Finite Group Databasebeta

← #46820518

Subgroups — #46820518

Order 34432.

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

normal subgroups only

This group has 496 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×#22575483C211yesmaximal, characteristic, complemented
2C2×M16×D269C211yesmaximal, characteristic, complemented
3#31771112C211yesmaximal, characteristic
4C2×#22575433C211yesmaximal, characteristic, complemented
5#31770862C212yesmaximal, complemented
6#31770850C212yesmaximal, complemented
7#31770845C212yesmaximal, complemented
8#31770833C212yesmaximal, complemented
9#31770645C211yesmaximal, characteristic, complemented
10#31770615C211yesmaximal, characteristic, complemented
11#417×C269C211yesmaximal, characteristic, complemented
12C22×Dic538C2211yescharacteristic
13C22×C4×D269C2211yescharacteristic
14#22575483—24no—
15C2×#16864392C2212yescomplemented
16C2×#16864390C2212yescomplemented
17C2×#16864386C2211yescharacteristic
18C2×#16864385C2212yes—
19M16×D269—24no—
20C2×C8×D269C2212yes—
21#22575434—21no—
22#22575433—21no—
23#22575433—21no—
24#22575433C2212yes—
25#22575433C2212yes—
26#22575429C2211yescharacteristic
27#22575429C2211yescharacteristic
28#22575424C2211yescharacteristic
29#22575424C2211yescharacteristic
30#22575421—21no—
31C4×C22⋊D269—21no—
32C4×C22⋊D269—21no—
33#22575412C2211yescharacteristic
34C2×C4⋊Dic269—21no—
35C2×#16864379—21no—
36#22575386C2212yescomplemented
37#22575384C2212yescomplemented
38#22575378C2212yescomplemented
39#22575378C2212yescomplemented
40C2×#41×C269C2211yescharacteristic
41C2×M16×C269C2211yescharacteristic
42C2×C4⋊C4×C269C2211yescharacteristic
43#103×C269C2212yescomplemented
44#102×C269C2212yescomplemented
45C23×D269D411yescharacteristic
46C2×Dic538—21no—
47C2×Dic538—22no—
48C2×Dic538—22no—
49C2×Dic538C2312yes—
50C2×C22⋊D269—21no—
51C2×C22⋊D269—21no—
52C22×Dic269—21no—
53C22×Dic269D411yescharacteristic
54C2×C4×D269—22no—
55C2×C4×D269—22no—
56C2×C4×D269D411yescharacteristic
57C2×C4×D269D411yescharacteristic
58C2×C4×D269C2311yescharacteristic
59C2×C4×D269C2311yescharacteristic
60#16864392—24no—
61#16864390—24no—
62#16864386D414yescomplemented
63#16864385—24no—
64C2×#14524130C2312yes—
65C8×D269—24no—
66C4⋊Dic269—22no—
67C4⋊Dic269—22no—
68C4⋊Dic269C2311yescharacteristic
69C4⋊Dic269C2311yescharacteristic
70#16864381—21no—
71#16864381—21no—
72C22⋊Dic269—21no—
73C22⋊Dic269—21no—
74C22⋊Dic269—21no—
75#16864379—22no—
76#16864379—22no—
77#16864379—21no—
78#16864379—21no—
79C4×Dic269—21no—
80#41×C269—22no—
81#41×C269—22no—
82C2×Q8×C269C2311yescharacteristic
83C2×D4×C269—21no—
84C2×D4×C269C2311yescharacteristic
85C22×C4×C269—21no—
86C22×C4×C269—21no—
87C22×C4×C269C2311yescharacteristic
88M16×C269—22no—
89C2×C8×C269C2312yes—
90C4⋊C4×C269—21no—
91C4⋊C4×C269C2312yes—
92C22×D269—22no—
93C22×D269—22no—
94C22×D269—41no—
95C22×D269C2×D411yescharacteristic
96C22×D269C2×D411yescharacteristic
97Dic538—22no—
98Dic538—22no—
99Dic538—42no—
100C22⋊D269—42no—
101C22⋊D269—42no—
102C2×Dic269—22no—
103C2×Dic269—21no—
104C2×Dic269—21no—
105C2×Dic269—22no—
106C2×Dic269—42no—
107C2×Dic269C2×D411yescharacteristic
108C2×Dic269C2×D411yescharacteristic
109C4×D269—22no—
110C4×D269—22no—
111C4×D269—21no—
112C4×D269—21no—
113C4×D269—21no—
114C4×D269—21no—
115#14524130C2×D414yes—
116C23×C269—21no—
117C23×C269C2×D411yescharacteristic
118Q8×C269—21no—
119Q8×C269—21no—
120D4×C269—21no—
121D4×C269—21no—
122D4×C269—42no—
123C2×C4×C269—22no—
124C2×C4×C269—21no—
125C2×C4×C269—42no—
126C2×C4×C269—42no—
127C2×C4×C269C2×D411yescharacteristic
128C2×C4×C269C2×D412yes—
129C2×C4×C269#4112yes—
130C2×C4×C269C2411yescharacteristic
131C8×C269—22no—
132C22×C269—21no—
133C22×C269—21no—
134C22×C269—41no—
135C22×C269—41no—
136C22×C269—41no—
137C22×C269#12112yes—
138C22×C269C22×D411yescharacteristic
139C2×D269—21no—
140C2×D269—21no—
141C2×D269—21no—
142C2×D269—21no—
143C2×D269—21no—
144C2×D269—44no—
145C2×D269#13712yescomplemented
146C4×C269—41no—
