D173C2×D411yescharacteristic
128C22×D173C2×D411yescharacteristic
129Dic346—21no—
130Dic346C2×D412yes—
131D692—21no—
132D692—41no—
133D692—42no—
134D692C2×D412yes—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—42no—
138C22⋊D173—81no—
139C22⋊D173—81no—
140C22⋊D173—81no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—21no—
144C2×Dic173—42no—
145C2×Dic173—42no—
146C2×Dic173C2×D411yescharacteristic
147C2×Dic173C2×D411yescharacteristic
148C4×D173—21no—
149C4×D173—21no—
150C4×D173—21no—
151C4×D173—21no—
152C4×D173—42no—
153C4×D173—81no—
154#13063436—21no—
155#13063436—21no—
156C23×C173—21no—
157C23×C173—21no—
158C23×C173—21no—
159Q8×C173—21no—
160Q8×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—21no—
166D4×C173—21no—
167D4×C173—42no—
168D4×C173—81no—
169C2×C4×C173—21no—
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173C2×D412yes—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173C22×D411yescharacteristic
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—22no—
188C2×D173—21no—
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—21no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—84no—
195C4×C173—21no—
196C4×C173—21no—
197C4×C173—41no—
198C4×C173C22×D411yescharacteristic
199C4×C173H(5,𝔽2)11yescharacteristic
200Dic173—21no—
201Dic173—21no—
202Dic173—22no—
203Dic173—41no—
204Dic173—41no—
205C2×C173—81no—
206C2×C173—81no—
207C2×C173—81no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—41no—
213D173—42no—
214C173#326811yescharacteristic, complemented, Sylow
215#3268—1731nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1731no—
217C2×#135—1731no—
218#550—1731no—
219#546—1731no—
220#496D17311yescharacteristic, complemented
221#468—1731no—
222#466—1731no—
223#464—1731no—
224#459—1731no—
225#452—1731no—
226#449—1731no—
227#449—1731no—
228#447—1731no—
229#440—1731no—
230#435—1731no—
231C2×#41—1731no—
232C2×#41—1731no—
233C2×#41—1731no—
234C22×D4—1731no—
235AGL(1,ℤ/8)—3464no—
236#135—3464no—
237C2×Q16—1731no—
238C2×SD16—1731no—
239C2×SD16—1731no—
240C2×SD16C2×D17311yescharacteristic, complemented
241C2×SD16C2×D17311yescharacteristic, complemented
242C2×D8—1731no—
243C2×D8—1731no—
244C2×D8C2×D17311yescharacteristic
245C2×D8C2×D17311yescharacteristic
246C2×M16—1731no—
247C22×C8—1731no—
248#127—3461no—
249#127C2×D17311yescharacteristic, complemented
250#124C2×D17311yescharacteristic
251#123—3461no—
252C22⋊Q8—1731no—
253#121—3461no—
254#121—3461no—
255#121—3461no—
256#121—1731no—
257#121—1731no—
258#121—1731no—
259#120—3461no—
260C4×Q8—1731no—
261C4×D4—3461no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265#107—1731no—
266C4⋊C8—1731no—
267#103—1731no—
268#103—1731no—
269#102—1731no—
270#102—1731no—
271#102—1731no—
272#102—1731no—
273C22⋊C8—1731no—
274C22⋊C8—1731no—
275#97C2×D17311yescharacteristic, complemented
276C24—3461no—
277#41—3462no—
278#41—3462no—
279#41—3462no—
280#41—6921no—
281#41—6921no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3461no—
288C2×D4—3462no—
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—1731no—
296C2×D4—1731no—
297C2×D4—6921no—
298C2×D4C22×D17311yescharacteristic
299C2×D4C22×D17311yescharacteristic
300C2×D4C22×D17311yescharacteristic
301C22×C4—3461no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—3461no—
305C22×C4—1731no—
306C22×C4—1731no—
307Q16—3462no—
308SD16—22no—
309SD16—41no—
310SD16—3462no—
311SD16—1734no—
312D8—22no—
313D8—41no—
314D8—3462no—
315D8—3462no—
316M16—3462no—
317C2×C8—3462no—
318C2×C8—1731no—
319C2×C8—1731no—
320C2×C8C22×D17311yescharacteristic
321C2×C8C22×D17311yescharacteristic
322C4⋊C4—3461no—
323C4⋊C4—3461no—
324C4⋊C4—1731no—
325C4⋊C4—1731no—
326C4⋊C4—1731no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—1731no—
335C22⋊C4—1731no—
336C42—3461no—
337C42C22×D17311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—3461no—
342C23—13841no—
343C23—1731no—
344C23—1731no—
345C23—6922no—
346C23—6922no—
347Q8—21no—
348Q8—21no—
349Q8—3461no—
350Q8—1732no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—3461no—
360D4—13841no—
361D4—13841no—
362D4—13841no—
363D4—1732no—
364D4—6921no—
365D4—6922no—
366D4—6922no—
367D4—6922no—
368D4—6922no—
369C2×C4—21no—
370C2×C4—3461no—
371C2×C4—3461no—
372C2×C4—3461no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—13841no—
378C2×C4—1731no—
379C2×C4—1731no—
380C2×C4—6922no—
381C2×C4—6922no—
382C2×C4—6922no—
383C2×C4D4×D17311yescharacteristic
384C2×C4D4×D17311yescharacteristic
385C2×C4C23×D17311yescharacteristic
386C8—21no—
387C8—3461no—
388C8—3461no—
389C8D4×D17312yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3461no—
400C22—3461no—
401C22—3461no—
402C22—3461no—
403C22—3461no—
404C22—3461no—
405C22—3462no—
406C22—13844no—
407C22—6921no—
408C22—6921no—
409C22C2×D4×D17311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—3461no—
414C4—3461no—
415C4—3462no—
416C4—6921no—
417C4—6921no—
418C4C2×D4×D17311yescharacteristic
419C4#1914972411yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—6921no—
424C2—6921no—
425C2—6922no—
426C2#2601212611yescharacteristic
427C2#2601228711yescharacteristic
428C2#2601229511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#496×C173C2
socleC22×C173C22×D4
hypercenter#496D173
hypocenterC173#3268
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149700C2semidirect
3. #26012161C2semidirect
4. #26012134C2semidirect
5. #26011734C2semidirect
6. #26011724C2semidirect
7. #26011710C2semidirect
8. #26011683C2semidirect
9. #26011650C2semidirect
10. #26011635C2semidirect
11. #26011633C2semidirect
12. #26011623C2semidirect
13. #26011593C2semidirect
14. #26011564C2semidirect
15. #496×C173C2semidirect
18. C2×#14583825C22semidirect
22. C2×#14583816C22semidirect
28. C2×C8×D173C22semidirect
37. #19149650C22semidirect
39. C4×Dic346C22semidirect
44. #19149613C22semidirect
45. C4⋊#13063436C22semidirect
46. #19149604C22semidirect
47. #19149603C22semidirect
50. #19149600C22semidirect
51. #19149600C22semidirect
53. #19149591C22semidirect
54. C2×SD16×C173C22semidirect
55. C2×SD16×C173C22semidirect
58. #127×C173C22semidirect
60. #97×C173C22semidirect
87. #14583815D4semidirect
214. C173#3268semidirect
220. #496D173semidirect
240. C2×SD16C2×D173semidirect
241. C2×SD16C2×D173semidirect
249. #127C2×D173semidirect
275. #97C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×#191497002
#260121612
#260121342
#260117342
#260117242
#260117102
#260116832
#260116502
#260116352
#260116332
#260116232
#260115932
#260115642
#496×C1732
#3268173