Finite Group Databasebeta

← #46849499

Subgroups — #46849499

Order 34688.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576183C211yesmaximal, characteristic, complemented
2#135×D271C211yesmaximal, characteristic, complemented
3#31774339C212yesmaximal, complemented
4#31774338C211yesmaximal, characteristic, complemented
5#31774335C211yesmaximal, characteristic, complemented
6#31774334C212yesmaximal, complemented
7#31774332C211yesmaximal, characteristic, complemented
8#31774330C212yesmaximal, complemented
9#31774328C212yesmaximal, complemented
10M32×D271C211yesmaximal, characteristic, complemented
11#510×C271C211yesmaximal, characteristic, complemented
12C2×#16864655—21no—
13#41×D271—21no—
14#22576183—21no—
15#22576183—21no—
16#22576183C2212yes—
17#22576183C2212yescomplemented
18#22576183C2211yescharacteristic, complemented
19#22576182C2211yescharacteristic
20#22576181—21no—
21#22576180—21no—
22#22576179C2211yescharacteristic, complemented
23Q16×D271C2211yescharacteristic, complemented
24C2×#16864641—21no—
25SD16×D271—21no—
26C2×D2168C2211yescharacteristic, complemented
27D8×D271C2211yescharacteristic, complemented
28C2×C8×D271C2211yescharacteristic
29#22576106C2212yescomplemented
30#22576105C2212yescomplemented
31#22576103C2212yescomplemented
32#22576103C2212yescomplemented
33D4336C2212yescomplemented
34#22576101C2212yescomplemented
35#22576100C2211yescharacteristic, complemented
36#22576099C2212yescomplemented
37C16×D271C2212yescomplemented
38#135×C271C2211yescharacteristic
39C2×Q16×C271C2211yescharacteristic, complemented
40Q32×C271C2212yescomplemented
41SD32×C271C2212yescomplemented
42M32×C271C2211yescharacteristic, complemented
43#16864655—21no—
44#16864655—22no—
45#16864655—22no—
46#16864655—41no—
47#16864655—41no—
48#16864654—21no—
49#16864654—41no—
50#16864653—21no—
51#16864653—41no—
52Q8×D271—21no—
53C2×D1084—21no—
54C2×D1084—41no—
55D4×D271—21no—
56D4×D271—41no—
57C2×C4×D271—41no—
58C2×C4×D271—41no—
59C2×C4×D271D411yescharacteristic
60Dic1084C2311yescharacteristic
61#16864644—21no—
62#16864643—21no—
63#16864642—21no—
64#16864641—21no—
65#16864641—22no—
66#16864641—22no—
67D2168—21no—
68D2168C2311yescharacteristic
69D2168C2312yes—
70#16864639—21no—
71C2×C271⋊C8D411yescharacteristic
72C8×D271D411yescharacteristic
73C8×D271D411yescharacteristic
74C8×D271C2311yescharacteristic
75C8×D271C2311yescharacteristic
76C271⋊C16C2312yes—
77#41×C271—21no—
78C2×Q8×C271—21no—
79Q16×C271—21no—
80Q16×C271C2311yescharacteristic
81Q16×C271C2312yes—
82SD16×C271—21no—
83D8×C271C2311yescharacteristic
84C2×C8×C271C2311yescharacteristic
85C16×C271C2312yes—
86C22×D271—41no—
87C22×D271—41no—
88C22×D271D811yescharacteristic
89Dic542—21no—
90Dic542—41no—
91D1084—21no—
92D1084—22no—
93D1084—41no—
94D1084—42no—
95D1084—42no—
96D1084—41no—
97C22⋊D271—41no—
98C22⋊D271—41no—
99C22⋊D271—41no—
100C2×Dic271—41no—
101C2×Dic271D811yescharacteristic
102C4×D271—41no—
103C4×D271—41no—
104C4×D271—41no—
105C4×D271—42no—
106C4×D271—42no—
107C4×D271D811yescharacteristic
108C4×D271D811yescharacteristic
109C4×D271C2×D411yescharacteristic
110C4×D271C2×D411yescharacteristic
111C271⋊C8C2×D411yescharacteristic
112C271⋊C8C2×D411yescharacteristic
113Q8×C271—21no—
114Q8×C271—22no—
115Q8×C271—41no—
116D4×C271—21no—
117D4×C271—41no—
118C2×C4×C271—41no—
119C2×C4×C271—41no—
120C2×C4×C271C2×D411yescharacteristic
121C8×C271C2×D411yescharacteristic
122C8×C271C2411yescharacteristic
123C22×C271—41no—
124C22×C271C2×D811yescharacteristic
125C2×D271—41no—
126C2×D271—41no—
127C2×D271—42no—
128C2×D271—82no—
129C2×D271—82no—
130C2×D271C2×D811yescharacteristic
131C2×D271C2×D811yescharacteristic
132C4×C271—41no—
133C4×C271—42no—
134C4×C271C2×D811yescharacteristic
135C4×C271C22×D411yescharacteristic
136Dic271—41no—
137Dic271C2×D811yescharacteristic
138Dic271C2×D811yescharacteristic
139C2×C271—21no—
140C2×C271—81no—
141C2×C271C22×D811yescharacteristic
142D271—21no—
143D271—21no—
144D271—81no—
145D271—82no—
146C271#341411yescharacteristic, complemented, Sylow
