Finite Group Databasebeta

← #46820505

Subgroups — #46820505

Order 34432.

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×#22575481C211yesmaximal, characteristic, complemented
2C2×M16×D269C211yesmaximal, characteristic, complemented
3#31771117C211yesmaximal, characteristic, complemented
4C2×#22575428C211yesmaximal, characteristic, complemented
5#31770859C212yesmaximal, complemented
6#31770852C212yesmaximal, complemented
7#31770842C212yesmaximal, complemented
8#31770835C212yesmaximal, complemented
9#31770609C211yesmaximal, characteristic, complemented
10#31770604C211yesmaximal, characteristic, complemented
11#428×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C22×D1076—21no—
14C22×D1076C2211yescharacteristic
15C22×C4×D269C2211yescharacteristic
16#22575481—24no—
17C2×#16864391C2212yescomplemented
18C2×D2152C2212yescomplemented
19C2×#16864386C2211yescharacteristic
20C2×#16864385C2212yescomplemented
21M16×D269—24no—
22C2×C8×D269C2212yescomplemented
23#22575437—21no—
24#22575433C2211yescharacteristic, complemented
25#22575433C2211yescharacteristic, complemented
26#22575428—21no—
27#22575428—21no—
28#22575428C2212yes—
29#22575428C2212yes—
30#22575428C2211yescharacteristic, complemented
31#22575428C2211yescharacteristic, complemented
32#22575423—21no—
33C4×D1076—21no—
34C4×D1076—21no—
35C2×#16864379—21no—
36#22575376C2212yescomplemented
37#22575373C2212yescomplemented
38#22575373C2212yescomplemented
39#22575373C2212yescomplemented
40C2×M16×C269C2211yescharacteristic
41#117×C269C2211yescharacteristic
42C2×C4⋊C4×C269C2211yescharacteristic
43#107×C269C2212yescomplemented
44#106×C269C2212yescomplemented
45C23×D269—21no—
46C23×D269D411yescharacteristic
47#16864402—22no—
48#16864402—22no—
49C2×Dic538C2311yescharacteristic
50C2×D1076—21no—
51C2×D1076—21no—
52C2×D1076—22no—
53C2×D1076—22no—
54C2×D1076—22no—
55C2×D1076—21no—
56C2×D1076—22no—
57C2×D1076—44no—
58C2×D1076C2312yes—
59C2×D1076C2311yescharacteristic
60C2×C22⋊D269—21no—
61C22×Dic269D411yescharacteristic
62C2×C4×D269—22no—
63C2×C4×D269—21no—
64C2×C4×D269—22no—
65C2×C4×D269D411yescharacteristic
66C2×C4×D269D411yescharacteristic
67C2×C4×D269C2311yescharacteristic
68C2×C4×D269C2311yescharacteristic
69#16864391—24no—
70D2152—24no—
71#16864386—22no—
72#16864385—24no—
73C2×#14524130C2312yes—
74C8×D269—24no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77#16864379—21no—
78#16864379—21no—
79#16864379—22no—
80#16864379—22no—
81#16864379—21no—
82#16864379—21no—
83C22×C4×C269—21no—
84C22×C4×C269C2311yescharacteristic
85M16×C269D414yescomplemented
86C2×C8×C269C2312yes—
87C4⋊C4×C269—21no—
88C4⋊C4×C269C2311yescharacteristic
89C4⋊C4×C269C2311yescharacteristic
90C4⋊C4×C269C2312yes—
91C22⋊C4×C269—21no—
92C42×C269—21no—
93C22×D269—22no—
94C22×D269—22no—
95C22×D269—21no—
96C22×D269—22no—
97C22×D269—41no—
98C22×D269—42no—
99C22×D269—42no—
100C22×D269—42no—
101C22×D269C2×D411yescharacteristic
102C22×D269C2×D411yescharacteristic
103Dic538—21no—
