Finite Group Databasebeta

← #41014578

Subgroups — #41014578

Order 27008.

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

normal subgroups only

This group has 368 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×#20269948C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269898C211yesmaximal, characteristic, complemented
4C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
5#27213003C211yesmaximal, characteristic, complemented
6#27212909C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic, complemented
8C22×C22⋊D211—21no—
9C22×C4×D211C2211yescharacteristic
10C22×C211⋊C8C2211yescharacteristic
11#20269948—22no—
12#20269948—22no—
13C2×C4⋊Dic211C2211yescharacteristic
14C2×C4⋊Dic211C2211yescharacteristic
15C2×C22⋊Dic211—21no—
16C2×(C2×D211)⋊C4—21no—
17#20269910—22no—
18#20269898—22no—
19(C2×D211)⋊C8—22no—
20C211⋊#95—21no—
21C22×D4×C211C2211yescharacteristic
22C22×C8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#102×C211—22no—
25C23×D211D411yescharacteristic
26C2×C22⋊D211—24no—
27C2×C22⋊D211—42no—
28C2×C22⋊D211—42no—
29C22×Dic211—21no—
30C22×Dic211—21no—
31C22×Dic211D411yescharacteristic
32C2×C4×D211—21no—
33C2×C4×D211—21no—
34C2×C4×D211—21no—
35C2×C4×D211—21no—
36C2×C4×D211—21no—
37C2×C4×D211—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8D412yescomplemented
41C4⋊Dic211—21no—
42C4⋊Dic211—21no—
43C4⋊Dic211—21no—
44C4⋊Dic211D412yes—
45C22⋊Dic211—24no—
46(C2×D211)⋊C4—42no—
47C24×C211—21no—
48C2×D4×C211—21no—
49C2×D4×C211—21no—
50C2×D4×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211D412yes—
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—
61C22×D211—21no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—21no—
67C22×D211D812yes—
68C22⋊D211—48no—
69C2×Dic211—21no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C4×D211—41no—
79C4×D211—41no—
80C4×D211—41no—
81C4×D211—41no—
82C211⋊C8—22no—
83C23×C211—22no—
84C23×C211—41no—
85C23×C211—41no—
86C23×C211—41no—
87C23×C211—41no—
88C23×C211—41no—
89C23×C211—41no—
90C23×C211C2×D411yescharacteristic
91D4×C211—22no—
92D4×C211—22no—
93D4×C211—41no—
94D4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211#4111yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C8×C211—41no—
105C22×C211—42no—
106C22×C211—42no—
107C22×C211—42no—
108C22×C211—81no—
109C22×C211—81no—
110C22×C211—81no—
111C22×C211—81no—
112C22×C211#12011yescharacteristic
113C22×C211#12111yescharacteristic
114C22×C211#12311yescharacteristic
115C22×C211C2×D811yescharacteristic
116C22×C211#13511yescharacteristic
117C22×C211AGL(1,ℤ/8)11yescharacteristic
118C22×C211#13711yescharacteristic
119C2×D211—22no—
120C2×D211—22no—
121C2×D211—22no—
122C2×D211—41no—
123C2×D211—41no—
124C2×D211—41no—
125C2×D211—41no—
126C4×C211—81no—
127C4×C211#12111yescharacteristic
128C4×C211#12111yescharacteristic
129C4×C211#12311yescharacteristic
130C4×C211#12411yescharacteristic
131Dic211—41no—
132Dic211—42no—
133Dic211—81no—
134C2×C211—82no—
135C2×C211#39411yescharacteristic
136C2×C211#44711yescharacteristic
137C2×C211#45211yescharacteristic
138C2×C211#46611yescharacteristic
139C2×C211#46811yescharacteristic
140C2×C211#48011yescharacteristic
141C2×C211#48211yescharacteristic
142D211—42no—
143C211#203311yescharacteristic, complemented, Sylow
