Finite Group Databasebeta

← #41013719

Subgroups — #41013719

Order 27008.

412 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 99 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×#20269952C211yesmaximal, characteristic
2C2×#20269935C211yesmaximal, characteristic, complemented
3C2×#20269899C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#340×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C22×D844C2211yescharacteristic
10C22×C4×D211C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269952—22no—
13#20269952C414yescomplemented
14#20269935—22no—
15#20269935—22no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×(C2×D211)⋊C4—21no—
18C2×(C2×D211)⋊C4—21no—
19C2×C4×Dic211—21no—
20#20269899—22no—
21#20269897—22no—
22(C2×D211)⋊C8—22no—
23C22×C8×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C23×D211—21no—
27C23×D211D411yescharacteristic
28C2×D844—24no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—42no—
35C2×D844—42no—
36C2×D844C2×C412yes—
37C22×Dic211—21no—
38C22×Dic211D411yescharacteristic
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211—21no—
43C2×C4×D211C2×C412yes—
44C2×C4×D211D412yes—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C2×C211⋊C8—21no—
48C4⋊Dic211—21no—
49C4⋊Dic211D412yescomplemented
50(C2×D211)⋊C4—24no—
51(C2×D211)⋊C4—24no—
52C4×Dic211—24no—
53C22×C4×C211—21no—
54C22×C4×C211—21no—
55C22×C4×C211C2311yescharacteristic
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C2×C8×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211—21no—
61C4⋊C4×C211C2×C412yes—
62C4⋊C4×C211D412yescomplemented
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—21no—
67C22×D211—21no—
68C22×D211—21no—
69C22×D211—22no—
70C22×D211—41no—
71C22×D211—41no—
72C22×D211—41no—
73C22×D211—41no—
74C22×D211—41no—
75C22×D211—41no—
76C22×D211C22⋊C412yes—
77D844—22no—
78D844—22no—
79D844—48no—
80D844—41no—
81D844—41no—
82C2×Dic211—21no—
83C2×Dic211—21no—
84C2×Dic211—22no—
85C2×Dic211—41no—
86C2×Dic211—41no—
87C2×Dic211C22⋊C412yes—
88C4×D211—22no—
89C4×D211—22no—
90C4×D211—22no—
91C4×D211—22no—
92C211⋊C8—41no—
93C23×C211C2×D411yescharacteristic
94C2×C4×C211—22no—
95C2×C4×C211—22no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211—41no—
99C2×C4×C211—41no—
100C2×C4×C211C22×C411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211C2×D411yescharacteristic
103C2×C4×C211C2×D411yescharacteristic
104C2×C4×C211#4111yescharacteristic
105C2×C4×C211#4111yescharacteristic
106C8×C211—41no—
107C22×C211C2×C22⋊C411yescharacteristic
108C22×C211#12011yescharacteristic
109C22×C211#12311yescharacteristic
110C22×C211#13511yescharacteristic
111C22×C211#13511yescharacteristic
112C22×C211AGL(1,ℤ/8)11yescharacteristic
113C22×C211AGL(1,ℤ/8)11yescharacteristic
114C2×D211—22no—
115C2×D211—22no—
116C2×D211—22no—
117C2×D211—41no—
118C2×D211—41no—
119C2×D211—41no—
120C2×D211—41no—
121C2×D211—42no—
122C2×D211—42no—
123C2×D211—42no—
124C2×D211—81no—
125C2×D211—81no—
126C2×D211—81no—
127C2×D211—81no—
128C4×C211—42no—
129C4×C211—42no—
130C4×C211C4×D411yescharacteristic
131C4×C211C4×D411yescharacteristic
132C4×C211#12111yescharacteristic
133C4×C211#12111yescharacteristic
134Dic211—22no—
135Dic211—42no—
136C2×C211#38611yescharacteristic
137C2×C211#41611yescharacteristic
138C2×C211#41811yescharacteristic
139C2×C211#44911yescharacteristic
140C2×C211#44911yescharacteristic
141C2×C211#48211yescharacteristic
142C2×C211#48211yescharacteristic
143D211—42no—
144D211—82no—
145C211#187511yescharacteristic, complemented, Sylow
146#1875—2111nomaximal, Sylow
147C2×#121—2111no—
148C2×#117—2111no—
