Finite Group Databasebeta

← #46845981

Subgroups — #46845981

Order 34688.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#22576186C211yesmaximal, characteristic, complemented
2C2×M16×D271C211yesmaximal, characteristic, complemented
3#31774409C211yesmaximal, characteristic, complemented
4C2×#22576131C211yesmaximal, characteristic, complemented
5#31774149C212yesmaximal, complemented
6#31774142C212yesmaximal, complemented
7#31774132C212yesmaximal, complemented
8#31774125C212yesmaximal, complemented
9#31773940C211yesmaximal, characteristic, complemented
10#31773899C211yesmaximal, characteristic, complemented
11#418×C271C211yesmaximal, characteristic, complemented
12C2×#16864654C2211yescharacteristic
13C22×C22⋊D271—21no—
14C22×C4×D271C2211yescharacteristic
15#22576186—24no—
16C2×#16864642C2212yescomplemented
17C2×#16864639C2212yescomplemented
18C2×#16864638C2211yescharacteristic
19C2×#16864637C2212yescomplemented
20M16×D271—24no—
21C2×C8×D271C2212yescomplemented
22#22576144—21no—
23#22576140C2211yescharacteristic, complemented
24#22576140C2211yescharacteristic, complemented
25#22576135C2211yescharacteristic, complemented
26#22576135C2211yescharacteristic, complemented
27#22576131—21no—
28#22576131—21no—
29#22576131C2212yes—
30#22576131C2212yes—
31#22576130—21no—
32C4×D1084—21no—
33C4×D1084—21no—
34C2×C4⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271—21no—
36#22576093C2212yescomplemented
37#22576091C2212yescomplemented
38#22576085C2212yescomplemented
39#22576082C2212yescomplemented
40C22×D4×C271C2211yescharacteristic
41C2×M16×C271C2211yescharacteristic
42#117×C271C2211yescharacteristic
43#102×C271C2212yescomplemented
44#102×C271C2212yescomplemented
45C23×D271D411yescharacteristic
46#16864654—22no—
47#16864654—22no—
48C2×Dic542C2311yescharacteristic
49C2×D1084—21no—
50C2×D1084—21no—
51C2×D1084C2311yescharacteristic
52C2×C22⋊D271—22no—
53C2×C22⋊D271—22no—
54C2×C22⋊D271—21no—
55C2×C22⋊D271—44no—
56C22×Dic271—21no—
57C22×Dic271D411yescharacteristic
58C2×C4×D271—22no—
59C2×C4×D271—21no—
60C2×C4×D271—22no—
61C2×C4×D271D411yescharacteristic
62C2×C4×D271D411yescharacteristic
63C2×C4×D271C2311yescharacteristic
64C2×C4×D271C2311yescharacteristic
65#16864642—24no—
66#16864639—24no—
67#16864638D414yescomplemented
68#16864637—24no—
69C2×C271⋊C8C2312yes—
70C8×D271—24no—
71C4⋊Dic271—21no—
72C4⋊Dic271—21no—
73C4⋊Dic271—21no—
74C4⋊Dic271C2312yes—
75C22⋊Dic271—22no—
76C22⋊Dic271—22no—
77(C2×D271)⋊C4—21no—
78(C2×D271)⋊C4—21no—
79(C2×D271)⋊C4—21no—
80(C2×D271)⋊C4—21no—
81C24×C271—21no—
82C2×D4×C271—22no—
83C2×D4×C271—21no—
84C2×D4×C271—22no—
85C2×D4×C271C2312yes—
86C22×C4×C271C2311yescharacteristic
87M16×C271—22no—
88C2×C8×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—41no—
97C22×D271C2×D411yescharacteristic
98C22×D271C2×D411yescharacteristic
99Dic542—21no—
100Dic542—21no—
101D1084—21no—
102D1084—21no—
103D1084—42no—
104D1084—42no—
105C22⋊D271—44no—
106C22⋊D271—42no—
107C22⋊D271—44no—
108C2×Dic271—22no—
109C2×Dic271—21no—
110C2×Dic271—22no—
111C2×Dic271—42no—
112C2×Dic271C2×D411yescharacteristic
113C2×Dic271C2×D411yescharacteristic
114C4×D271—22no—
115C4×D271—22no—
116C4×D271—21no—
117C4×D271—21no—
118C4×D271—21no—
119C4×D271—21no—
120C4×D271—42no—
121C271⋊C8C2×D414yescomplemented
122C23×C271—22no—
123C23×C271—42no—
124C23×C271—42no—
125C23×C271—42no—
126C23×C271C2×D411yescharacteristic
127D4×C271—22no—
128D4×C271—22no—
129D4×C271—42no—
130C2×C4×C271—21no—
131C2×C4×C271—21no—
132C2×C4×C271C2×D411yescharacteristic
133C2×C4×C271C2×D412yes—
134C2×C4×C271#4112yes—
135C2×C4×C271C2411yescharacteristic
136C8×C271—22no—
137C22×C271—21no—
138C22×C271—21no—
139C22×C271—42no—
140C22×C271—42no—
141C22×C271—42no—
142C22×C271—84no—
143C22×C271#12112yes—
144C22×C271C22×D411yescharacteristic
145C2×D271—21no—
146C2×D271—21no—
147C2×D271—21no—
148C2×D271—21no—
149C2×D271—21no—
150C2×D271—44no—
151C2×D271—41no—
152C2×D271—41no—
153C2×D271—41no—
154C2×D271AGL(1,ℤ/8)12yescomplemented
155C4×C271—41no—
156C4×C271—41no—
157C4×C271#12112yes—
