Finite Group Databasebeta

← #41013858

Subgroups — #41013858

Order 27008.

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

normal subgroups only

This group has 390 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×M16×D211C211yesmaximal, characteristic, complemented
2C2×#20269948C211yesmaximal, characteristic
3C2×#20269894C211yesmaximal, characteristic, complemented
4#27213109C211yesmaximal, characteristic, complemented
5#27212914C211yesmaximal, characteristic, complemented
6#27212914C211yesmaximal, characteristic, complemented
7C2×#100×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211C412yes—
9C22×C4×D211C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11C2×#16809946C2211yescharacteristic
12C2×#16809945—21no—
13M16×D211—22no—
14M16×D211—22no—
15C2×C8×D211—21no—
16#20269948—21no—
17#20269948—21no—
18#20269948—21no—
19#20269948—21no—
20C2×C4⋊Dic211C2211yescharacteristic
21C2×C22⋊Dic211C412yes—
22#20269894—22no—
23(C2×D211)⋊C8—21no—
24(C2×D211)⋊C8—21no—
25C22×D4×C211C2211yescharacteristic
26C2×M16×C211C2211yescharacteristic
27C2×M16×C211C2211yescharacteristic
28#100×C211—22no—
29C23×D211C2×C411yescharacteristic
30C2×C22⋊D211—22no—
31C2×C22⋊D211—22no—
32C2×C22⋊D211—44no—
33C22×Dic211C2×C411yescharacteristic
34C22×Dic211C2×C412yes—
35C2×C4×D211—21no—
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C4×D211—21no—
39C2×C4×D211D411yescharacteristic
40C2×C4×D211D411yescharacteristic
41C2×C4×D211D411yescharacteristic
42C2×C4×D211D411yescharacteristic
43#16809946—21no—
44#16809946—21no—
45#16809946—21no—
46#16809946D412yescomplemented
47#16809945—24no—
48C2×C211⋊C8—21no—
49C2×C211⋊C8—21no—
50C8×D211—24no—
51C4⋊Dic211—21no—
52C4⋊Dic211—21no—
53C22⋊Dic211—22no—
54C22⋊Dic211—22no—
55C24×C211C2×C412yes—
56C2×D4×C211—21no—
57C2×D4×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C22×C4×C211C2311yescharacteristic
63M16×C211—21no—
64M16×C211—21no—
65M16×C211—21no—
66M16×C211D412yescomplemented
67C2×C8×C211—21no—
68C2×C8×C211—21no—
69C22×D211—21no—
70C22×D211—21no—
71C22×D211—22no—
72C22×D211—41no—
73C22×D211C22⋊C411yescharacteristic
74C22×D211C22⋊C411yescharacteristic
75C22⋊D211—44no—
76C22⋊D211—44no—
77C2×Dic211—21no—
78C2×Dic211—21no—
79C2×Dic211—22no—
80C2×Dic211—42no—
81C2×Dic211C22⋊C411yescharacteristic
82C2×Dic211C22⋊C411yescharacteristic
83C4×D211—21no—
84C4×D211—21no—
85C4×D211—21no—
86C4×D211—21no—
87C4×D211—21no—
88C4×D211—21no—
89C4×D211—21no—
90C4×D211—21no—
91C211⋊C8—22no—
92C211⋊C8—41no—
93C23×C211—22no—
94C23×C211—22no—
95C23×C211—22no—
96C23×C211—42no—
97C23×C211—42no—
98C23×C211C22×C411yescharacteristic
99D4×C211—41no—
100D4×C211—41no—
101D4×C211—41no—
102D4×C211—41no—
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211C2×D411yescharacteristic
105C2×C4×C211C2×D411yescharacteristic
106C2×C4×C211C2×D411yescharacteristic
107C2×C4×C211#4111yescharacteristic
108C2×C4×C211#4111yescharacteristic
109C8×C211—22no—
110C8×C211—41no—
111C22×C211—21no—
112C22×C211—21no—
113C22×C211—42no—
114C22×C211—42no—
115C22×C211—42no—
116C22×C211—82no—
117C22×C211—82no—
118C22×C211C2×C22⋊C411yescharacteristic
119C22×C211C4×D411yescharacteristic
120C22×C211C4×D411yescharacteristic
121C2×D211—21no—
122C2×D211—21no—
123C2×D211—21no—
124C2×D211—21no—
125C2×D211—21no—
126C2×D211—42no—
127C2×D211—42no—
128C2×D211#10012yescomplemented
129C4×C211#12011yescharacteristic
130C4×C211#12111yescharacteristic
131C4×C211#12111yescharacteristic
132C4×C211#12311yescharacteristic
133Dic211—21no—
134Dic211—21no—
135Dic211—42no—
136C2×C211—21no—
137C2×C211—21no—
