Finite Group Databasebeta

← #41014381

Subgroups — #41014381

Order 27008.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20269954C211yesmaximal, characteristic
2#27213004C211yesmaximal, characteristic
3C22⋊C4×Dic211C211yesmaximal, characteristic
4#27212913C211yesmaximal, characteristic
5#27212912C211yesmaximal, characteristic
6#27212912C211yesmaximal, characteristic
7#342×C211C211yesmaximal, characteristic
8C22×Dic422C411yescharacteristic, complemented
9C23×Dic211C2211yescharacteristic
10#20269954—21no—
11#20269954—21no—
12#20269954—21no—
13#20269954—21no—
14C2×C4⋊Dic211C411yescharacteristic, complemented
15C2×Dic211⋊C4—21no—
16C2×Dic211⋊C4C411yescharacteristic, complemented
17C2×Dic211⋊C4C411yescharacteristic, complemented
18C2×C22⋊Dic211C2211yescharacteristic
19C2×C22⋊Dic211C2211yescharacteristic
20C2×C22⋊Dic211C2211yescharacteristic
21C2×C4×Dic211—21no—
22C211⋊#95—21no—
23C211⋊#95—21no—
24C2×C22⋊C4×C211C2211yescharacteristic
25C2×C22⋊C4×C211C2211yescharacteristic
26C2×C22⋊C4×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—42no—
30C22×Dic211—21no—
31C22×Dic211—21no—
32C22×Dic211—42no—
33C22×Dic211C2×C411yescharacteristic
34C22×Dic211C2×C411yescharacteristic
35C22×Dic211C2×C411yescharacteristic
36C22×Dic211C2×C411yescharacteristic
37C22×Dic211D411yescharacteristic
38C22×Dic211D411yescharacteristic
39C22×Dic211D411yescharacteristic
40C22×Dic211D411yescharacteristic
41C4⋊Dic211—21no—
42C4⋊Dic211—21no—
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45Dic211⋊C4—21no—
46Dic211⋊C4—21no—
47Dic211⋊C4—42no—
48C22⋊Dic211—21no—
49C22⋊Dic211—21no—
50C22⋊Dic211—21no—
51C22⋊Dic211—21no—
52C22⋊Dic211—21no—
53C22⋊Dic211Q812yescomplemented
54C4×Dic211—42no—
55C24×C211C2311yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2×C411yescharacteristic
59C22×C4×C211C2×C411yescharacteristic
60C22⋊C4×C211—21no—
61C22⋊C4×C211—21no—
62C22⋊C4×C211—21no—
63C22⋊C4×C211—21no—
64C22⋊C4×C211—21no—
65C22⋊C4×C211Q812yescomplemented
66Dic422—42no—
67Dic422—42no—
68C2×Dic211—22no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—22no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C2×Dic211—41no—
79C2×Dic211—41no—
80C2×Dic211—41no—
81C2×Dic211—41no—
82C2×Dic211—41no—
83C2×Dic211—41no—
84C2×Dic211C22⋊C411yescharacteristic
85C2×Dic211C22⋊C411yescharacteristic
86C2×Dic211C22⋊C411yescharacteristic
87C2×Dic211C22⋊C411yescharacteristic
88C23×C211—41no—
89C23×C211—41no—
90C23×C211C22×C411yescharacteristic
91C23×C211C2×D411yescharacteristic
92C23×C211C2×D411yescharacteristic
93C23×C211C2×Q811yescharacteristic
94C23×C211C2×Q811yescharacteristic
95C23×C211#4111yescharacteristic
96C23×C211#4111yescharacteristic
97C2×C4×C211—21no—
98C2×C4×C211—21no—
99C2×C4×C211—22no—
100C2×C4×C211—41no—
101C2×C4×C211—41no—
102C2×C4×C211—41no—
103C2×C4×C211—41no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211—41no—
107C22×C211—21no—
108C22×C211—21no—
109C22×C211—21no—
110C22×C211—21no—
111C22×C211—21no—
112C22×C211—21no—
113C22×C211—41no—
114C22×C211—41no—
115C22×C211—41no—
116C22×C211—41no—
117C22×C211C2×C22⋊C411yescharacteristic
118C22×C211C4×Q811yescharacteristic
119C22×C211C4×Q811yescharacteristic
120C22×C211C22⋊Q811yescharacteristic
121C22×C211C22⋊Q811yescharacteristic
122C22×C211#12411yescharacteristic
123C22×C211C4⋊Q811yescharacteristic
124C4×C211—42no—
125C4×C211—41no—
126C4×C211—41no—
127C4×C211—81no—
128Dic211—21no—
129Dic211—21no—
130Dic211—21no—
131Dic211—42no—
132Dic211—41no—
133Dic211—41no—
134Dic211—81no—