158C4×C271C22×D411yescharacteristic
159C4×C271C2×#4111yescharacteristic
160Dic271—21no—
161Dic271—21no—
162Dic271—41no—
163Dic271—42no—
164C2×C271—22no—
165C2×C271—82no—
166C2×C271C2×#12111yescharacteristic
167C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C271#57511yescharacteristic
169D271—22no—
170D271—41no—
171D271—81no—
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#428—2711no—
183#418D27111yescharacteristic, complemented
184#417—2711no—
185C2×#41—2711no—
186C22×D4—5421no—
187C22×D4C2×D27111yescharacteristic
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⋊C4—2711no—
211C2×C22⋊C4—5421no—
212#107—2712no—
213#106—2712no—
214#103—2712no—
215#102—2712no—
216#102C2×D27112yescomplemented
217#102C2×D27112yescomplemented
218C24—21no—
219C24—2711no—
220#41—5422no—
221#41—5422no—
222C2×Q8—2711no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—2711no—
227C2×D4—5421no—
228C2×D4—5421no—
229C2×D4—5422no—
230C2×D4—5422no—
231C2×D4—5421no—
232C2×D4—10844no—
233C2×D4C22×D27112yes—
234C22×C4—2711no—
235C22×C4—2711no—
236C22×C4—2711no—
237C22×C4—2711no—
238C22×C4—2711no—
239C22×C4—5422no—
240C22×C4—5422no—
241C22×C4—5421no—
242C22×C4—5421no—
243C22×C4C22×D27111yescharacteristic
244SD16—5424no—
245D8—5424no—
246M16—22no—
247M16—2714no—
248M16—5424no—
249C2×C8—2712no—
250C2×C8—5424no—
251C2×C8C22×D27112yes—
252C4⋊C4—2712no—
253C4⋊C4—5421no—
254C4⋊C4—5421no—
255C4⋊C4—5421no—
256C4⋊C4C22×D27111yescharacteristic
257C4⋊C4C22×D27111yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—5421no—
260C22⋊C4—5421no—
261C22⋊C4—5421no—
262C22⋊C4—5421no—
263C22⋊C4—5422no—
264C22⋊C4—5422no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—2711no—
271C23—2711no—
272C23—5422no—
273C23—5422no—
274C23—5421no—
275C23—10841no—
276C23D4×D27111yescharacteristic
277Q8—5421no—
278Q8—5421no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—5421no—
283D4—5421no—
284D4—10842no—
285D4—10842no—
286D4—10844no—
287D4—10844no—
288D4—10842no—
289C2×C4—21no—
290C2×C4—21no—
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—5422no—
301C2×C4—5421no—
302C2×C4—10842no—
303C2×C4—10842no—
304C2×C4D4×D27111yescharacteristic
305C2×C4D4×D27112yes—
306C2×C4#1686465412yes—
307C2×C4C23×D27111yescharacteristic
308C8—22no—
309C8—2714no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—2712no—
317C22—5421no—
318C22—5421no—
319C22—5421no—
320C22—5421no—
321C22—5421no—
322C22—10844no—
323C22—10841no—
324C22—10841no—
325C22—10841no—
326C22#2257613312yes—
327C22C2×D4×D27111yescharacteristic
328C4—41no—
329C4—41no—
330C4—5421no—
331C4—5421no—
332C4—10841no—
333C4—10842no—
334C4#2257613312yes—
335C4C2×D4×D27111yescharacteristic
336C4C2×#1686465411yescharacteristic
337C2—22no—
338C2—82no—
339C2—5422no—
340C2—10841no—
341C2—21681no—
342C2C2×#2257613311yescharacteristic
343C2AGL(1,ℤ/8)×D27111yescharacteristic
344C2#3177469311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#418×C271C2
socleC22×C271C22×D4
hypercenter#418D271
hypocenterC271#3055
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. C2×M16×D271C2semidirect
3. #31774409C2semidirect
4. C2×#22576131C2semidirect
5. #31774149C2semidirect
6. #31774142C2semidirect
7. #31774132C2semidirect
8. #31774125C2semidirect
9. #31773940C2semidirect
10. #31773899C2semidirect
11. #418×C271C2semidirect
16. C2×#16864642C22semidirect
17. C2×#16864639C22semidirect
19. C2×#16864637C22semidirect
21. C2×C8×D271C22semidirect
23. #22576140C22semidirect
24. #22576140C22semidirect
25. #22576135C22semidirect
26. #22576135C22semidirect
36. #22576093C22semidirect
37. #22576091C22semidirect
38. #22576085C22semidirect
39. #22576082C22semidirect
43. #102×C271C22semidirect
44. #102×C271C22semidirect
67. #16864638D4semidirect
121. C271⋊C8C2×D4semidirect
154. C2×D271AGL(1,ℤ/8)semidirect
172. C271#3055semidirect
183. #418D271semidirect
216. #102C2×D271semidirect
217. #102C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
C2×M16×D2712
#317744092
C2×#225761312
#317741492
#317741422
#317741322
#317741252
#317739402
#317738992
#418×C2712
#3055271