144#2033—2111nomaximal, Sylow
145C2×#121—2111no—
146C2×#107—2111no—
147C2×#102—2111no—
148C2×#102D21111yescharacteristic, complemented
149C2×C22⋊C8—2111no—
150#380—2111no—
151#340—2111no—
152C22×D4—4221no—
153C22×D4C2×D21111yescharacteristic
154C23×C4—2111no—
155C22×C8—2111no—
156C22×C8C2×D21111yescharacteristic
157#121—4222no—
158#121—4222no—
159C2×C4⋊C4—2111no—
160C2×C4⋊C4—2111no—
161C2×C4⋊C4C2×D21111yescharacteristic
162C2×C22⋊C4—4221no—
163C2×C22⋊C4—4221no—
164#107—4222no—
165#102—22no—
166#102—4222no—
167C22⋊C8—4222no—
168#95—4221no—
169C24—21no—
170C24—2111no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—8442no—
177C2×D4—8442no—
178C2×D4—4224no—
179C2×D4C22⋊D21112yes—
180C22×C4—21no—
181C22×C4—4221no—
182C22×C4—4221no—
183C22×C4—4221no—
184C22×C4—4221no—
185C22×C4—4221no—
186C22×C4—4221no—
187C22×C4—4221no—
188C22×C4—4221no—
189C22×C4—2111no—
190C22×C4C22×D21111yescharacteristic
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8—2112no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—4221no—
200C4⋊C4—4221no—
201C4⋊C4—4221no—
202C4⋊C4—2112no—
203C22⋊C4—8442no—
204C22⋊C4—4224no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—4221no—
213C23—4221no—
214C23—4221no—
215C23—4221no—
216C23—4221no—
217C23—4221no—
218C23—2112no—
219C23D4×D21111yescharacteristic
220D4—22no—
221D4—22no—
222D4—41no—
223D4—41no—
224D4—8448no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—8441no—
229C2×C4—8441no—
230C2×C4—8441no—
231C2×C4—8441no—
232C2×C4—8441no—
233C2×C4—8441no—
234C2×C4—8441no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—4221no—
238C2×C4—4222no—
239C2×C4—4221no—
240C2×C4—4221no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4C2×C22⋊D21111yescharacteristic
244C2×C4#1680996111yescharacteristic
245C2×C4#1680996211yescharacteristic
246C2×C4#1680996211yescharacteristic
247C8—41no—
248C8—4222no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—8441no—
257C22—8441no—
258C22—8441no—
259C22—8441no—
260C22—4222no—
261C22—4222no—
262C22—4222no—
263C22#2026994511yescharacteristic
264C22#2026995011yescharacteristic
265C22#2026995911yescharacteristic
266C22D8×D21111yescharacteristic
267C22#2026999811yescharacteristic
268C22#2027000111yescharacteristic
269C22#2027000611yescharacteristic
270C4—81no—
271C4—8441no—
272C4—8442no—
273C4—16881no—
274C4#2026994911yescharacteristic
275C4#2026995011yescharacteristic
276C4#2026996011yescharacteristic
277C4#2026996311yescharacteristic
278C2—82no—
279C2—8442no—
280C2#2721306111yescharacteristic
281C2#2721327811yescharacteristic
282C2#2721330511yescharacteristic
283C2#2721338111yescharacteristic
284C2#2721339111yescharacteristic
285C2#2721345811yescharacteristic
286C2#2721346911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269948C2
hypercenterC2×#102D211
hypocenterC211#2033
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×#20269910C2semidirect
3. C2×#20269898C2semidirect
4. C2×(C2×D211)⋊C8C2semidirect
5. #27213003C2semidirect
6. #27212909C2semidirect
7. C2×#102×C211C2semidirect
40. C2×C211⋊C8D4semidirect
143. C211#2033semidirect
148. C2×#102D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699482
C2×#202699102
C2×#202698982
C2×(C2×D211)⋊C82
#272130032
#272129092
C2×#102×C2112
#2033211