135Dic211#9912yes—
136C2×C211—21no—
137C2×C211—21no—
138C2×C211—21no—
139C2×C211—21no—
140C2×C211—21no—
141C2×C211—21no—
142C2×C211#39111yescharacteristic
143C2×C211C2×#9911yescharacteristic
144C2×C211#41011yescharacteristic
145C211#199611yescharacteristic, complemented, Sylow
146#1996—2111nomaximal, Sylow
147C2×C22⋊Q8—2111no—
148#380—2111no—
149C4×C22⋊C4—2111no—
150#342—2111no—
151#342—2111no—
152#342—2111no—
153#342D21111yescharacteristic
154C22×Q8—2111no—
155C23×C4—2111no—
156C22⋊Q8—4221no—
157C22⋊Q8—4221no—
158C22⋊Q8—4221no—
159C22⋊Q8—4221no—
160C2×C4⋊C4—4221no—
161C2×C4⋊C4—2111no—
162C2×C4⋊C4—2111no—
163C2×C4⋊C4—2111no—
164C2×C22⋊C4—2111no—
165C2×C22⋊C4—2111no—
166C2×C22⋊C4—2111no—
167C2×C22⋊C4C2×D21111yescharacteristic
168C2×C22⋊C4C2×D21111yescharacteristic
169C2×C22⋊C4C2×D21111yescharacteristic
170C2×C42—4221no—
171#95—4221no—
172#95—4221no—
173C24C22×D21111yescharacteristic
174C2×Q8—8442no—
175C2×Q8—4221no—
176C2×Q8—4221no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—8442no—
180C22×C4—4221no—
181C22×C4—4221no—
182C22×C4—2111no—
183C22×C4—2111no—
184C22×C4—2111no—
185C22×C4—2111no—
186C22×C4—2111no—
187C22×C4—2111no—
188C22×C4—2111no—
189C22×C4—2111no—
190C22×C4C4×D21111yescharacteristic
191C22×C4C4×D21111yescharacteristic
192C4⋊C4—8442no—
193C4⋊C4—4221no—
194C4⋊C4—4221no—
195C4⋊C4—4221no—
196C4⋊C4—4221no—
197C4⋊C4—4221no—
198C4⋊C4—4221no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—4221no—
205C22⋊C4—4221no—
206C22⋊C4—4221no—
207C22⋊C4—4221no—
208C22⋊C4—4221no—
209C22⋊C4—2112no—
210C22⋊C4Dic42212yescomplemented
211C42—8442no—
212C23—41no—
213C23—41no—
214C23C2×C4×D21111yescharacteristic
215C23D4×D21111yescharacteristic
216C23D4×D21111yescharacteristic
217C23Q8×D21111yescharacteristic
218C23C2×Dic42211yescharacteristic
219C23#1680996111yescharacteristic
220C23#1680996211yescharacteristic
221Q8—8442no—
222Q8—8442no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—8441no—
234C2×C4—8441no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—8441no—
238C2×C4—8441no—
239C2×C4—8441no—
240C2×C4—8441no—
241C2×C4—8441no—
242C2×C4—8441no—
243C2×C4—4221no—
244C2×C4—4221no—
245C2×C4—4222no—
246C2×C4—4222no—
247C2×C4—4222no—
248C2×C4—4222no—
249C2×C4—2111no—
250C2×C4—2111no—
251C2×C4—2111no—
252C2×C4—2111no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22C22⋊C4×D21111yescharacteristic
264C22#2026994211yescharacteristic
265C22C4×Dic42211yescharacteristic
266C22#2026995411yescharacteristic
267C22#2026995511yescharacteristic
268C22#2026996311yescharacteristic
269C22#2026997611yescharacteristic
270C4—42no—
271C4—41no—
272C4—41no—
273C4—81no—
274C4—8442no—
275C4—8441no—
276C4—8441no—
277C4—16881no—
278C4—4221no—
279C4—4221no—
280C4—4221no—
281C4—2112no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—21no—
288C2#2721305011yescharacteristic
289C2#99×D21111yescharacteristic
290C2#2721311811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211C2×C22⋊C4
prime radical, cosocleC24×C211C23
hypercenter#342D211
hypocenterC211#1996
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
8. C22×Dic422C4semidirect
14. C2×C4⋊Dic211C4semidirect
16. C2×Dic211⋊C4C4semidirect
17. C2×Dic211⋊C4C4semidirect
53. C22⋊Dic211Q8semidirect
65. C22⋊C4×C211Q8semidirect
145. C211#1996semidirect
210. C22⋊C4Dic422semidirect

Maximal subgroups

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

subgroupindex
C2×#202699542
#272130042
C22⋊C4×Dic2112
#272129132
#272129122
#342×C2112
#1996211