Finite Group Databasebeta

← #41014580

Subgroups — #41014580

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×#20269951C211yesmaximal, characteristic
2C2×#20269910C211yesmaximal, characteristic, complemented
3C2×#20269897C211yesmaximal, characteristic, complemented
4C2×#20269897C211yesmaximal, characteristic, complemented
5C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7#380×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C22⋊D211—21no—
10C22×C211⋊C8C2211yescharacteristic
11C22×C211⋊C8C2211yescharacteristic
12#20269951—22no—
13#20269951—22no—
14C2×C4⋊Dic211C2211yescharacteristic
15C2×(C2×D211)⋊C4—21no—
16#20269910—22no—
17#20269897—22no—
18#20269897—22no—
19C22⋊(C211⋊C8)—22no—
20C23×C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C22⋊C4×C211—21no—
24#95×C211—21no—
25C23×D211—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844D412yes—
32C2×C22⋊D211—24no—
33C2×C22⋊D211—42no—
34C2×C22⋊D211—42no—
35C22×Dic211—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8—21no—
41C2×C211⋊C8D412yescomplemented
42C4⋊Dic211—21no—
43C4⋊Dic211D412yes—
44(C2×D211)⋊C4—24no—
45C24×C211D411yescharacteristic
46C22×C4×C211—21no—
47C22×C4×C211—21no—
48C22×C4×C211—21no—
49C22×C4×C211—21no—
50C22×C4×C211—21no—
51C22×C4×C211—21no—
52C22×C4×C211—21no—
53C22×C4×C211—21no—
54C22×C4×C211D411yescharacteristic
55C22×C4×C211C2311yescharacteristic
56C4⋊C4×C211—21no—
57C4⋊C4×C211—21no—
58C4⋊C4×C211—21no—
59C4⋊C4×C211—21no—
60C22⋊C4×C211—42no—
61C22×D211—22no—
62C22×D211—41no—
63C22×D211—41no—
64C22×D211—41no—
65C22×D211—41no—
66C22×D211—41no—
67C22×D211—41no—
68D844—22no—
69D844—22no—
70D844—41no—
71D844—41no—
72C22⋊D211—48no—
73C2×Dic211—22no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C211⋊C8—22no—
77C211⋊C8—41no—
78C23×C211—21no—
79C23×C211—21no—
80C23×C211—21no—
81C23×C211—21no—
82C23×C211—21no—
83C23×C211—21no—
84C23×C211D812yes—
85C23×C211C2×D411yescharacteristic
86C2×C4×C211—21no—
87C2×C4×C211—21no—
88C2×C4×C211—21no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211—41no—
96C2×C4×C211—41no—
97C2×C4×C211—41no—
98C2×C4×C211—41no—
99C2×C4×C211C2×D411yescharacteristic
100C2×C4×C211C2×D411yescharacteristic
101C2×C4×C211C2×D411yescharacteristic
102C2×C4×C211#4111yescharacteristic
103C2×C4×C211#4111yescharacteristic
104C2×C4×C211#4111yescharacteristic
105C22×C211—22no—
106C22×C211—22no—
107C22×C211—22no—
108C22×C211—41no—
109C22×C211—41no—
110C22×C211—41no—
111C22×C211—41no—
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—42no—
120C2×D211—42no—
121C2×D211—42no—
122C2×D211—81no—
123C2×D211—81no—
124C2×D211—81no—
125C2×D211—81no—
126C4×C211—41no—
127C4×C211—81no—
128C4×C211—81no—
129C4×C211#12111yescharacteristic
130C4×C211#12111yescharacteristic
131C4×C211#12311yescharacteristic
132C4×C211#12411yescharacteristic
133Dic211—42no—
134C2×C211—42no—
135C2×C211#39411yescharacteristic
136C2×C211#44711yescharacteristic
137C2×C211#45211yescharacteristic
138C2×C211#46611yescharacteristic
139C2×C211#46811yescharacteristic
140C2×C211#48011yescharacteristic
141C2×C211#48211yescharacteristic
142D211—82no—
143C211#203311yescharacteristic, complemented, Sylow
144#2033—2111nomaximal, Sylow
145C2×#121—2111no—
146C2×#107—2111no—
147C2×#102—2111no—
148C2×#102—2111no—
149C2×C22⋊C8—2111no—
150#380D21111yescharacteristic, complemented
151#340—2111no—
152C22×D4—4221no—
153C22×D4—2111no—
154C23×C4C2×D21111yescharacteristic
155C22×C8—2111no—
156C22×C8—2111no—
157#121—4222no—
158#121—4222no—
159C2×C4⋊C4—2111no—
160C2×C4⋊C4C2×D21111yescharacteristic
161C2×C4⋊C4C2×D21111yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—4221no—
164#107—4222no—
165#102—4222no—
166#102—4222no—
167C22⋊C8—4222no—
168#95—21no—
169C24—4221no—
170C24C22⋊D21111yescharacteristic
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—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—4221no—
189C22×C4C22⋊D21111yescharacteristic
190C22×C4C22×D21111yescharacteristic
191C2×C8—4221no—
192C2×C8—4221no—
193C2×C8—4221no—
194C2×C8—4221no—
195C2×C8—4221no—
196C2×C8—2112no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—4221no—
202C4⋊C4—2112no—
203C22⋊C4—42no—
204C22⋊C4—4224no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—8441no—
212C23—8441no—
213C23—8441no—
214C23—8441no—
215C23—8441no—
216C23—8441no—
217C23—4222no—
218C23#1680994712yes—
219C23C2×C22⋊D21111yescharacteristic
220D4—8448no—
221D4—8441no—
222D4—8441no—
223D4—4222no—
224D4—4222no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—8441no—
239C2×C4—8441no—
240C2×C4—4222no—
241C2×C4D4×D21111yescharacteristic
242C2×C4D4×D21111yescharacteristic
243C2×C4D4×D21111yescharacteristic
244C2×C4#1680996111yescharacteristic
245C2×C4#1680996211yescharacteristic
246C2×C4#1680996211yescharacteristic
247C8—8441no—
248C8—4222no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—8442no—
257C22—8442no—
258C22—8442no—
259C22—16881no—
260C22—16881no—
261C22—16881no—
262C22—16881no—
263C22#2026994511yescharacteristic
264C22#2026994911yescharacteristic
265C22#2026996011yescharacteristic
266C22C2×#1680994711yescharacteristic
267C22#2026999711yescharacteristic
268C22#2027000211yescharacteristic
269C22#2027000811yescharacteristic
270C4—41no—
271C4—81no—
272C4—81no—
273C4—8442no—
274C4#2026995011yescharacteristic
275C4#2026995011yescharacteristic
276C4#2026995911yescharacteristic
277C4#2026996311yescharacteristic
278C2—42no—
279C2—16882no—
280C2#2721306111yescharacteristic
281C2#2721327911yescharacteristic
282C2#2721330611yescharacteristic
283C2#2721338011yescharacteristic
284C2#2721339411yescharacteristic
285C2#2721345911yescharacteristic
286C2#2721347011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#380×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269951C2
hypercenter#380D211
hypocenterC211#2033
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20269910C2semidirect
3. C2×#20269897C2semidirect
4. C2×#20269897C2semidirect
5. C2×C22⋊(C211⋊C8)C2semidirect
6. #27212908C2semidirect
7. #380×C211C2semidirect
41. C2×C211⋊C8D4semidirect
143. C211#2033semidirect
150. #380D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699512
C2×#202699102
C2×#202698972
C2×C22⋊(C211⋊C8)2
#272129082
#380×C2112
#2033211