Finite Group Databasebeta

← #46845977

Subgroups — #46845977

Order 34688.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D271C211yesmaximal, characteristic, complemented
2#31774404C211yesmaximal, characteristic, complemented
3C2×#22576131C211yesmaximal, characteristic, complemented
4#31774147C212yesmaximal, complemented
5#31774140C212yesmaximal, complemented
6#31774130C212yesmaximal, complemented
7#31774123C212yesmaximal, complemented
8#31773939C211yesmaximal, characteristic, complemented
9#31773902C211yesmaximal, characteristic, complemented
10#31773898C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C271C211yesmaximal, characteristic, complemented
12C22×C22⋊D271—21no—
13C22×C4×D271C2211yescharacteristic
14C2×#16864638C2211yescharacteristic
15C2×#16864637C2212yescomplemented
16M16×D271—24no—
17C2×C8×D271C2212yescomplemented
18#22576141—21no—
19#22576136C2211yescharacteristic, complemented
20#22576136C2211yescharacteristic, complemented
21#22576131—21no—
22#22576131—21no—
23#22576131C2212yes—
24#22576131C2212yes—
25#22576131C2211yescharacteristic, complemented
26#22576131C2211yescharacteristic, complemented
27#22576128—21no—
28C4×C22⋊D271—21no—
29C4×C22⋊D271—21no—
30#22576119C2211yescharacteristic
31C2×C4⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271—21no—
33#22576094C2212yescomplemented
34#22576092C2212yescomplemented
35#22576084C2212yescomplemented
36#22576081C2212yescomplemented
37#22576081C2212yescomplemented
38#22576081C2212yescomplemented
39C2×#41×C271C2211yescharacteristic
40C22×D4×C271C2211yescharacteristic
41AGL(1,ℤ/8)×C271—24no—
42C2×SD16×C271C2212yescomplemented
43C2×D8×C271C2212yescomplemented
44C2×M16×C271C2211yescharacteristic
45C23×D271D411yescharacteristic
46C2×C22⋊D271—21no—
47C2×C22⋊D271—21no—
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50C2×C22⋊D271—44no—
51C22×Dic271—21no—
52C22×Dic271D411yescharacteristic
53C2×C4×D271—22no—
54C2×C4×D271—22no—
55C2×C4×D271D411yescharacteristic
56C2×C4×D271D411yescharacteristic
57C2×C4×D271C2311yescharacteristic
58C2×C4×D271C2311yescharacteristic
59#16864638—22no—
60#16864637—24no—
61C2×C271⋊C8C2312yes—
62C8×D271—24no—
63C4⋊Dic271—21no—
64C4⋊Dic271C2312yes—
65C4⋊Dic271C2311yescharacteristic
66C4⋊Dic271C2311yescharacteristic
67Dic271⋊C4—21no—
68Dic271⋊C4—21no—
69C22⋊Dic271—22no—
70C22⋊Dic271—22no—
71C22⋊Dic271—21no—
72C22⋊Dic271—21no—
73C22⋊Dic271—21no—
74(C2×D271)⋊C4—21no—
75(C2×D271)⋊C4—21no—
76C4×Dic271—21no—
77C24×C271—21no—
78#41×C271—22no—
79#41×C271—22no—
80C2×Q8×C271C2311yescharacteristic
81C2×D4×C271—21no—
82C2×D4×C271—22no—
83C2×D4×C271—21no—
84C2×D4×C271—22no—
85C2×D4×C271C2312yes—
86C2×D4×C271C2311yescharacteristic
87C22×C4×C271—21no—
88C22×C4×C271C2311yescharacteristic
89SD16×C271—24no—
90D8×C271—24no—
91M16×C271D414yescomplemented
92C2×C8×C271C2312yes—
93C22×D271—22no—
94C22×D271—22no—
95C22×D271—41no—
96C22×D271C2×D411yescharacteristic
97C22×D271C2×D411yescharacteristic
98C22⋊D271—42no—
99C22⋊D271—44no—
100C22⋊D271—42no—
