Finite Group Databasebeta

← #46809804

Subgroups — #46809804

Order 34432.

368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#31771163C211yesmaximal, characteristic, complemented
2#31770741C211yesmaximal, characteristic, complemented
3#31770738C211yesmaximal, characteristic, complemented
4#31770515C211yesmaximal, characteristic, complemented
5#31770338C211yesmaximal, characteristic, complemented
6#31770328C211yesmaximal, characteristic, complemented
7#327×C269C211yesmaximal, characteristic, complemented
8C22×D1076—21no—
9C22×D1076C2211yescharacteristic, complemented
10C2×D4×D269—21no—
11C2×D2152—21no—
12C2×#16864387—21no—
13#22575438—21no—
14#22575435—21no—
15#22575427C2211yescharacteristic
16#22575425—21no—
17#22575423—21no—
18#22575421—21no—
19C2×#16864379—21no—
20C2×#16864379C2211yescharacteristic
21#22575375—21no—
22#22575373—21no—
23C22⋊#14524130C2211yescharacteristic, complemented
24#121×C269C2211yescharacteristic
25C22⋊C8×C269C2211yescharacteristic, complemented
26#95×C269C2211yescharacteristic, complemented
27C23×D269—21no—
28C23×D269—21no—
29C23×D269D411yescharacteristic
30C2×D1076—21no—
31C2×D1076—21no—
32C2×D1076—21no—
33C2×D1076—42no—
34C2×D1076—42no—
35C2×D1076—42no—
36C2×D1076—42no—
37C2×D1076D411yescharacteristic, complemented
38C2×C22⋊D269—21no—
39C2×C22⋊D269—21no—
40C2×C22⋊D269—21no—
41D4×D269—42no—
42D4×D269—42no—
43C22×Dic269D411yescharacteristic
44C2×C4×D269—21no—
45D2152—42no—
46#16864387—42no—
47C2×#14524130—21no—
48C4⋊Dic269D411yescharacteristic, complemented
49#16864381—21no—
50C22⋊Dic269—21no—
51#16864379—21no—
52#16864379—21no—
53#16864379—21no—
54#16864379—21no—
55#16864379—42no—
56C2×D4×C269—21no—
57C2×D4×C269D411yescharacteristic, complemented
58C22×C4×C269—21no—
59C22×C4×C269C2311yescharacteristic
60C2×C8×C269—21no—
61C4⋊C4×C269D411yescharacteristic, complemented
62C22⋊C4×C269—21no—
63C22×D269—21no—
64C22×D269—21no—
65C22×D269—22no—
66C22×D269—21no—
67C22×D269—42no—
68C22×D269—41no—
69C22×D269—41no—
70C22×D269—41no—
71C22×D269—41no—
72C22×D269—41no—
73C22×D269—41no—
74C22×D269—41no—
75C22×D269—41no—
76C22×D269—81no—
77C22×D269—81no—
78C22×D269D812yescomplemented
79D1076—22no—
80D1076—22no—
81D1076—41no—
82D1076—41no—
83D1076—41no—
84D1076—42no—
85D1076—48no—
86C22⋊D269—42no—
87C22⋊D269—41no—
88C22⋊D269—41no—
89C22⋊D269—81no—
90C2×Dic269—21no—
91C2×Dic269—21no—
92C2×Dic269—41no—
93C4×D269—42no—
94#14524130—41no—
95C23×C269—21no—
96C23×C269C2×D411yescharacteristic
97D4×C269—41no—
98D4×C269—41no—
99D4×C269—41no—
100C2×C4×C269—22no—
101C2×C4×C269—21no—
102C2×C4×C269—41no—
103C2×C4×C269—41no—
104C2×C4×C269—41no—
105C2×C4×C269C2×D411yescharacteristic
106C2×C4×C269C2×D411yescharacteristic
107C8×C269—41no—
108C22×C269—21no—
109C22×C269—21no—
110C22×C269—41no—
111C22×C269—41no—
112C22×C269—41no—
113C22×C269#12011yescharacteristic
114C22×C269C2×D811yescharacteristic
115C22×C269AGL(1,ℤ/8)11yescharacteristic
116C2×D269—22no—
117C2×D269—41no—
118C2×D269—41no—
119C2×D269—42no—
120C2×D269—42no—
121C2×D269—42no—
122C2×D269—41no—
123C2×D269—41no—
124C2×D269—42no—
125C2×D269—41no—
126C2×D269—81no—
127C2×D269—84no—
128C2×D269—81no—
129C2×D269—81no—
130C2×D269—81no—
131C2×D269—81no—
132C4×C269—21no—
133C4×C269—21no—
134C4×C269—42no—
135C4×C269—41no—
136Dic269—41no—
137Dic269—41no—
138C2×C269—21no—
139C2×C269—21no—
140C2×C269—81no—
141C2×C269#44711yescharacteristic
142C2×C269#45311yescharacteristic
143C2×C269B(4,𝔽2)11yescharacteristic
144D269—42no—
145D269—82no—
146D269—81no—
147C269#159511yescharacteristic, complemented, Sylow
148#1595—2691nomaximal, Sylow
149#534—2691no—
150#447—2691no—
