Finite Group Databasebeta

← #46820516

Subgroups — #46820516

Order 34432.

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

normal subgroups only

This group has 496 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×#22575486C211yesmaximal, characteristic, complemented
2C2×M16×D269C211yesmaximal, characteristic, complemented
3#31771117C211yesmaximal, characteristic
4C2×#22575433C211yesmaximal, characteristic, complemented
5#31770864C212yesmaximal, complemented
6#31770852C212yesmaximal, complemented
7#31770847C212yesmaximal, complemented
8#31770835C212yesmaximal, complemented
9#31770613C211yesmaximal, characteristic, complemented
10#31770604C211yesmaximal, characteristic, complemented
11#428×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×Dic538C2211yescharacteristic
14C22×C4×D269C2211yescharacteristic
15#22575486—24no—
16C2×Dic1076C2212yescomplemented
17C2×#16864391C2212yescomplemented
18C2×#16864386C2211yescharacteristic
19C2×#16864385C2212yes—
20M16×D269—24no—
21C2×C8×D269C2212yes—
22#22575437—21no—
23#22575433—21no—
24#22575433—21no—
25#22575433C2212yes—
26#22575433C2211yescharacteristic
27#22575433C2212yes—
28#22575433C2211yescharacteristic
29#22575428C2211yescharacteristic
30#22575428C2211yescharacteristic
31#22575423—21no—
32C4×D1076—21no—
33C4×D1076—21no—
34C2×C4⋊Dic269—21no—
35C2×#16864379—21no—
36#22575376C2212yescomplemented
37#22575376C2212yescomplemented
38#22575376C2212yescomplemented
39#22575373C2212yescomplemented
40C2×M16×C269C2211yescharacteristic
41#117×C269C2211yescharacteristic
42C2×C4⋊C4×C269C2211yescharacteristic
43#107×C269C2212yescomplemented
44#106×C269C2212yescomplemented
45C23×D269D411yescharacteristic
46#16864402—22no—
47#16864402—22no—
48C2×Dic538—21no—
49C2×Dic538—22no—
50C2×Dic538—22no—
51C2×Dic538C2312yes—
52C2×Dic538C2311yescharacteristic
53C2×D1076—21no—
54C2×D1076—21no—
55C2×D1076C2311yescharacteristic
56C2×C22⋊D269—21no—
57C22×Dic269—21no—
58C22×Dic269D411yescharacteristic
59C2×C4×D269—22no—
60C2×C4×D269—21no—
61C2×C4×D269—22no—
62C2×C4×D269D411yescharacteristic
63C2×C4×D269D411yescharacteristic
64C2×C4×D269C2311yescharacteristic
65C2×C4×D269C2311yescharacteristic
66Dic1076—24no—
67#16864391—24no—
68#16864386—22no—
69#16864385—24no—
70C2×#14524130C2312yes—
71C8×D269—24no—
72C4⋊Dic269—22no—
73C4⋊Dic269—22no—
74C4⋊Dic269—21no—
75C4⋊Dic269—21no—
76#16864379—22no—
77#16864379—22no—
78#16864379—21no—
79#16864379—21no—
80#16864379—21no—
81#16864379—21no—
82C22×C4×C269—21no—
83C22×C4×C269C2311yescharacteristic
84M16×C269D414yescomplemented
85C2×C8×C269C2312yes—
86C4⋊C4×C269—21no—
87C4⋊C4×C269C2312yes—
88C4⋊C4×C269C2311yescharacteristic
89C4⋊C4×C269C2311yescharacteristic
90C22⋊C4×C269—21no—
91C42×C269—21no—
92C22×D269—22no—
93C22×D269—22no—
94C22×D269—21no—