138C2×C211—82no—
139C2×C211#38611yescharacteristic
140C2×C211C2×#10011yescharacteristic
141C2×C211#41311yescharacteristic
142D211—22no—
143D211—41no—
144C211#190011yescharacteristic, complemented, Sylow
145#1900—2111nomaximal, Sylow
146C22×M16—2111no—
147C2×#121—2111no—
148C2×#100—2111no—
149C2×#100D21111yescharacteristic, complemented
150#407—2111no—
151#343—2111no—
152#343—2111no—
153C22×D4—2112no—
154C22×D4C2×D21111yescharacteristic
155C23×C4—2111no—
156C2×M16—4221no—
157C2×M16—4222no—
158C2×M16—4222no—
159C2×M16—2111no—
160C2×M16—2111no—
161C2×M16C2×D21111yescharacteristic
162C2×M16C2×D21111yescharacteristic
163C22×C8—4221no—
164#121—4221no—
165#121—4221no—
166#121—4221no—
167#121—4221no—
168C2×C4⋊C4—2111no—
169C2×C22⋊C4—2112no—
170#100—22no—
171#100—4222no—
172C22⋊C8—4221no—
173C22⋊C8—4221no—
174C24—2111no—
175C24C4×D21112yes—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—8444no—
183C2×D4—4222no—
184C2×D4—4222no—
185C22×C4—4221no—
186C22×C4—4221no—
187C22×C4—4221no—
188C22×C4—4221no—
189C22×C4—2112no—
190C22×C4—2111no—
191C22×C4—2111no—
192C22×C4—2111no—
193C22×C4—2111no—
194C22×C4—2111no—
195C22×C4C22×D21111yescharacteristic
196M16—21no—
197M16—21no—
198M16—21no—
199M16—4224no—
200M16—4221no—
201M16—4221no—
202M16—4221no—
203M16—2112no—
204M16C22⋊D21112yescomplemented
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—4224no—
208C2×C8—4221no—
209C2×C8—4221no—
210C4⋊C4—4221no—
211C4⋊C4—4221no—
212C22⋊C4—4222no—
213C22⋊C4—4222no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—8441no—
220C23—4222no—
221C23—4221no—
222C23—4221no—
223C23—2111no—
224C23—2111no—
225C23C2×C4×D21111yescharacteristic
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—8444no—
231D4—8444no—
232C2×C4—8442no—
233C2×C4—4222no—
234C2×C4—4221no—
235C2×C4—4221no—
236C2×C4—4221no—
237C2×C4—4221no—
238C2×C4—4221no—
239C2×C4—4221no—
240C2×C4—4221no—
241C2×C4—4221no—
242C2×C4—4221no—
243C2×C4—4221no—
244C2×C4—2111no—
245C2×C4—2111no—
246C2×C4D4×D21111yescharacteristic
247C2×C4D4×D21111yescharacteristic
248C2×C4D4×D21111yescharacteristic
249C2×C4C2×C22⋊D21111yescharacteristic
250C2×C4#1680996111yescharacteristic
251C2×C4#1680996211yescharacteristic
252C8—22no—
253C8—41no—
254C8—8441no—
255C8—4222no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—82no—
262C22—82no—
263C22—8442no—
264C22—8442no—
265C22—4221no—
266C22—4221no—
267C22—4221no—
268C22—4221no—
269C22—4221no—
270C22—2112no—
271C22C22⋊C4×D21111yescharacteristic
272C22(C22⋊D211)⋊C411yescharacteristic
273C22C4×C22⋊D21111yescharacteristic
274C4—8442no—
275C4—4221no—
276C4—4221no—
277C4#2026994511yescharacteristic
278C4#2026994811yescharacteristic
279C4#2026995011yescharacteristic
280C4#2026995911yescharacteristic
281C2—21no—
282C2—21no—
283C2—82no—
284C2—8441no—
285C2—4222no—
286C2#2721302211yescharacteristic
287C2#100×D21111yescharacteristic
288C2#2721312711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#100×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#20269948C2
cosocleC22×C4×C211C23
hypercenterC2×#100D211
hypocenterC211#1900
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×M16×D211C2semidirect
3. C2×#20269894C2semidirect
4. #27213109C2semidirect
5. #27212914C2semidirect
6. #27212914C2semidirect
7. C2×#100×C211C2semidirect
46. #16809946D4semidirect
66. M16×C211D4semidirect
128. C2×D211#100semidirect
144. C211#1900semidirect
149. C2×#100D211semidirect
204. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D2112
C2×#202699482
C2×#202698942
#272131092
#272129142
C2×#100×C2112
#1900211