104Dic538—21no—
105D1076—21no—
106D1076—22no—
107D1076—22no—
108D1076—21no—
109D1076—42no—
110D1076—44no—
111D1076—42no—
112D1076—44no—
113D1076—42no—
114C22⋊D269—42no—
115C2×Dic269—22no—
116C2×Dic269—21no—
117C2×Dic269C2×D411yescharacteristic
118C2×Dic269C2×D411yescharacteristic
119C4×D269—22no—
120C4×D269—22no—
121C4×D269—21no—
122C4×D269—21no—
123C4×D269—21no—
124C4×D269—21no—
125C4×D269—42no—
126#14524130—22no—
127C23×C269C2×D411yescharacteristic
128C2×C4×C269—21no—
129C2×C4×C269—21no—
130C2×C4×C269—22no—
131C2×C4×C269—42no—
132C2×C4×C269C2×D411yescharacteristic
133C2×C4×C269C2×D412yes—
134C2×C4×C269#4112yes—
135C2×C4×C269C2411yescharacteristic
136C8×C269C2×D414yescomplemented
137C22×C269—21no—
138C22×C269—21no—
139C22×C269#12112yes—
140C22×C269C22×D411yescharacteristic
141C2×D269—21no—
142C2×D269—21no—
143C2×D269—21no—
144C2×D269—21no—
145C2×D269—21no—
146C2×D269—44no—
147C2×D269—41no—
148C2×D269—42no—
149C2×D269—42no—
150C2×D269—41no—
151C2×D269—41no—
152C2×D269—42no—
153C2×D269—84no—
154C2×D269AGL(1,ℤ/8)12yescomplemented
155C4×C269—42no—
156C4×C269—41no—
157C4×C269—41no—
158C4×C269#12112yes—
159C4×C269C22×D411yescharacteristic
160C4×C269C2×#4111yescharacteristic
161Dic269—21no—
162Dic269—21no—
163Dic269—41no—
164C2×C269—22no—
165C2×C269C2×#12111yescharacteristic
166C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C269#57511yescharacteristic
168D269—22no—
169D269—41no—
170D269—81no—
171D269—82no—
172C269#305511yescharacteristic, complemented, Sylow
173#3055—2691nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2691no—
175C22×M16—2691no—
176#525—2691no—
177C2×#121—2691no—
178#469—2692no—
179#468—2692no—
180#466—2692no—
181#465—2692no—
182#428D26911yescharacteristic, complemented
183#418—2691no—
184#417—2691no—
185C2×#41—2691no—
186C22×D4—2691no—
187C22×D4—5381no—
188C23×C4—2691no—
189AGL(1,ℤ/8)—5384no—
190C2×SD16—2692no—
191C2×D8—2692no—
192C2×M16—2691no—
193C2×M16—2692no—
194C2×M16—5384no—
195C2×M16C2×D26911yescharacteristic
196C22×C8—2692no—
197#123—5381no—
198C22⋊Q8—2691no—
199C22⋊Q8—2691no—
200#121—2692no—
201#121—2692no—
202#121—2691no—
203#121—2691no—
204#121—5381no—
205#121—5381no—
206#120—5381no—
207C4×D4—5381no—
208C4×D4—5381no—
209#117C2×D26911yescharacteristic
210C2×C4⋊C4C2×D26911yescharacteristic
211C2×C22⋊C4—5381no—
212#107C2×D26912yescomplemented
213#106C2×D26912yescomplemented
214#103—2692no—
215#102—2692no—
216#102—2692no—
217#102—2692no—
218C24—2691no—
219C24—5381no—
220#41—5382no—
221#41—5382no—
222C2×Q8—2691no—
223C2×D4—2691no—
224C2×D4—2692no—
225C2×D4—5381no—
226C2×D4—5381no—
227C2×D4—5382no—
228C2×D4—5382no—
229C2×D4—5381no—
230C2×D4—5381no—
231C2×D4—5382no—
232C2×D4—5382no—
233C2×D4—10764no—
234C22×C4—21no—
235C22×C4—2691no—
236C22×C4—2691no—
237C22×C4—2691no—
238C22×C4—2691no—
239C22×C4—2691no—
240C22×C4—5382no—
241C22×C4—5382no—
242C22×C4—5381no—
243C22×C4C22×D26911yescharacteristic
244SD16—5384no—
245D8—5384no—
246M16—5382no—
247M16—5384no—
