Finite Group Databasebeta

← #36365109

Subgroups — #36365109

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×#19149698C211yesmaximal, characteristic, complemented
3#26012161C211yesmaximal, characteristic, complemented
4#26012136C211yesmaximal, characteristic, complemented
5#26011878C211yesmaximal, characteristic, complemented
6#26011736C211yesmaximal, characteristic, complemented
7#26011726C211yesmaximal, characteristic, complemented
8#26011714C211yesmaximal, characteristic, complemented
9#26011654C211yesmaximal, characteristic, complemented
10#26011637C211yesmaximal, characteristic, complemented
11#26011633C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011591C211yesmaximal, characteristic, complemented
14#26011561C211yesmaximal, characteristic, complemented
15#459×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×#14583825C2211yescharacteristic, complemented
19C2×D4×D173C2211yescharacteristic
20#19149703—24no—
21#19149698—24no—
22C2×Dic692C2211yescharacteristic, complemented
23C2×#14583815C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic, complemented
26C2×#14583814C2211yescharacteristic, complemented
27C2×#14583811C2211yescharacteristic
28C2×#14583811C2211yescharacteristic
29C2×#14583809C2211yescharacteristic
30C2×C8×D173C2211yescharacteristic, complemented
31#19149675—21no—
32#19149667C2211yescharacteristic
33#19149663—21no—
34#19149659C2211yescharacteristic
35#19149654C2211yescharacteristic
36#19149653—21no—
37#19149651—21no—
38#19149651—21no—
39#19149650C2211yescharacteristic
40#19149650C2211yescharacteristic, complemented
41#19149649—21no—
42C4×Dic346C2211yescharacteristic, complemented
43C4×D692C2211yescharacteristic
44D4×Dic173—21no—
45C2×#14583803—21no—
46#19149612C2211yescharacteristic, complemented
47#19149604C2211yescharacteristic, complemented
48#19149602C2211yescharacteristic, complemented
49#19149601C2211yescharacteristic
50#19149600C2211yescharacteristic
51#19149591C2211yescharacteristic
52#19149591C2211yescharacteristic, complemented
53#19149590C2211yescharacteristic, complemented
54C2×D8×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#127×C173C2211yescharacteristic, complemented
57C4×D4×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic, complemented
59#102×C173C2211yescharacteristic, complemented
60#102×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583825—22no—
63#14583825—22no—
64#14583825—22no—
65#14583825—41no—
66#14583825—41no—
67#14583825—41no—
68C2×Dic346C2311yescharacteristic
69C2×Dic346C2311yescharacteristic
70C2×D692—21no—
71C2×D692D411yescharacteristic
72C2×D692C2311yescharacteristic
73C2×C22⋊D173—21no—
74C2×C22⋊D173—21no—
75C2×C22⋊D173—21no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173—21no—
78C2×C22⋊D173—21no—
79D4×D173—22no—
80D4×D173—41no—
81C22×Dic173—21no—
82C22×Dic173—21no—
83C22×Dic173—21no—
84C2×C4×D173C2311yescharacteristic
85C2×C4×D173C2311yescharacteristic
86Dic692—22no—
87#14583815—22no—
88#14583814—22no—
89#14583814—41no—
90#14583814D414yescomplemented
91#14583811—22no—
92#14583811—41no—
93#14583809—22no—
94C2×#13063436C2311yescharacteristic
95C2×#13063436C2311yescharacteristic
96C8×D173—22no—
97C4⋊Dic173—21no—
98C4⋊Dic173D411yescharacteristic
99C4⋊Dic173C2311yescharacteristic
100#14583805—21no—
101C22⋊Dic173—21no—
102C22⋊Dic173—21no—
103#14583803—21no—
104#14583803—21no—
105#14583803—21no—
106#14583803—21no—
107#14583803D411yescharacteristic
108#14583803D411yescharacteristic
109C4×Dic173—21no—
110C2×D4×C173—21no—
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C2×D4×C173C2311yescharacteristic
114C2×D4×C173C2311yescharacteristic
115C22×C4×C173—21no—
116D8×C173—22no—
117D8×C173—22no—
118C2×C8×C173C2311yescharacteristic
119C2×C8×C173C2311yescharacteristic
120C4⋊C4×C173C2311yescharacteristic
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—
130Dic346—21no—
131Dic346—21no—
132Dic346C2×D412yes—
133D692—21no—
134D692—21no—
135D692—41no—
136D692—81no—
137C22⋊D173—42no—
138C22⋊D173—42no—
139C22⋊D173—42no—
140C22⋊D173—42no—
141C22⋊D173—81no—
142C22⋊D173—81no—
143C2×Dic173—21no—
144C2×Dic173—21no—
145C2×Dic173—21no—
146C2×Dic173—42no—
147C2×Dic173—42no—
148C2×Dic173—42no—
149C2×Dic173C2×D411yescharacteristic
150C2×Dic173C2×D411yescharacteristic
151C4×D173—21no—
152C4×D173—21no—
153C4×D173—21no—
154C4×D173—21no—
155#13063436—21no—
