Finite Group Databasebeta

← #46845984

Subgroups — #46845984

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×#22576188C211yesmaximal, characteristic, complemented
2C2×M16×D271C211yesmaximal, characteristic, complemented
3#31774409C211yesmaximal, characteristic, complemented
4C2×#22576135C211yesmaximal, characteristic, complemented
5#31774151C212yesmaximal, complemented
6#31774144C212yesmaximal, complemented
7#31774134C212yesmaximal, complemented
8#31774127C212yesmaximal, complemented
9#31773901C211yesmaximal, characteristic, complemented
10#31773896C211yesmaximal, characteristic, complemented
11#428×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×D1084—21no—
14C22×D1084C2211yescharacteristic
15C22×C4×D271C2211yescharacteristic
16#22576188—24no—
17C2×#16864643C2212yescomplemented
18C2×D2168C2212yescomplemented
19C2×#16864638C2211yescharacteristic
20C2×#16864637C2212yescomplemented
21M16×D271—24no—
22C2×C8×D271C2212yescomplemented
23#22576144—21no—
24#22576140C2211yescharacteristic, complemented
25#22576140C2211yescharacteristic, complemented
26#22576135—21no—
27#22576135—21no—
28#22576135C2212yes—
29#22576135C2212yes—
30#22576135C2211yescharacteristic, complemented
31#22576135C2211yescharacteristic, complemented
32#22576130—21no—
33C4×D1084—21no—
34C4×D1084—21no—
35C2×(C2×D271)⋊C4—21no—
36#22576083C2212yescomplemented
37#22576080C2212yescomplemented
38#22576080C2212yescomplemented
39#22576080C2212yescomplemented
40C2×M16×C271C2211yescharacteristic
41#117×C271C2211yescharacteristic
42C2×C4⋊C4×C271C2211yescharacteristic
43#107×C271C2212yescomplemented
44#106×C271C2212yescomplemented
45C23×D271—21no—
46C23×D271D411yescharacteristic
47#16864654—22no—
48#16864654—22no—
49C2×Dic542C2311yescharacteristic
50C2×D1084—21no—
51C2×D1084—21no—
52C2×D1084—22no—
53C2×D1084—22no—
54C2×D1084—22no—
55C2×D1084—21no—
56C2×D1084—22no—
57C2×D1084—44no—
58C2×D1084C2312yes—
59C2×D1084C2311yescharacteristic
60C2×C22⋊D271—21no—
61C22×Dic271D411yescharacteristic
62C2×C4×D271—22no—
63C2×C4×D271—21no—
64C2×C4×D271—22no—
65C2×C4×D271D411yescharacteristic
66C2×C4×D271D411yescharacteristic
67C2×C4×D271C2311yescharacteristic
68C2×C4×D271C2311yescharacteristic
69#16864643—24no—
70D2168—24no—
71#16864638—22no—
72#16864637—24no—
73C2×C271⋊C8C2312yes—
74C8×D271—24no—
75C4⋊Dic271—21no—
76C4⋊Dic271—21no—
77(C2×D271)⋊C4—21no—
78(C2×D271)⋊C4—21no—
79(C2×D271)⋊C4—22no—
80(C2×D271)⋊C4—22no—
81(C2×D271)⋊C4—21no—
82(C2×D271)⋊C4—21no—
83C22×C4×C271—21no—
84C22×C4×C271C2311yescharacteristic
85M16×C271D414yescomplemented
86C2×C8×C271C2312yes—
87C4⋊C4×C271—21no—
88C4⋊C4×C271C2312yes—
89C4⋊C4×C271C2311yescharacteristic
90C4⋊C4×C271C2311yescharacteristic
91C22⋊C4×C271—21no—
92C42×C271—21no—
93C22×D271—22no—
94C22×D271—22no—
95C22×D271—21no—
96C22×D271—22no—
97C22×D271—41no—
98C22×D271—42no—
99C22×D271—42no—
100C22×D271—42no—
101C22×D271C2×D411yescharacteristic
102C22×D271C2×D411yescharacteristic
103Dic542—21no—
104Dic542—21no—
105D1084—21no—
106D1084—22no—
107D1084—22no—
108D1084—21no—
109D1084—42no—
110D1084—44no—
111D1084—42no—
112D1084—44no—
113D1084—42no—
114C22⋊D271—42no—
115C2×Dic271—22no—
116C2×Dic271—21no—
117C2×Dic271C2×D411yescharacteristic
118C2×Dic271C2×D411yescharacteristic
119C4×D271—22no—
120C4×D271—22no—
121C4×D271—21no—
122C4×D271—21no—
123C4×D271—21no—
124C4×D271—21no—
125C4×D271—42no—
126C271⋊C8—22no—
127C23×C271C2×D411yescharacteristic
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271—22no—
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#12112yes—
140C22×C271C22×D411yescharacteristic
141C2×D271—21no—
142C2×D271—21no—
143C2×D271—21no—
144C2×D271—21no—
145C2×D271—21no—
146C2×D271—44no—
147C2×D271—41no—
148C2×D271—42no—
149C2×D271—42no—
150C2×D271—41no—
151C2×D271—41no—
152C2×D271—42no—
153C2×D271—84no—
154C2×D271AGL(1,ℤ/8)12yescomplemented
155C4×C271—42no—
156C4×C271—41no—
157C4×C271—41no—
158C4×C271#12112yes—
159C4×C271C22×D411yescharacteristic
160C4×C271C2×#4111yescharacteristic
161Dic271—21no—
162Dic271—21no—
163Dic271—41no—
164C2×C271—22no—
165C2×C271C2×#12111yescharacteristic
166C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C271#57511yescharacteristic