151#447—2691no—
152#392—2691no—
153#327—2691no—
154#327D26911yescharacteristic, complemented
155#323—2691no—
156C22×D4—2691no—
157C22×D4—5381no—
158C22×D4—5381no—
159C2×D8—5381no—
160C2×D8—5381no—
161#123—5381no—
162#123—5381no—
163#121—2691no—
164#121—5381no—
165#121C2×D26911yescharacteristic
166#120—5381no—
167#120—5381no—
168C2×C22⋊C4—2691no—
169C2×C22⋊C4—5381no—
170#102—5381no—
171#102—5381no—
172C22⋊C8—2691no—
173C22⋊C8C2×D26911yescharacteristic, complemented
174#95C2×D26911yescharacteristic, complemented
175C24—2691no—
176C24—5381no—
177C24—5381no—
178C2×D4—21no—
179C2×D4—2691no—
180C2×D4—5381no—
181C2×D4—5381no—
182C2×D4—5381no—
183C2×D4—5381no—
184C2×D4—5381no—
185C2×D4—5381no—
186C2×D4—10762no—
187C2×D4—10762no—
188C2×D4—10762no—
189C2×D4—10762no—
190C2×D4—10762no—
191C2×D4—10762no—
192C2×D4D107611yescharacteristic, complemented
193C22×C4—21no—
194C22×C4—2691no—
195C22×C4—5381no—
196C22×C4C22×D26911yescharacteristic
197D8—10762no—
198D8—10762no—
199C2×C8—21no—
200C2×C8—5381no—
201C4⋊C4—2691no—
202C4⋊C4—5381no—
203C4⋊C4D107611yescharacteristic, complemented
204C22⋊C4—21no—
205C22⋊C4—5381no—
206C22⋊C4—5381no—
207C22⋊C4—5381no—
208C22⋊C4—5381no—
209C22⋊C4—5381no—
210C22⋊C4—10762no—
211C23—21no—
212C23—2692no—
213C23—5382no—
214C23—5381no—
215C23—5381no—
216C23—5381no—
217C23—10761no—
218C23—10761no—
219C23—10761no—
220C23—10761no—
221C23—10761no—
222C23—10762no—
223C23—10761no—
224C23—10761no—
225C23—10761no—
226C23—21521no—
227C23—21521no—
228C23D4×D26911yescharacteristic
229D4—41no—
230D4—41no—
231D4—41no—
232D4—5382no—
233D4—5382no—
234D4—10761no—
235D4—10761no—
236D4—10761no—
237D4—10762no—
238D4—10768no—
239D4—10762no—
240D4—10761no—
241D4—10761no—
242D4—21521no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—5381no—
249C2×C4—5381no—
250C2×C4—10762no—
251C2×C4—10761no—
252C2×C4D4×D26911yescharacteristic
253C2×C4C2×D107611yescharacteristic
254C8—41no—
255C8—10761no—
256C22—21no—
257C22—21no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—5382no—
262C22—10762no—
263C22—10762no—
264C22—10762no—
265C22—10761no—
266C22—10761no—
267C22—10761no—
268C22—10761no—
269C22—10762no—
270C22—10761no—
271C22—21521no—
272C22—21521no—
273C22—21521no—
274C22—21524no—
275C22—21521no—
276C22—21521no—
277C22#2257542311yescharacteristic
278C22D8×D26911yescharacteristic
279C22#2257548011yescharacteristic
280C4—21no—
281C4—21no—
282C4—42no—
283C4—41no—
284C4—10761no—
285C4—10761no—
286C2—21no—
287C2—21no—
288C2—81no—
289C2—10762no—
290C2—21522no—
291C2—21521no—
292C2#3177074011yescharacteristic
293C2#3177077411yescharacteristic
294C2#3177079311yescharacteristic
subgroupG/H
centerC22#22575423
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#327×C269C2
socleC22×C269#120
hypercenter#327D269
hypocenterC269#1595
LCSG ▷ C22×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
1. #31771163C2semidirect
2. #31770741C2semidirect
3. #31770738C2semidirect
4. #31770515C2semidirect
5. #31770338C2semidirect
6. #31770328C2semidirect
7. #327×C269C2semidirect
9. C22×D1076C22semidirect
23. C22⋊#14524130C22semidirect
25. C22⋊C8×C269C22semidirect
26. #95×C269C22semidirect
37. C2×D1076D4semidirect
48. C4⋊Dic269D4semidirect
57. C2×D4×C269D4semidirect
61. C4⋊C4×C269D4semidirect
78. C22×D269D8semidirect
147. C269#1595semidirect
154. #327D269semidirect
173. C22⋊C8C2×D269semidirect
174. #95C2×D269semidirect
192. C2×D4D1076semidirect
203. C4⋊C4D1076semidirect

Maximal subgroups

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

subgroupindex
#317711632
#317707412
#317707382
#317705152
#317703382
#317703282
#327×C2692
#1595269