Finite Group Databasebeta

← #36364388

Subgroups — #36364388

Order 22144.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012001C211yesmaximal, characteristic
2C22⋊Q8×D173C211yesmaximal, characteristic, complemented
3#26011834C211yesmaximal, characteristic, complemented
4#26011830C211yesmaximal, characteristic, complemented
5#26011828C211yesmaximal, characteristic, complemented
6#26011808C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011747C211yesmaximal, characteristic, complemented
9#26011735C211yesmaximal, characteristic, complemented
10#26011654C211yesmaximal, characteristic, complemented
11#26011650C211yesmaximal, characteristic, complemented
12#26011648C211yesmaximal, characteristic, complemented
13#26011628C211yesmaximal, characteristic, complemented
14#26011454C211yesmaximal, characteristic, complemented
15#468×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19C2×Dic692C2211yescharacteristic, complemented
20C2×#14583816C2211yescharacteristic, complemented
21C2×#14583814C2211yescharacteristic, complemented
22C2×#14583810C2211yescharacteristic
23#19149661—21no—
24#19149659C2211yescharacteristic
25#19149659C2211yescharacteristic
26#19149658C2211yescharacteristic
27#19149658C2211yescharacteristic
28#19149657—21no—
29#19149656—21no—
30#19149655C2211yescharacteristic
31#19149650C2211yescharacteristic
32#19149647—21no—
33D4×Dic173—21no—
34#19149639—21no—
35#19149637C2211yescharacteristic
36C4⋊C4×D173—21no—
37C4⋊C4×D173C2211yescharacteristic
38C22⋊C4×D173—21no—
39#19149612C2211yescharacteristic, complemented
40#19149611C2211yescharacteristic, complemented
41#19149610C2211yescharacteristic, complemented
42#19149604C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149602C2211yescharacteristic, complemented
46#19149602C2211yescharacteristic, complemented
47#19149600C2211yescharacteristic, complemented
48#19149591C2211yescharacteristic
49#19149591C2211yescharacteristic
50#19149591C2211yescharacteristic
51#19149591C2211yescharacteristic
52C2×SD16×C173C2211yescharacteristic, complemented
53C2×M16×C173C2211yescharacteristic
54C22⋊Q8×C173C2211yescharacteristic
55#121×C173C2211yescharacteristic
56#107×C173C2211yescharacteristic, complemented
57#103×C173C2211yescharacteristic, complemented
58#102×C173C2211yescharacteristic, complemented
59C23×D173D411yescharacteristic
60#14583825—41no—
61#14583825—41no—
62C2×Dic346—21no—
63C2×Dic346C2311yescharacteristic
64C2×Dic346C2311yescharacteristic
65Q8×D173—41no—
66Q8×D173—41no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C22×Dic173—21no—
70C22×Dic173D411yescharacteristic
71C2×C4×D173—21no—
72C2×C4×D173—21no—
73C2×C4×D173—41no—
74C2×C4×D173—41no—
75C2×C4×D173D411yescharacteristic
76C2×C4×D173D411yescharacteristic
77C2×C4×D173C2311yescharacteristic
78C2×C4×D173C2311yescharacteristic
79Dic692—41no—
80#14583816—41no—
81#14583814—41no—
82#14583810—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C4⋊Dic173—21no—
86C4⋊Dic173—21no—
87C4⋊Dic173C2311yescharacteristic
88C4⋊Dic173C2311yescharacteristic
89#14583805—21no—
90#14583805—21no—
91#14583805—21no—
92#14583805—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100C4×Dic173—21no—
101C2×Q8×C173C2311yescharacteristic
102C2×D4×C173—21no—
103C2×D4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105SD16×C173—41no—
106M16×C173—41no—
107C2×C8×C173C2311yescharacteristic
108C2×C8×C173C2311yescharacteristic
109C4⋊C4×C173—21no—
110C4⋊C4×C173C2311yescharacteristic
111C4⋊C4×C173C2311yescharacteristic
112C22⋊C4×C173—21no—
113C22⋊C4×C173—21no—
114C22×D173—22no—
115C22×D173—41no—
116C22×D173—41no—
117C22×D173C2×D411yescharacteristic
118C22×D173C2×D411yescharacteristic
119Dic346—21no—
120Dic346—21no—
121Dic346—21no—
122Dic346—21no—
123Dic346—42no—
124C22⋊D173—42no—
125C22⋊D173—42no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—41no—
131C2×Dic173—42no—
132C2×Dic173C2×D411yescharacteristic
133C2×Dic173C2×D411yescharacteristic
134C4×D173—21no—
135C4×D173—21no—
136C4×D173—21no—
137C4×D173—21no—
138C4×D173—41no—
139C4×D173—41no—
140C4×D173—42no—
141C4×D173—42no—
142#13063436—21no—
143#13063436—21no—
144C23×C173—21no—
145C23×C173C2×D411yescharacteristic
146Q8×C173—21no—
147Q8×C173—21no—
148D4×C173—21no—
149D4×C173—21no—
150D4×C173—42no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2411yescharacteristic
158C8×C173—21no—
159C8×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173C22×D411yescharacteristic
167C2×D173—21no—
168C2×D173—21no—
169C2×D173—21no—
170C2×D173—21no—
171C2×D173—21no—
172C2×D173—44no—
173C2×D173#13712yes—
174C4×C173—21no—
175C4×C173—41no—
176C4×C173—41no—
177C4×C173—41no—
178C4×C173H(5,𝔽2)11yescharacteristic
179C4×C173H(5,𝔽2)11yescharacteristic