168D271—22no—
169D271—41no—
170D271—81no—
171D271—82no—
172C271#305511yescharacteristic, complemented, Sylow
173#3055—2711nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2711no—
175C22×M16—2711no—
176#525—2711no—
177C2×#121—2711no—
178#469—2712no—
179#468—2712no—
180#466—2712no—
181#465—2712no—
182#428D27111yescharacteristic, complemented
183#418—2711no—
184#417—2711no—
185C2×#41—2711no—
186C22×D4—2711no—
187C22×D4—5421no—
188C23×C4—2711no—
189AGL(1,ℤ/8)—5424no—
190C2×SD16—2712no—
191C2×D8—2712no—
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#117C2×D27111yescharacteristic
210C2×C4⋊C4C2×D27111yescharacteristic
211C2×C22⋊C4—5421no—
212#107C2×D27112yescomplemented
213#106C2×D27112yescomplemented
214#103—2712no—
215#102—2712no—
216#102—2712no—
217#102—2712no—
218C24—2711no—
219C24—5421no—
220#41—5422no—
221#41—5422no—
222C2×Q8—2711no—
223C2×D4—2711no—
224C2×D4—2712no—
225C2×D4—5421no—
226C2×D4—5421no—
227C2×D4—5422no—
228C2×D4—5422no—
229C2×D4—5421no—
230C2×D4—5421no—
231C2×D4—5422no—
232C2×D4—5422no—
233C2×D4—10844no—
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—5424no—
245D8—5424no—
246M16—5422no—
247M16—5424no—
248M16D108414yescomplemented
249C2×C8—2712no—
250C2×C8—5424no—
251C2×C8C22×D27112yes—
252C4⋊C4—21no—
253C4⋊C4—5421no—
254C4⋊C4—5421no—
255C4⋊C4C22×D27111yescharacteristic
256C4⋊C4C22×D27111yescharacteristic
257C4⋊C4C22×D27112yes—
258C22⋊C4—21no—
259C22⋊C4—5422no—
260C22⋊C4—5422no—
261C22⋊C4—5421no—
262C22⋊C4—5421no—
263C22⋊C4—5421no—
264C22⋊C4—5421no—
265C42—21no—
266C23—2711no—
267C23—2711no—
268C23—5422no—
269C23—5422no—
270C23—5421no—
271C23—5422no—
272C23—10841no—
273C23—10842no—
274C23—10842no—
275C23—10842no—
276C23D4×D27111yescharacteristic
277Q8—5421no—
278Q8—5421no—
279D4—5421no—
280D4—5422no—
281D4—5422no—
282D4—5421no—
283D4—10844no—
284D4—10842no—
285D4—10844no—
286D4—10842no—
287D4—10842no—
288D4—10842no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—2711no—
294C2×C4—2711no—
295C2×C4—5422no—
296C2×C4—5422no—
297C2×C4—5422no—
298C2×C4—5421no—
299C2×C4—5421no—
300C2×C4—5421no—
301C2×C4—5421no—
302C2×C4—5421no—
303C2×C4—10842no—
304C2×C4D4×D27111yescharacteristic
305C2×C4C2×D108412yes—
306C2×C4#1686465512yes—
307C2×C4C23×D27111yescharacteristic
308C8—5422no—
309C8C2×D108414yescomplemented
310C22—21no—
311C22—21no—
312C22—2712no—
313C22—5421no—
314C22—5421no—
315C22—5421no—
316C22—5421no—
317C22—5421no—
318C22—10844no—
319C22—10841no—
320C22—10842no—
321C22—10842no—
322C22—10841no—
323C22—10841no—
324C22—10842no—
325C22—21684no—
326C22#2257613512yes—
327C22C2×D4×D27111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—5421no—
332C4—5421no—
333C4—10841no—
334C4#2257613512yes—
335C4C22×D108411yescharacteristic
336C4C2×#1686465511yescharacteristic
337C2—22no—
338C2—5422no—
339C2—10841no—
340C2—21682no—
341C2—21681no—
342C2C2×#2257613511yescharacteristic
343C2AGL(1,ℤ/8)×D27111yescharacteristic
344C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#428×C271C2
socleC22×C271C22×D4
hypercenter#428D271
hypocenterC271#3055
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22576188C2semidirect
2. C2×M16×D271C2semidirect
3. #31774409C2semidirect
4. C2×#22576135C2semidirect
5. #31774151C2semidirect
6. #31774144C2semidirect
7. #31774134C2semidirect
8. #31774127C2semidirect
9. #31773901C2semidirect
10. #31773896C2semidirect
11. #428×C271C2semidirect
17. C2×#16864643C22semidirect
18. C2×D2168C22semidirect
20. C2×#16864637C22semidirect
22. C2×C8×D271C22semidirect
24. #22576140C22semidirect
25. #22576140C22semidirect
30. #22576135C22semidirect
31. #22576135C22semidirect
36. #22576083C22semidirect
37. #22576080C22semidirect
38. #22576080C22semidirect
39. #22576080C22semidirect
43. #107×C271C22semidirect
44. #106×C271C22semidirect
85. M16×C271D4semidirect
136. C8×C271C2×D4semidirect
154. C2×D271AGL(1,ℤ/8)semidirect
172. C271#3055semidirect
182. #428D271semidirect
212. #107C2×D271semidirect
213. #106C2×D271semidirect
248. M16D1084semidirect
309. C8C2×D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761882
C2×M16×D2712
#317744092
C2×#225761352
#317741512
#317741442
#317741342
#317741272
#317739012
#317738962
#428×C2712
#3055271