Finite Group Databasebeta

← #44400141

Subgroups — #44400141

Order 29056.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#20300258C211yesmaximal, characteristic
2#28409505C211yesmaximal, characteristic, complemented
3C2×#20300202C211yesmaximal, characteristic, complemented
4#28408887C211yesmaximal, characteristic, complemented
5#28408887C211yesmaximal, characteristic, complemented
6#28408798C211yesmaximal, characteristic, complemented
7#342×C227C211yesmaximal, characteristic, complemented
8C24×D227C2211yescharacteristic
9C22×D908—21no—
10C22×C22⋊D227C2211yescharacteristic
11#20300258—22no—
12#20300258C414yescomplemented
13C4×C22⋊D227—22no—
14(C22⋊D227)⋊C4—22no—
15#20300244—21no—
16C2×C22⋊Dic227C2211yescharacteristic
17C2×C22⋊Dic227C2211yescharacteristic
18C2×(C2×D227)⋊C4—21no—
19C2×(C2×D227)⋊C4—22no—
20C2×(C2×D227)⋊C4—22no—
21C22⋊C4×D227—21no—
22#20300202—22no—
23C2×C22⋊C4×C227C2211yescharacteristic
24C2×C22⋊C4×C227C2211yescharacteristic
25C2×C22⋊C4×C227C2211yescharacteristic
26C23×D227—21no—
27C23×D227—21no—
28C23×D227—21no—
29C23×D227—21no—
30C23×D227—21no—
31C23×D227—21no—
32C23×D227—44no—
33C23×D227C2×C412yescomplemented
34C23×D227D411yescharacteristic
35C2×D908—24no—
36C2×D908—44no—
37C2×C22⋊D227—21no—
38C2×C22⋊D227—21no—
39C2×C22⋊D227—21no—
40C2×C22⋊D227—21no—
41C2×C22⋊D227C2×C412yes—
42C2×C22⋊D227D412yes—
43C22×Dic227—21no—
44C22×Dic227D411yescharacteristic
45C22×Dic227D412yescomplemented
46C2×C4×D227—22no—
47Dic227⋊C4—22no—
48C22⋊Dic227—21no—
49C22⋊Dic227—21no—
50C22⋊Dic227—21no—
51C22⋊Dic227—21no—
52(C2×D227)⋊C4—22no—
53(C2×D227)⋊C4—24no—
54(C2×D227)⋊C4—44no—
55(C2×D227)⋊C4—44no—
56C4×Dic227—22no—
57C24×C227C2311yescharacteristic
58C22×C4×C227—21no—
59C22×C4×C227—21no—
60C22×C4×C227D412yescomplemented
61C22⋊C4×C227—21no—
62C22⋊C4×C227—21no—
63C22⋊C4×C227—21no—
64C22⋊C4×C227—21no—
65C22⋊C4×C227—21no—
66C22⋊C4×C227C2×C412yescomplemented
67C22×D227—24no—
68C22×D227—24no—
69C22×D227—24no—
70C22×D227—22no—
71C22×D227—22no—
72C22×D227—22no—
73C22×D227—21no—
74C22×D227—21no—
75C22×D227—44no—
76C22×D227—44no—
77C22×D227—44no—
78C22×D227—44no—
79C22×D227—42no—
80C22×D227—42no—
81C22×D227—42no—
82C22×D227—41no—
83C22×D227—41no—
84C22×D227—41no—
85C22×D227—41no—
86C22×D227—42no—
87C22×D227C22⋊C412yescomplemented
88D908—48no—
89C22⋊D227—22no—
90C22⋊D227—22no—
91C22⋊D227—22no—
92C22⋊D227—22no—
93C2×Dic227—21no—
94C2×Dic227—21no—
95C2×Dic227—22no—
96C2×Dic227—42no—
97C2×Dic227—42no—
98C2×Dic227—41no—
99C2×Dic227C22⋊C412yescomplemented
100C4×D227—82no—
101C23×C227—42no—
102C23×C227C22×C411yescharacteristic
103C23×C227C2×D411yescharacteristic
104C23×C227C2×D411yescharacteristic
105C23×C227C2×D411yescharacteristic
106C23×C227C2×D411yescharacteristic
107C23×C227#4111yescharacteristic
108C23×C227#4111yescharacteristic
109C2×C4×C227—22no—
110C2×C4×C227—22no—
111C2×C4×C227—42no—
112C2×C4×C227—42no—
113C2×C4×C227—42no—
114C2×C4×C227—41no—
115C22×C227—22no—
116C22×C227—22no—
117C22×C227—22no—
118C22×C227—41no—
119C22×C227—41no—
120C22×C227—41no—
121C22×C227—41no—
122C22×C227C2×C22⋊C411yescharacteristic
123C22×C227C4×D411yescharacteristic
124C22×C227C4×D411yescharacteristic
125C22×C227#12011yescharacteristic
126C22×C227#12111yescharacteristic
127C22×C227#12111yescharacteristic
128C22×C227#12311yescharacteristic
129C2×D227—24no—
130C2×D227—24no—
131C2×D227—24no—
132C2×D227—22no—
133C2×D227—44no—
134C2×D227—42no—
135C2×D227—44no—
136C2×D227—42no—
137C2×D227—42no—
138C2×D227—44no—
139C2×D227—42no—
140C2×D227—42no—
141C2×D227—44no—
142C2×D227—42no—
143C2×D227—81no—
144C4×C227—42no—
145C4×C227—42no—
146C4×C227—81no—
147Dic227—22no—
148Dic227—42no—
149Dic227—81no—
150C2×C227—21no—
151C2×C227—21no—
152C2×C227—21no—
153C2×C227—21no—
154C2×C227—21no—
155C2×C227—21no—
156C2×C227#38611yescharacteristic