147#3414—2711nomaximal, Sylow
148C2×#135—2711no—
149C2×#135—2711no—
150#510—2711no—
151#510—2712no—
152#510D27111yescharacteristic, complemented
153#509—2711no—
154#509—2712no—
155#509—2711no—
156#508—2712no—
157#508—2712no—
158C2×M32—2711no—
159C2×#41—5421no—
160C2×#41—5421no—
161#135—2712no—
162#135—2711no—
163#135—2712no—
164#135—2711no—
165#135—2711no—
166#135—5421no—
167#135—5421no—
168#135—5421no—
169#135—5421no—
170#135C2×D27111yescharacteristic
171C2×Q16—2711no—
172C2×Q16C2×D27111yescharacteristic, complemented
173C2×SD16—5421no—
174C2×SD16—5421no—
175C2×D8—2711no—
176C2×D8—2711no—
177C22×C8—2711no—
178Q32—2712no—
179Q32C2×D27112yescomplemented
180SD32—2712no—
181SD32—2712no—
182SD32—2712no—
183SD32C2×D27112yescomplemented
184D16—2712no—
185D16—2712no—
186M32—2711no—
187M32—2712no—
188M32C2×D27111yescharacteristic, complemented
189C2×C16—2712no—
190#41—21no—
191#41—5421no—
192#41—5422no—
193#41—5421no—
194#41—5422no—
195#41—5421no—
196#41—10841no—
197#41—10841no—
198#41—10841no—
199#41—10841no—
200C2×Q8—21no—
201C2×Q8—5421no—
202C2×D4—5421no—
203C2×D4—5421no—
204C2×D4—10841no—
205C2×D4—10841no—
206C22×C4—2711no—
207C22×C4—10841no—
208C22×C4—10841no—
209Q16—21no—
210Q16—2711no—
211Q16—5421no—
212Q16C22×D27111yescharacteristic
213Q16C22×D27112yes—
214SD16—21no—
215SD16—5421no—
216SD16—5422no—
217SD16—5421no—
218SD16—5422no—
219SD16—5421no—
220D8—2712no—
221D8—2711no—
222D8—5421no—
223D8—5421no—
224D8C22×D27111yescharacteristic
225C2×C8—2711no—
226C2×C8—2711no—
227C2×C8—2711no—
228C2×C8—2711no—
229C2×C8—2711no—
230C2×C8C22×D27111yescharacteristic
231C16—2712no—
232C16C22×D27112yes—
233C23—2711no—
234C23—10841no—
235C23—10841no—
236Q8—21no—
237Q8—22no—
238Q8—41no—
239Q8—5421no—
240Q8—10841no—
241D4—21no—
242D4—41no—
243D4—5421no—
244D4—5422no—
245D4—10841no—
246D4—10842no—
247D4—10842no—
248D4—10841no—
249D4—10841no—
250D4—10841no—
251D4—10841no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—2711no—
255C2×C4—2711no—
256C2×C4—2711no—
257C2×C4—2711no—
258C2×C4—2711no—
259C2×C4—10841no—
260C2×C4—10842no—
261C2×C4—10842no—
262C2×C4—10841no—
263C2×C4—10841no—
264C2×C4—10841no—
265C2×C4D4×D27111yescharacteristic
266C8—2711no—
267C8—2711no—
268C8D4×D27111yescharacteristic
269C8C23×D27111yescharacteristic
270C22—41no—
271C22—2711no—
272C22—2711no—
273C22—10841no—
274C22—10842no—
275C22—10841no—
276C22—21682no—
277C22—21682no—
278C22D8×D27111yescharacteristic
279C4—41no—
280C4—42no—
281C4—2711no—
282C4—2711no—
283C4—10841no—
284C4D8×D27111yescharacteristic
285C4C2×D4×D27111yescharacteristic
286C2—21no—
287C2—81no—
288C2—5421no—
289C2—5421no—
290C2—21682no—
291C2—21681no—
292C2C2×D8×D27111yescharacteristic
subgroupG/H
centerC2C2×D8×D271
commutator, cosocleC8×C271C24
FrattiniC8C23×D271
Fitting#510×C271C2
socleC2×C271C22×D8
hypercenter#510D271
hypocenterC271#3414
LCSG ▷ C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. C2×#22576183C2semidirect
2. #135×D271C2semidirect
3. #31774339C2semidirect
4. #31774338C2semidirect
5. #31774335C2semidirect
6. #31774334C2semidirect
7. #31774332C2semidirect
8. #31774330C2semidirect
9. #31774328C2semidirect
10. M32×D271C2semidirect
11. #510×C271C2semidirect
17. #22576183C22semidirect
18. #22576183C22semidirect
22. #22576179C22semidirect
23. Q16×D271C22semidirect
26. C2×D2168C22semidirect
27. D8×D271C22semidirect
29. #22576106C22semidirect
30. #22576105C22semidirect
31. #22576103C22semidirect
32. #22576103C22semidirect
33. D4336C22semidirect
34. #22576101C22semidirect
35. #22576100C22semidirect
36. #22576099C22semidirect
37. C16×D271C22semidirect
39. C2×Q16×C271C22semidirect
40. Q32×C271C22semidirect
41. SD32×C271C22semidirect
42. M32×C271C22semidirect
146. C271#3414semidirect
152. #510D271semidirect
172. C2×Q16C2×D271semidirect
179. Q32C2×D271semidirect
183. SD32C2×D271semidirect
188. M32C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761832
#135×D2712
#317743392
#317743382
#317743352
#317743342
#317743322
#317743302
#317743282
M32×D2712
#510×C2712
#3414271