101C22⋊D271—44no—
102C2×Dic271—22no—
103C2×Dic271—21no—
104C2×Dic271—21no—
105C2×Dic271—22no—
106C2×Dic271—42no—
107C2×Dic271C2×D411yescharacteristic
108C2×Dic271C2×D411yescharacteristic
109C4×D271—22no—
110C4×D271—22no—
111C4×D271—21no—
112C4×D271—21no—
113C4×D271—21no—
114C4×D271—21no—
115C271⋊C8—22no—
116C23×C271—21no—
117C23×C271—22no—
118C23×C271—42no—
119C23×C271—42no—
120C23×C271—42no—
121C23×C271C2×D411yescharacteristic
122Q8×C271—21no—
123Q8×C271—21no—
124D4×C271—21no—
125D4×C271—22no—
126D4×C271—22no—
127D4×C271—21no—
128D4×C271—42no—
129D4×C271—42no—
130C2×C4×C271—21no—
131C2×C4×C271—42no—
132C2×C4×C271C2×D411yescharacteristic
133C2×C4×C271C2×D412yes—
134C2×C4×C271#4112yes—
135C2×C4×C271C2411yescharacteristic
136C8×C271C2×D414yescomplemented
137C22×C271—21no—
138C22×C271—21no—
139C22×C271—41no—
140C22×C271—42no—
141C22×C271—42no—
142C22×C271—41no—
143C22×C271—42no—
144C22×C271—41no—
145C22×C271—84no—
146C22×C271#12112yes—
147C22×C271C22×D411yescharacteristic
148C2×D271—21no—
149C2×D271—21no—
150C2×D271—21no—
151C2×D271—21no—
152C2×D271—21no—
153C2×D271—44no—
154C2×D271AGL(1,ℤ/8)12yescomplemented
155C4×C271—41no—
156C4×C271#12112yes—
157C4×C271C22×D411yescharacteristic
158C4×C271C2×#4111yescharacteristic
159Dic271—21no—
160Dic271—21no—
161Dic271—42no—
162Dic271—41no—
163Dic271—41no—
164C2×C271—22no—
165C2×C271—81no—
166C2×C271—82no—
167C2×C271C2×#12111yescharacteristic
168C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C271#57511yescharacteristic
170D271—22no—
171D271—41no—
172C271#305511yescharacteristic, complemented, Sylow
173#3055—2711nomaximal, Sylow
174C2×AGL(1,ℤ/8)D27111yescharacteristic, complemented
175C22×M16—2711no—
176#525—2711no—
177C2×#121—2711no—
178#469—2712no—
179#468—2712no—
180#466—2712no—
181#465—2712no—
182#428—2711no—
183#418—2711no—
184#417—2711no—
185C2×#41C2×D27111yescharacteristic
186C22×D4—5421no—
187C22×D4C2×D27111yescharacteristic
188C23×C4—2711no—
189AGL(1,ℤ/8)—24no—
190C2×SD16C2×D27112yescomplemented
191C2×D8C2×D27112yescomplemented
192C2×M16—2711no—
193C2×M16—2712no—
194C2×M16—5424no—
195C2×M16C2×D27111yescharacteristic
196C22×C8—2712no—
197#123—5421no—
198C22⋊Q8—2711no—
199C22⋊Q8—2711no—
200#121—2712no—
201#121—2712no—
202#121—2711no—
203#121—2711no—
204#121—5421no—
205#121—5421no—
206#120—5421no—
207C4×D4—5421no—
208C4×D4—5421no—
209#117—2711no—
210C2×C4⋊C4—2711no—
211C2×C22⋊C4—5421no—
212#107—2712no—
213#106—2712no—
214#103—2712no—
215#102—2712no—
216#102—2712no—
217#102—2712no—
218C24—21no—
219C24—2711no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D27111yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—5421no—
228C2×D4—5421no—
229C2×D4—5422no—
230C2×D4—5422no—
231C2×D4—10844no—
232C2×D4C22×D27111yescharacteristic
233C2×D4C22×D27112yes—
234C22×C4—21no—
235C22×C4—2711no—
236C22×C4—2711no—
237C22×C4—2711no—
238C22×C4—2711no—
239C22×C4—2711no—
240C22×C4—5422no—
241C22×C4—5422no—
242C22×C4—5421no—
243C22×C4C22×D27111yescharacteristic
244SD16—24no—
245D8—24no—