157C2×C227B(4,𝔽2)12yes—
158D227—44no—
159D227—42no—
160C227#188911yescharacteristic, complemented, Sylow
161#1889—2271nomaximal, Sylow
162C2×#120—2271no—
163#518—2271no—
164C2×#99—2271no—
165#379—2271no—
166#379—2271no—
167#342—2271no—
168#342D22711yescharacteristic, complemented
169C25—2271no—
170C22×D4—4541no—
171C22×D4—2271no—
172#120—4542no—
173#120—2274no—
174C4×D4—4542no—
175C4×D4—4542no—
176#117—4541no—
177C2×C22⋊C4—4541no—
178C2×C22⋊C4—4542no—
179C2×C22⋊C4—4542no—
180C2×C22⋊C4—4541no—
181C2×C22⋊C4—2271no—
182C2×C22⋊C4—2271no—
183C2×C22⋊C4C2×D22711yescharacteristic
184C2×C22⋊C4C2×D22711yescharacteristic
185C2×C22⋊C4C2×D22711yescharacteristic
186#99—4542no—
187C24—9084no—
188C24—4541no—
189C24—4541no—
190C24—4541no—
191C24—4541no—
192C24—4541no—
193C24—4541no—
194C24—2272no—
195C24—2271no—
196C24C22×D22711yescharacteristic
197C2×D4—9084no—
198C2×D4—4544no—
199C2×D4—4541no—
200C2×D4—4541no—
201C2×D4—4541no—
202C2×D4—4541no—
203C2×D4—2272no—
204C2×D4—2272no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—4542no—
208C22×C4—4541no—
209C22×C4—2272no—
210C22×C4—2271no—
211C22×C4D90812yescomplemented
212C4⋊C4—4542no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—9084no—
219C22⋊C4—9084no—
220C22⋊C4—4542no—
221C22⋊C4—4544no—
222C22⋊C4—4541no—
223C22⋊C4—4541no—
224C22⋊C4—4541no—
225C22⋊C4—4541no—
226C22⋊C4C4×D22712yescomplemented
227C42—4542no—
228C23—42no—
229C23—9082no—
230C23—9082no—
231C23—9082no—
232C23—9082no—
233C23—9084no—
234C23—9084no—
235C23—9084no—
236C23—9084no—
237C23—9081no—
238C23—9081no—
239C23—9081no—
240C23—9081no—
241C23—4544no—
242C23—4544no—
243C23—4544no—
244C23—4541no—
245C23—4542no—
246C23—4542no—
247C23—4542no—
248C23—4541no—
249C23—2272no—
250C23C2×C4×D22711yescharacteristic
251C23D4×D22711yescharacteristic
252C23D4×D22711yescharacteristic
253C23D4×D22711yescharacteristic
254C23C2×D90811yescharacteristic
255C23#1681388411yescharacteristic
256C23#1681388511yescharacteristic
257D4—9088no—
258D4—4542no—
259D4—4542no—
260D4—4542no—
261D4—4542no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—18162no—
269C2×C4—9082no—
270C2×C4—9081no—
271C2×C4—9082no—
272C2×C4—4542no—
273C2×C4—4541no—
274C2×C4—4541no—
275C2×C4—2272no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—18161no—
284C22—9082no—
285C22—9082no—
286C22—9084no—
287C22—9084no—
288C22—9084no—
289C22—9082no—
290C22—9082no—
291C22—9082no—
292C22—9082no—
293C22—9084no—
294C22—4544no—
295C22—4544no—
296C22—4544no—
297C22—4542no—
298C22C22⋊C4×D22711yescharacteristic
299C22#2030025011yescharacteristic
300C22C4×D90811yescharacteristic
301C22#2030025811yescharacteristic
302C22#2030026111yescharacteristic
303C22#2030026311yescharacteristic
304C22#2030027311yescharacteristic
305C4—42no—
306C4—42no—
307C4—81no—
308C4—18161no—
309C4—9082no—
310C4—4542no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—9082no—
318C2—9084no—
319C2#2840891011yescharacteristic
320C2#2840922012yes—
subgroupG/H
centerC22#20300258
commutatorC23×C227C22×C4
FrattiniC24C22×D227
Fitting#342×C227C2
socleC22×C227#120
prime radicalC2×#20300258C2
cosocleC24×C227C23
hypercenter#342D227
hypocenterC227#1889
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. #28409505C2semidirect
3. C2×#20300202C2semidirect
4. #28408887C2semidirect
5. #28408887C2semidirect
6. #28408798C2semidirect
7. #342×C227C2semidirect
12. #20300258C4semidirect
33. C23×D227C2×C4semidirect
45. C22×Dic227D4semidirect
60. C22×C4×C227D4semidirect
66. C22⋊C4×C227C2×C4semidirect
87. C22×D227C22⋊C4semidirect
99. C2×Dic227C22⋊C4semidirect
160. C227#1889semidirect
168. #342D227semidirect
211. C22×C4D908semidirect
226. C22⋊C4C4×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002582
#284095052
C2×#203002022
#284088872
#284087982
#342×C2272
#1889227