Finite Group Databasebeta

← #41014582

Subgroups — #41014582

Order 27008.

368 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20269899C211yesmaximal, characteristic, complemented
3C2×#20269897C211yesmaximal, characteristic, complemented
4C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
5#27213003C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#107×C211C211yesmaximal, characteristic, complemented
8C22×D844—21no—
9C22×D844C2211yescharacteristic
10C22×C4×D211C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269952—22no—
13#20269952—22no—
14C2×C4⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17#20269899—22no—
18#20269897—22no—
19(C2×D211)⋊C8—22no—
20C211⋊#95—21no—
21C22×C8×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#107×C211—22no—
25C23×D211—21no—
26C23×D211D411yescharacteristic
27C2×D844—24no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—42no—
34C2×D844—42no—
35C2×D844D412yes—
36C22×Dic211—21no—
37C22×Dic211D411yescharacteristic
38C2×C4×D211—21no—
39C2×C4×D211—21no—
40C2×C4×D211—21no—
41C2×C4×D211—21no—
42C2×C4×D211—21no—
43C2×C4×D211—21no—
44C2×C211⋊C8—21no—
45C2×C211⋊C8—21no—
46C2×C211⋊C8—21no—
47C4⋊Dic211—21no—
48C4⋊Dic211—21no—
49(C2×D211)⋊C4—24no—
50(C2×D211)⋊C4—42no—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C2×C8×C211D412yescomplemented
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211—21no—
60C4⋊C4×C211D412yes—
61C22×D211—21no—
62C22×D211—21no—
63C22×D211—21no—
64C22×D211—21no—
65C22×D211—21no—
66C22×D211—21no—
67C22×D211—22no—
68C22×D211—41no—
69C22×D211—41no—
70C22×D211—41no—
71C22×D211—41no—
72C22×D211—41no—
73C22×D211—41no—
74C22×D211D812yes—
75D844—22no—
76D844—22no—
77D844—48no—
78D844—41no—
79D844—41no—
80C2×Dic211—21no—
81C2×Dic211—21no—
82C2×Dic211—21no—
83C2×Dic211—41no—
84C2×Dic211—41no—
85C2×Dic211—41no—
86C4×D211—41no—
87C4×D211—41no—
88C4×D211—41no—
89C4×D211—41no—
90C211⋊C8—41no—
91C23×C211C2×D411yescharacteristic
92C2×C4×C211—22no—
93C2×C4×C211—41no—
94C2×C4×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—22no—
105C22×C211#12011yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12311yescharacteristic
108C22×C211C2×D811yescharacteristic
109C22×C211#13511yescharacteristic
110C22×C211AGL(1,ℤ/8)11yescharacteristic
111C22×C211#13711yescharacteristic
112C2×D211—22no—
113C2×D211—22no—
114C2×D211—22no—
115C2×D211—41no—
116C2×D211—41no—
117C2×D211—41no—
118C2×D211—41no—
119C2×D211—42no—
120C2×D211—42no—
121C2×D211—42no—
122C2×D211—81no—
123C2×D211—81no—
124C2×D211—81no—
125C2×D211—81no—
126C4×C211—42no—
127C4×C211—81no—
128C4×C211#12111yescharacteristic
129C4×C211#12111yescharacteristic
130C4×C211#12311yescharacteristic
131C4×C211#12411yescharacteristic
132Dic211—41no—
133Dic211—81no—
134C2×C211#39411yescharacteristic
135C2×C211#44711yescharacteristic
136C2×C211#45211yescharacteristic
137C2×C211#46611yescharacteristic
138C2×C211#46811yescharacteristic
139C2×C211#48011yescharacteristic
140C2×C211#48211yescharacteristic
141D211—42no—
142D211—82no—
143C211#203311yescharacteristic, complemented, Sylow
144#2033—2111nomaximal, Sylow
145C2×#121—2111no—
146C2×#107D21111yescharacteristic, complemented
147C2×#102—2111no—
148C2×#102—2111no—
149C2×C22⋊C8—2111no—
150#380—2111no—