246M16—5422no—
247M16—5424no—
248M16C22⋊D27114yescomplemented
249C2×C8—2712no—
250C2×C8—5424no—
251C2×C8C22×D27112yes—
252C4⋊C4—2711no—
253C4⋊C4—2711no—
254C4⋊C4—2712no—
255C4⋊C4—5421no—
256C4⋊C4—5421no—
257C4⋊C4—5421no—
258C22⋊C4—5421no—
259C22⋊C4—5421no—
260C22⋊C4—5422no—
261C22⋊C4—5422no—
262C22⋊C4—5421no—
263C22⋊C4—5421no—
264C22⋊C4—5421no—
265C42—5421no—
266C23—21no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—2711no—
272C23—2711no—
273C23—5422no—
274C23—5422no—
275C23—10841no—
276C23D4×D27111yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—22no—
281D4—22no—
282D4—21no—
283D4—42no—
284D4—42no—
285D4—10844no—
286D4—10842no—
287D4—10844no—
288D4—10842no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—2711no—
292C2×C4—2711no—
293C2×C4—5422no—
294C2×C4—5422no—
295C2×C4—5422no—
296C2×C4—5421no—
297C2×C4—5421no—
298C2×C4—5421no—
299C2×C4—5421no—
300C2×C4—5421no—
301C2×C4—5421no—
302C2×C4—5422no—
303C2×C4—10842no—
304C2×C4D4×D27111yescharacteristic
305C2×C4C2×C22⋊D27112yes—
306C2×C4#1686465312yes—
307C2×C4C23×D27111yescharacteristic
308C8—5422no—
309C8C2×C22⋊D27114yescomplemented
310C22—21no—
311C22—21no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—84no—
319C22—2712no—
320C22—5421no—
321C22—5421no—
322C22—5421no—
323C22—5421no—
324C22—5421no—
325C22—10844no—
326C22#2257613112yes—
327C22C2×D4×D27111yescharacteristic
328C4—41no—
329C4—5421no—
330C4—5421no—
331C4—10842no—
332C4—10841no—
333C4—10841no—
334C4#2257613112yes—
335C4C22×C22⋊D27111yescharacteristic
336C4C2×#1686465311yescharacteristic
337C2—22no—
338C2—82no—
339C2—81no—
340C2—5422no—
341C2—10841no—
342C2C2×#2257613111yescharacteristic
343C2AGL(1,ℤ/8)×D27111yescharacteristic
344C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×AGL(1,ℤ/8)×C271C2
socleC22×C271C22×D4
hypercenterC2×AGL(1,ℤ/8)D271
hypocenterC271#3055
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×M16×D271C2semidirect
2. #31774404C2semidirect
3. C2×#22576131C2semidirect
4. #31774147C2semidirect
5. #31774140C2semidirect
6. #31774130C2semidirect
7. #31774123C2semidirect
8. #31773939C2semidirect
9. #31773902C2semidirect
10. #31773898C2semidirect
11. C2×AGL(1,ℤ/8)×C271C2semidirect
15. C2×#16864637C22semidirect
17. C2×C8×D271C22semidirect
19. #22576136C22semidirect
20. #22576136C22semidirect
25. #22576131C22semidirect
26. #22576131C22semidirect
33. #22576094C22semidirect
34. #22576092C22semidirect
35. #22576084C22semidirect
36. #22576081C22semidirect
37. #22576081C22semidirect
38. #22576081C22semidirect
42. C2×SD16×C271C22semidirect
43. C2×D8×C271C22semidirect
91. M16×C271D4semidirect
136. C8×C271C2×D4semidirect
154. C2×D271AGL(1,ℤ/8)semidirect
172. C271#3055semidirect
174. C2×AGL(1,ℤ/8)D271semidirect
190. C2×SD16C2×D271semidirect
191. C2×D8C2×D271semidirect
248. M16C22⋊D271semidirect
309. C8C2×C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2712
#317744042
C2×#225761312
#317741472
#317741402
#317741302
#317741232
#317739392
#317739022
#317738982
C2×AGL(1,ℤ/8)×C2712
#3055271