156#13063436C2×D412yes—
157C23×C173—21no—
158C23×C173—21no—
159C23×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—21no—
163D4×C173—21no—
164D4×C173—21no—
165D4×C173—42no—
166D4×C173—81no—
167D4×C173C2×D412yes—
168C2×C4×C173—21no—
169C2×C4×C173—81no—
170C2×C4×C173C2×D411yescharacteristic
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173—21no—
175C22×C173—22no—
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—21no—
188C2×D173—21no—
189C2×D173—21no—
190C2×D173—21no—
191C2×D173—41no—
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—42no—
206C2×C173—81no—
207C2×C173—81no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—41no—
213D173—81no—
214C173#326811yescharacteristic, complemented, Sylow
215#3268—1731nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1731no—
217C2×#135—1731no—
218#550—1731no—
219#546—1731no—
220#496—1731no—
221#468—1731no—
222#466—1731no—
223#464—1731no—
224#459D17311yescharacteristic, complemented
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×SD16—1731no—
241C2×SD16—1731no—
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#124—1731no—
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×D4C2×D17311yescharacteristic
264C2×C22⋊C4—3461no—
265#107—1731no—
266C4⋊C8C2×D17311yescharacteristic, complemented
267#103—1731no—
268#103—1731no—
269#102—1731no—
270#102—1731no—
271#102C2×D17311yescharacteristic, complemented
272#102C2×D17311yescharacteristic, complemented
273C22⋊C8—1731no—
274C22⋊C8—1731no—
275#97—1731no—
276C24—3461no—
277#41—3462no—
278#41—3462no—
279#41—3462no—
280#41—6921no—
281#41—6921no—
282#41—6921no—
283C2×Q8—1731no—
284C2×Q8—1731no—
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—21no—
302C22×C4—3461no—
303C22×C4—3461no—
304C22×C4—3461no—
305C22×C4—1731no—
306C22×C4—1731no—
307Q16—3462no—
308SD16—3462no—
309SD16—3462no—
310SD16—1734no—
311SD16—6921no—
312D8—22no—
313D8—22no—
314D8—3462no—
315D8—6921no—
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⋊C4C22×D17311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—3461no—
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—3461no—
348Q8—3461no—
349Q8—3461no—
350Q8—1732no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—42no—
357D4—81no—
358D4—3461no—
359D4—3461no—
360D4—13841no—
361D4—13841no—
362D4—13841no—
363D4—6921no—
364D4—6922no—
365D4—6922no—
366D4—6922no—
367D4—6922no—
368D4D4×D17312yes—
369C2×C4—21no—
370C2×C4—81no—
371C2×C4—3461no—
372C2×C4—3461no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
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—21no—
388C8—3461no—
389C8—1732no—
390C22—22no—
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—13844no—
406C22—6921no—
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#1914972511yescharacteristic
420C2—42no—
421C2—81no—
422C2—81no—
423C2—13841no—
424C2—6921no—
425C2—6921no—
426C2#2601212511yescharacteristic
427C2#2601228711yescharacteristic
428C2#2601229211yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#459×C173C2
socleC22×C173C22×D4
hypercenter#459D173
hypocenterC173#3268
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#19149703C2semidirect
2. C2×#19149698C2semidirect
3. #26012161C2semidirect
4. #26012136C2semidirect
5. #26011878C2semidirect
6. #26011736C2semidirect
7. #26011726C2semidirect
8. #26011714C2semidirect
9. #26011654C2semidirect
10. #26011637C2semidirect
11. #26011633C2semidirect
12. #26011623C2semidirect
13. #26011591C2semidirect
14. #26011561C2semidirect
15. #459×C173C2semidirect
18. C2×#14583825C22semidirect
22. C2×Dic692C22semidirect
25. C2×#14583814C22semidirect
26. C2×#14583814C22semidirect
30. C2×C8×D173C22semidirect
40. #19149650C22semidirect
42. C4×Dic346C22semidirect
46. #19149612C22semidirect
47. #19149604C22semidirect
48. #19149602C22semidirect
52. #19149591C22semidirect
53. #19149590C22semidirect
56. #127×C173C22semidirect
58. C4⋊C8×C173C22semidirect
59. #102×C173C22semidirect
60. #102×C173C22semidirect
90. #14583814D4semidirect
214. C173#3268semidirect
224. #459D173semidirect
249. #127C2×D173semidirect
266. C4⋊C8C2×D173semidirect
271. #102C2×D173semidirect
272. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497032
C2×#191496982
#260121612
#260121362
#260118782
#260117362
#260117262
#260117142
#260116542
#260116372
#260116332
#260116232
#260115912
#260115612
#459×C1732
#3268173