149C2×#102—2111no—
150C2×#102—2111no—
151C2×C22⋊C8—2111no—
152#340—2111no—
153#340D21111yescharacteristic, complemented
154C22×D4—4221no—
155C22×D4—2111no—
156C23×C4—2111no—
157C22×C8—2111no—
158C22×C8C2×D21111yescharacteristic
159#121—4222no—
160#121—2114no—
161#117—4222no—
162#117—4222no—
163C2×C4⋊C4—2111no—
164C2×C4⋊C4C2×D21111yescharacteristic
165C2×C4⋊C4C2×D21111yescharacteristic
166C2×C22⋊C4—4221no—
167C2×C22⋊C4—4221no—
168C2×C42—4221no—
169#102—4222no—
170#102—4222no—
171C22⋊C8—4222no—
172C24—4221no—
173C24—2111no—
174C2×D4—8442no—
175C2×D4—8442no—
176C2×D4—4224no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—4221no—
180C2×D4—4221no—
181C2×D4—4221no—
182C2×D4—2112no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—4221no—
186C22×C4—4221no—
187C22×C4—4221no—
188C22×C4—4221no—
189C22×C4—4221no—
190C22×C4—2111no—
191C22×C4—2112no—
192C22×C4—2112no—
193C22×C4C22×D21111yescharacteristic
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—4221no—
198C2×C8—4221no—
199C2×C8—4221no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—4221no—
203C4⋊C4—2112no—
204C4⋊C4C4×D21112yes—
205C4⋊C4D84412yescomplemented
206C22⋊C4—4224no—
207C22⋊C4—4224no—
208C42—4224no—
209C23—8441no—
210C23—8441no—
211C23—8441no—
212C23—8441no—
213C23—8441no—
214C23—8441no—
215C23—4222no—
216C23—4221no—
217C23—4221no—
218C23—4221no—
219C23—4221no—
220C23—4221no—
221C23—4221no—
222C23—2112no—
223C23D4×D21111yescharacteristic
224D4—8448no—
225D4—8441no—
226D4—8441no—
227D4—4222no—
228D4—4222no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—4222no—
238C2×C4—4221no—
239C2×C4—4221no—
240C2×C4—4222no—
241C2×C4—4222no—
242C2×C4—4222no—
243C2×C4—4222no—
244C2×C4—2112no—
245C2×C4C2×C4×D21111yescharacteristic
246C2×C4D4×D21111yescharacteristic
247C2×C4D4×D21111yescharacteristic
248C2×C4C2×D84411yescharacteristic
249C2×C4#1680996211yescharacteristic
250C2×C4#1680996311yescharacteristic
251C8—41no—
252C8—8441no—
253C22—8441no—
254C22—8441no—
255C22—8441no—
256C22—8441no—
257C22—8442no—
258C22—8442no—
259C22—8442no—
260C22—16881no—
261C22—16881no—
262C22—16881no—
263C22—16881no—
264C22—4222no—
265C22—4222no—
266C22—4222no—
267C22C22⋊C4×D21111yescharacteristic
268C22#2026994711yescharacteristic
269C22#2026996211yescharacteristic
270C22#2026999811yescharacteristic
271C22#2027000011yescharacteristic
272C22#2027000111yescharacteristic
273C22#2027000411yescharacteristic
274C4—42no—
275C4—42no—
276C4—8442no—
277C4—4222no—
278C4#2026993911yescharacteristic
279C4C4×D84411yescharacteristic
280C4#2026995011yescharacteristic
281C4#2026995211yescharacteristic
282C2—8442no—
283C2—16882no—
284C2#2721302411yescharacteristic
285C2#2721314111yescharacteristic
286C2#2721314911yescharacteristic
287C2#2721329211yescharacteristic
288C2#2721329411yescharacteristic
289C2#2721347211yescharacteristic
290C2#2721347311yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#340×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269952C2
cosocleC22×C4×C211C23
hypercenter#340D211
hypocenterC211#1875
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#20269935C2semidirect
3. C2×#20269899C2semidirect
4. C2×#20269897C2semidirect
5. C2×(C2×D211)⋊C8C2semidirect
6. #27212908C2semidirect
7. #340×C211C2semidirect
13. #20269952C4semidirect
49. C4⋊Dic211D4semidirect
62. C4⋊C4×C211D4semidirect
145. C211#1875semidirect
153. #340D211semidirect
205. C4⋊C4D844semidirect

Maximal subgroups

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

subgroupindex
C2×#202699522
C2×#202699352
C2×#202698992
C2×#202698972
C2×(C2×D211)⋊C82
#272129082
#340×C2112
#1875211