147C4×C269—42no—
148C4×C269#12112yes—
149C4×C269C22×D411yescharacteristic
150C4×C269C2×#4111yescharacteristic
151Dic269—21no—
152Dic269—21no—
153Dic269—42no—
154Dic269—41no—
155Dic269—41no—
156C2×C269—22no—
157C2×C269—81no—
158C2×C269C2×#12111yescharacteristic
159C2×C269C2×#13711yescharacteristic
160C2×C269#57511yescharacteristic
161D269—22no—
162D269—41no—
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#428—2691no—
174#419—2691no—
175#417D26911yescharacteristic, complemented
176C2×#41C2×D26911yescharacteristic
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—2691no—
189C22⋊Q8—2691no—
190C22⋊Q8—2692no—
191C22⋊Q8—2692no—
192C22⋊Q8—5381no—
193C22⋊Q8—5381no—
194#121—2691no—
195#121—2691no—
196#120—5381no—
197C4×D4—5381no—
198C4×D4—5381no—
199#117—2691no—
200C2×C4⋊C4—5381no—
201C2×C4⋊C4C2×D26911yescharacteristic
202C2×C22⋊C4—5381no—
203#107—2692no—
204#106—2692no—
205#103—2692no—
206#103—2692no—
207#103C2×D26912yescomplemented
208#102C2×D26912yescomplemented
209C24—2691no—
210#41—22no—
211#41—22no—
212C2×Q8—2692no—
213C2×Q8—5381no—
214C2×Q8—5382no—
215C2×Q8—5382no—
216C2×Q8C22×D26911yescharacteristic
217C2×D4—21no—
218C2×D4—5381no—
219C2×D4—5381no—
220C2×D4C22×D26911yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—2691no—
224C22×C4—2691no—
225C22×C4—2691no—
226C22×C4—2691no—
227C22×C4—2691no—
228C22×C4—5382no—
229C22×C4—5382no—
230C22×C4—5381no—
231C22×C4C22×D26911yescharacteristic
232Q16—5384no—
233SD16—5384no—
234M16—22no—
235M16—2694no—
236M16—5384no—
237C2×C8—2692no—
238C2×C8—5384no—
239C2×C8C22×D26912yes—
240C4⋊C4—21no—
241C4⋊C4—2691no—
242C4⋊C4—2691no—
243C4⋊C4—5382no—
244C4⋊C4—5382no—
245C4⋊C4—5381no—
246C4⋊C4—5381no—
247C4⋊C4C22×D26912yes—
248C22⋊C4—5382no—
249C22⋊C4—5382no—
250C22⋊C4—5381no—
251C22⋊C4—5381no—
252C22⋊C4—5381no—
253C22⋊C4—5381no—
254C22⋊C4—5381no—
255C42—5381no—
256C23—21no—
257C23—2691no—
258C23—2691no—
259C23—5382no—
260C23—5382no—
261C23—10761no—
262C23D4×D26911yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—5382no—
266Q8—5382no—
267Q8—10762no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—10762no—
272D4—10762no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
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—5381no—
288C2×C4—5382no—
289C2×C4—10762no—
290C2×C4D4×D26911yescharacteristic
291C2×C4D4×D26912yes—
292C2×C4#1686440212yes—
293C2×C4C23×D26911yescharacteristic
294C8—22no—
295C8—2694no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—2692no—
302C22—5381no—
303C22—5381no—
304C22—5381no—
305C22—5381no—
306C22—5381no—
307C22—10764no—
308C22#2257542612yes—
309C22C2×D4×D26911yescharacteristic
310C4—42no—
311C4—41no—
312C4—5381no—
313C4—5381no—
314C4—10761no—
315C4—10761no—
316C4—10762no—
317C4#2257542612yes—
318C4C2×D4×D26911yescharacteristic
319C4C2×#1686440211yescharacteristic
320C2—22no—
321C2—81no—
322C2—5382no—
323C2—10761no—
324C2C2×#2257542611yescharacteristic
325C2#137×D26911yescharacteristic
326C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#417×C269C2
socleC22×C269C22×D4
prime radical#31771112C2
hypercenter#417D269
hypocenterC269#3056
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22575483C2semidirect
2. C2×M16×D269C2semidirect
4. C2×#22575433C2semidirect
5. #31770862C2semidirect
6. #31770850C2semidirect
7. #31770845C2semidirect
8. #31770833C2semidirect
9. #31770645C2semidirect
10. #31770615C2semidirect
11. #417×C269C2semidirect
15. C2×#16864392C22semidirect
16. C2×#16864390C22semidirect
36. #22575386C22semidirect
37. #22575384C22semidirect
38. #22575378C22semidirect
39. #22575378C22semidirect
43. #103×C269C22semidirect
44. #102×C269C22semidirect
62. #16864386D4semidirect
145. C2×D269#137semidirect
163. C269#3056semidirect
175. #417D269semidirect
207. #103C2×D269semidirect
208. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754832
C2×M16×D2692
#317711122
C2×#225754332
#317708622
#317708502
#317708452
#317708332
#317706452
#317706152
#417×C2692
#3056269