95C22×D269—41no—
96C22×D269C2×D411yescharacteristic
97C22×D269C2×D411yescharacteristic
98Dic538—22no—
99Dic538—21no—
100Dic538—22no—
101Dic538—21no—
102Dic538—42no—
103D1076—21no—
104D1076—21no—
105D1076—42no—
106D1076—42no—
107C22⋊D269—42no—
108C2×Dic269—22no—
109C2×Dic269—21no—
110C2×Dic269—22no—
111C2×Dic269—42no—
112C2×Dic269C2×D411yescharacteristic
113C2×Dic269C2×D411yescharacteristic
114C4×D269—22no—
115C4×D269—22no—
116C4×D269—21no—
117C4×D269—21no—
118C4×D269—21no—
119C4×D269—21no—
120C4×D269—42no—
121#14524130—22no—
122C23×C269C2×D411yescharacteristic
123C2×C4×C269—21no—
124C2×C4×C269—21no—
125C2×C4×C269—22no—
126C2×C4×C269—42no—
127C2×C4×C269C2×D411yescharacteristic
128C2×C4×C269C2×D412yes—
129C2×C4×C269#4112yes—
130C2×C4×C269C2411yescharacteristic
131C8×C269C2×D414yes—
132C22×C269—21no—
133C22×C269—21no—
134C22×C269#12112yes—
135C22×C269C22×D411yescharacteristic
136C2×D269—21no—
137C2×D269—21no—
138C2×D269—21no—
139C2×D269—21no—
140C2×D269—21no—
141C2×D269—44no—
142C2×D269—41no—
143C2×D269—41no—
144C2×D269—41no—
145C2×D269#13712yescomplemented
146C4×C269—41no—
147C4×C269—42no—
148C4×C269—41no—
149C4×C269#12112yes—
150C4×C269C22×D411yescharacteristic
151C4×C269C2×#4111yescharacteristic
152Dic269—21no—
153Dic269—21no—
154Dic269—41no—
155Dic269—42no—
156C2×C269—22no—
157C2×C269C2×#12111yescharacteristic
158C2×C269C2×#13711yescharacteristic
159C2×C269#57511yescharacteristic
160D269—22no—
161D269—41no—
162D269—81no—
163C269#305611yescharacteristic, complemented, Sylow
164#3056—2691nomaximal, Sylow
165C2×#137—2691no—
166C22×M16—2691no—
167#525—2691no—
168C2×C22⋊Q8—2691no—
169#470—2692no—
170#468—2692no—
171#467—2692no—
172#465—2692no—
173#428D26911yescharacteristic, complemented
174#419—2691no—
175#417—2691no—
176C2×#41—2691no—
177C22×Q8—2691no—
178C23×C4—2691no—
179#137—5384no—
180C2×Q16—2692no—
181C2×SD16—2692no—
182C2×M16—2691no—
183C2×M16—2692no—
184C2×M16—5384no—
185C2×M16C2×D26911yescharacteristic
186C22×C8—2692no—
187#123—5381no—
188C22⋊Q8—2692no—
189C22⋊Q8—2692no—
190C22⋊Q8—2691no—
191C22⋊Q8—2691no—
192C22⋊Q8—5381no—
193C22⋊Q8—5381no—
194#121—2691no—
195#121—2691no—
196#120—5381no—
197C4×D4—5381no—
198C4×D4—5381no—
199#117C2×D26911yescharacteristic
200C2×C4⋊C4—5381no—
201C2×C4⋊C4C2×D26911yescharacteristic
202C2×C22⋊C4—5381no—
203#107C2×D26912yescomplemented
204#106C2×D26912yescomplemented
205#103—2692no—
206#103—2692no—
207#103—2692no—
208#102—2692no—
209C24—2691no—
210#41—5382no—
211#41—5382no—
212C2×Q8—2692no—
213C2×Q8—2691no—
214C2×Q8—5381no—
215C2×Q8—5382no—
216C2×Q8—5382no—
217C2×D4—2691no—
218C2×D4—5381no—
219C2×D4—5381no—
220C2×D4—5381no—
221C22×C4—21no—
222C22×C4—2691no—
223C22×C4—2691no—
224C22×C4—2691no—
225C22×C4—2691no—
226C22×C4—2691no—
227C22×C4—5382no—
228C22×C4—5382no—