248M16D107614yescomplemented
249C2×C8—2692no—
250C2×C8—5384no—
251C2×C8C22×D26912yes—
252C4⋊C4—21no—
253C4⋊C4—5381no—
254C4⋊C4—5381no—
255C4⋊C4C22×D26911yescharacteristic
256C4⋊C4C22×D26911yescharacteristic
257C4⋊C4C22×D26912yes—
258C22⋊C4—21no—
259C22⋊C4—5382no—
260C22⋊C4—5382no—
261C22⋊C4—5381no—
262C22⋊C4—5381no—
263C22⋊C4—5381no—
264C22⋊C4—5381no—
265C42—21no—
266C23—2691no—
267C23—2691no—
268C23—5382no—
269C23—5382no—
270C23—5381no—
271C23—5382no—
272C23—10761no—
273C23—10762no—
274C23—10762no—
275C23—10762no—
276C23D4×D26911yescharacteristic
277Q8—5381no—
278Q8—5381no—
279D4—5381no—
280D4—5382no—
281D4—5382no—
282D4—5381no—
283D4—10764no—
284D4—10762no—
285D4—10764no—
286D4—10762no—
287D4—10762no—
288D4—10762no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—2691no—
294C2×C4—2691no—
295C2×C4—5382no—
296C2×C4—5382no—
297C2×C4—5382no—
298C2×C4—5381no—
299C2×C4—5381no—
300C2×C4—5381no—
301C2×C4—5381no—
302C2×C4—5381no—
303C2×C4—10762no—
304C2×C4D4×D26911yescharacteristic
305C2×C4C2×D107612yes—
306C2×C4#1686440312yes—
307C2×C4C23×D26911yescharacteristic
308C8—5382no—
309C8C2×D107614yescomplemented
310C22—21no—
311C22—21no—
312C22—2692no—
313C22—5381no—
314C22—5381no—
315C22—5381no—
316C22—5381no—
317C22—5381no—
318C22—10764no—
319C22—10761no—
320C22—10762no—
321C22—10761no—
322C22—10762no—
323C22—10762no—
324C22—10761no—
325C22—21524no—
326C22#2257542812yes—
327C22C2×D4×D26911yescharacteristic
328C4—41no—
329C4—42no—
330C4—41no—
331C4—5381no—
332C4—5381no—
333C4—10761no—
334C4#2257542812yes—
335C4C22×D107611yescharacteristic
336C4C2×#1686440311yescharacteristic
337C2—22no—
338C2—5382no—
339C2—10761no—
340C2—21522no—
341C2—21521no—
342C2C2×#2257542811yescharacteristic
343C2AGL(1,ℤ/8)×D26911yescharacteristic
344C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
hypercenter#428D269
hypocenterC269#3055
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×#22575481C2semidirect
2. C2×M16×D269C2semidirect
3. #31771117C2semidirect
4. C2×#22575428C2semidirect
5. #31770859C2semidirect
6. #31770852C2semidirect
7. #31770842C2semidirect
8. #31770835C2semidirect
9. #31770609C2semidirect
10. #31770604C2semidirect
11. #428×C269C2semidirect
17. C2×#16864391C22semidirect
18. C2×D2152C22semidirect
20. C2×#16864385C22semidirect
22. C2×C8×D269C22semidirect
24. #22575433C22semidirect
25. #22575433C22semidirect
30. #22575428C22semidirect
31. #22575428C22semidirect
36. #22575376C22semidirect
37. #22575373C22semidirect
38. #22575373C22semidirect
39. #22575373C22semidirect
43. #107×C269C22semidirect
44. #106×C269C22semidirect
85. M16×C269D4semidirect
136. C8×C269C2×D4semidirect
154. C2×D269AGL(1,ℤ/8)semidirect
172. C269#3055semidirect
182. #428D269semidirect
212. #107C2×D269semidirect
213. #106C2×D269semidirect
248. M16D1076semidirect
309. C8C2×D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×M16×D2692
#317711172
C2×#225754282
#317708592
#317708522
#317708422
#317708352
#317706092
#317706042
#428×C2692
#3055269