151#340—2111no—
152C22×D4—4221no—
153C22×D4—2111no—
154C23×C4—2111no—
155C22×C8—2111no—
156C22×C8C2×D21111yescharacteristic
157#121—4222no—
158#121—4222no—
159C2×C4⋊C4—2111no—
160C2×C4⋊C4C2×D21111yescharacteristic
161C2×C4⋊C4C2×D21111yescharacteristic
162C2×C22⋊C4—4221no—
163C2×C22⋊C4—4221no—
164#107—22no—
165#102—4222no—
166#102—4222no—
167C22⋊C8—4222no—
168#95—4221no—
169C24—4221no—
170C24—2111no—
171C2×D4—8442no—
172C2×D4—8442no—
173C2×D4—4224no—
174C2×D4—4221no—
175C2×D4—4221no—
176C2×D4—4221no—
177C2×D4—4221no—
178C2×D4—4221no—
179C2×D4—2112no—
180C22×C4—21no—
181C22×C4—21no—
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—4221no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8D84412yescomplemented
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—4221no—
201C4⋊C4—4221no—
202C4⋊C4C22⋊D21112yes—
203C22⋊C4—8442no—
204C22⋊C4—4224no—
205C23—8441no—
206C23—8441no—
207C23—8441no—
208C23—8441no—
209C23—8441no—
210C23—8441no—
211C23—4222no—
212C23—4221no—
213C23—4221no—
214C23—4221no—
215C23—4221no—
216C23—4221no—
217C23—4221no—
218C23—2112no—
219C23D4×D21111yescharacteristic
220D4—8448no—
221D4—8441no—
222D4—8441no—
223D4—4222no—
224D4—4222no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—8441no—
232C2×C4—8441no—
233C2×C4—8441no—
234C2×C4—8441no—
235C2×C4—8441no—
236C2×C4—8441no—
237C2×C4—8441no—
238C2×C4—4221no—
239C2×C4—4221no—
240C2×C4—4221no—
241C2×C4D4×D21111yescharacteristic
242C2×C4C2×C22⋊D21111yescharacteristic
243C2×C4C2×D84411yescharacteristic
244C2×C4#1680996211yescharacteristic
245C2×C4#1680996311yescharacteristic
246C2×C4#1680996311yescharacteristic
247C8—22no—
248C8—8441no—
249C22—8441no—
250C22—8441no—
251C22—8441no—
252C22—8441no—
253C22—8442no—
254C22—8442no—
255C22—8442no—
256C22—16881no—
257C22—16881no—
258C22—16881no—
259C22—16881no—
260C22—4222no—
261C22—4222no—
262C22—4222no—
263C22#2026994511yescharacteristic
264C22#2026995211yescharacteristic
265C22#2026996211yescharacteristic
266C22D8×D21111yescharacteristic
267C22#2027000011yescharacteristic
268C22#2027000111yescharacteristic
269C22#2027000911yescharacteristic
270C4—42no—
271C4—81no—
272C4—8441no—
273C4—16881no—
274C4#2026995111yescharacteristic
275C4#2026995211yescharacteristic
276C4#2026996211yescharacteristic
277C4#2026996611yescharacteristic
278C2—8442no—
279C2—16882no—
280C2#2721306311yescharacteristic
281C2#2721327811yescharacteristic
282C2#2721330711yescharacteristic
283C2#2721338311yescharacteristic
284C2#2721339311yescharacteristic
285C2#2721346111yescharacteristic
286C2#2721347311yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#107×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269952C2
hypercenterC2×#107D211
hypocenterC211#2033
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#20269899C2semidirect
3. C2×#20269897C2semidirect
4. C2×(C2×D211)⋊C8C2semidirect
5. #27213003C2semidirect
6. #27212908C2semidirect
7. C2×#107×C211C2semidirect
56. C2×C8×C211D4semidirect
143. C211#2033semidirect
146. C2×#107D211semidirect
196. C2×C8D844semidirect

Maximal subgroups

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

subgroupindex
C2×#202699522
C2×#202698992
C2×#202698972
C2×(C2×D211)⋊C82
#272130032
#272129082
C2×#107×C2112
#2033211