229C22×C4—5381no—
230C22×C4—5381no—
231C22×C4C22×D26911yescharacteristic
232Q16—5384no—
233SD16—5384no—
234M16—5382no—
235M16—5384no—
236M16D107614yescomplemented
237C2×C8—2692no—
238C2×C8—5384no—
239C2×C8C22×D26912yes—
240C4⋊C4—21no—
241C4⋊C4—5382no—
242C4⋊C4—5382no—
243C4⋊C4—5381no—
244C4⋊C4—5381no—
245C4⋊C4C22×D26912yes—
246C4⋊C4C22×D26911yescharacteristic
247C4⋊C4C22×D26911yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—5381no—
250C22⋊C4—5381no—
251C22⋊C4—5382no—
252C22⋊C4—5382no—
253C22⋊C4—5381no—
254C22⋊C4—5381no—
255C42—21no—
256C23—2691no—
257C23—2691no—
258C23—5382no—
259C23—5382no—
260C23—5381no—
261C23—10761no—
262C23D4×D26911yescharacteristic
263Q8—5382no—
264Q8—5381no—
265Q8—5382no—
266Q8—5381no—
267Q8—10762no—
268D4—5381no—
269D4—5381no—
270D4—10762no—
271D4—10762no—
272D4—10762no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—2691no—
278C2×C4—2691no—
279C2×C4—5382no—
280C2×C4—5382no—
281C2×C4—5382no—
282C2×C4—5381no—
283C2×C4—5381no—
284C2×C4—5381no—
285C2×C4—5381no—
286C2×C4—5381no—
287C2×C4—5382no—
288C2×C4—10762no—
289C2×C4—10762no—
290C2×C4D4×D26911yescharacteristic
291C2×C4C2×D107612yes—
292C2×C4#1686440312yes—
293C2×C4C23×D26911yescharacteristic
294C8—5382no—
295C8C2×D107614yes—
296C22—21no—
297C22—21no—
298C22—2692no—
299C22—5381no—
300C22—5381no—
301C22—5381no—
302C22—5381no—
303C22—5381no—
304C22—10764no—
305C22—10761no—
306C22—10761no—
307C22—10761no—
308C22#2257542812yes—
309C22C2×D4×D26911yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—5381no—
314C4—5381no—
315C4—10761no—
316C4—10762no—
317C4#2257542812yes—
318C4C22×D107611yescharacteristic
319C4C2×#1686440311yescharacteristic
320C2—22no—
321C2—5382no—
322C2—10761no—
323C2—21521no—
324C2C2×#2257542811yescharacteristic
325C2#137×D26911yescharacteristic
326C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
prime radical#31771117C2
hypercenter#428D269
hypocenterC269#3056
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22575486C2semidirect
2. C2×M16×D269C2semidirect
4. C2×#22575433C2semidirect
5. #31770864C2semidirect
6. #31770852C2semidirect
7. #31770847C2semidirect
8. #31770835C2semidirect
9. #31770613C2semidirect
10. #31770604C2semidirect
11. #428×C269C2semidirect
16. C2×Dic1076C22semidirect
17. C2×#16864391C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575376C22semidirect
39. #22575373C22semidirect
43. #107×C269C22semidirect
44. #106×C269C22semidirect
84. M16×C269D4semidirect
145. C2×D269#137semidirect
163. C269#3056semidirect
173. #428D269semidirect
203. #107C2×D269semidirect
204. #106C2×D269semidirect
236. M16D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754862
C2×M16×D2692
#317711172
C2×#225754332
#317708642
#317708522
#317708472
#317708352
#317706132
#317706042
#428×C2692
#3056269