180Dic173—21no—
181Dic173—21no—
182Dic173—41no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186C2×C173—41no—
187C2×C173—81no—
188C2×C173#53411yescharacteristic
189C2×C173C2×#13711yescharacteristic
190C2×C173#57511yescharacteristic
191D173—22no—
192D173—41no—
193C173#319611yescharacteristic, complemented, Sylow
194#3196—1731nomaximal, Sylow
195#525—1731no—
196C2×C22⋊Q8—1731no—
197#485—1731no—
198#485—1731no—
199#484—1731no—
200#481—1731no—
201#470—1731no—
202#470—1731no—
203#468—1731no—
204#468D17311yescharacteristic, complemented
205#452—1731no—
206#452—1731no—
207#451—1731no—
208#448—1731no—
209#407—1731no—
210C2×#41—1731no—
211C22×Q8—1731no—
212C23×C4—1731no—
213C2×Q16—1731no—
214C2×Q16—1731no—
215C2×SD16—1731no—
216C2×SD16C2×D17311yescharacteristic, complemented
217C2×M16—1731no—
218C2×M16C2×D17311yescharacteristic
219#123—3461no—
220C22⋊Q8—3461no—
221C22⋊Q8—3461no—
222C22⋊Q8—1731no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227C22⋊Q8C2×D17311yescharacteristic
228#121—1731no—
229#121C2×D17311yescharacteristic
230#120—3461no—
231C4×D4—3461no—
232C4×D4—3461no—
233#117—1731no—
234C2×C4⋊C4—3461no—
235C2×C4⋊C4—1731no—
236C2×C22⋊C4—3461no—
237#107—1731no—
238#107C2×D17311yescharacteristic, complemented
239#106—1731no—
240#106—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#103—1731no—
245#103—1731no—
246#103C2×D17311yescharacteristic, complemented
247#102—1731no—
248#102C2×D17311yescharacteristic, complemented
249C22⋊C8—1731no—
250C22⋊C8—1731no—
251C22⋊C8—1731no—
252C22⋊C8—1731no—
253C24—1731no—
254#41—6921no—
255#41—6921no—
256C2×Q8—3461no—
257C2×Q8—1731no—
258C2×Q8—1731no—
259C2×Q8—6921no—
260C2×Q8—6921no—
261C2×Q8C22×D17311yescharacteristic
262C2×D4—21no—
263C2×D4—3461no—
264C2×D4—3461no—
265C2×D4C22×D17311yescharacteristic
266C22×C4—3461no—
267C22×C4—3461no—
268C22×C4—3461no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—6921no—
275C22×C4—6921no—
276C22×C4C22×D17311yescharacteristic
277Q16—6921no—
278Q16—6921no—
279SD16—41no—
280SD16—6921no—
281M16—41no—
282M16—6921no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8C22×D17311yescharacteristic
286C2×C8C22×D17311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—1731no—
295C4⋊C4—1731no—
296C4⋊C4C22×D17311yescharacteristic
297C4⋊C4C22×D17311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—3461no—
307C42—3461no—
308C23—21no—
309C23—3462no—
310C23—1731no—
311C23—1731no—
312C23—6921no—
313C23—6921no—
314C23D4×D17311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—3461no—
318Q8—3461no—
319Q8—3461no—
320Q8—3461no—
321Q8—6922no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—6922no—
326D4—6922no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—6921no—
343C2×C4—6921no—
344C2×C4—6922no—
345C2×C4—6922no—
346C2×C4—6921no—
347C2×C4—6922no—
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3461no—
361C22—3461no—
362C22—3461no—
363C22—3461no—
364C22—3461no—
365C22—1732no—
366C22—6924no—
367C22C2×D4×D17311yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—3461no—
373C4—3461no—
374C4—6921no—
375C4—6921no—
376C4—6921no—
377C4—6921no—
378C4#1914972411yescharacteristic
379C4#1914972511yescharacteristic
380C2—41no—
381C2—81no—
382C2—3462no—
383C2—6921no—
384C2#2601204911yescharacteristic
385C2#137×D17311yescharacteristic
386C2#2601228711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#468×C173C2
socleC22×C173C22×D4
prime radical#26012001C2
hypercenter#468D173
hypocenterC173#3196
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. C22⋊Q8×D173C2semidirect
3. #26011834C2semidirect
4. #26011830C2semidirect
5. #26011828C2semidirect
6. #26011808C2semidirect
7. #26011749C2semidirect
8. #26011747C2semidirect
9. #26011735C2semidirect
10. #26011654C2semidirect
11. #26011650C2semidirect
12. #26011648C2semidirect
13. #26011628C2semidirect
14. #26011454C2semidirect
15. #468×C173C2semidirect
19. C2×Dic692C22semidirect
20. C2×#14583816C22semidirect
21. C2×#14583814C22semidirect
39. #19149612C22semidirect
40. #19149611C22semidirect
41. #19149610C22semidirect
42. #19149604C22semidirect
43. #19149604C22semidirect
44. #19149603C22semidirect
45. #19149602C22semidirect
46. #19149602C22semidirect
47. #19149600C22semidirect
52. C2×SD16×C173C22semidirect
56. #107×C173C22semidirect
57. #103×C173C22semidirect
58. #102×C173C22semidirect
193. C173#3196semidirect
204. #468D173semidirect
216. C2×SD16C2×D173semidirect
238. #107C2×D173semidirect
246. #103C2×D173semidirect
248. #102C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120012
C22⋊Q8×D1732
#260118342
#260118302
#260118282
#260118082
#260117492
#260117472
#260117352
#260116542
#260116502
#260116482
#260116282
#260114542
#468×C1732
#3196173