Finite Group Databasebeta

← #41014496

Subgroups — #41014496

Order 27008.

370 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×#16809952C211yesmaximal, characteristic
2C2×#20269975C211yesmaximal, characteristic
3C2×#20269902C211yesmaximal, characteristic
4C2×#20269901C211yesmaximal, characteristic
5#27213053C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C22×Q16×C211C211yesmaximal, characteristic
8C22×Dic422C2211yescharacteristic
9C2×#16809952—22no—
10C2×#16809952—22no—
11C2×#16809952—22no—
12C22×C211⋊C8C2211yescharacteristic
13#20269975—22no—
14#20269975—22no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4—22no—
17C2×C4×Dic211C2211yescharacteristic
18#20269902—22no—
19#20269901—22no—
20C211⋊#95—22no—
21C22×Q8×C211C2211yescharacteristic
22C22×Q8×C211C2211yescharacteristic
23C2×Q16×C211—22no—
24C2×Q16×C211—22no—
25C2×Q16×C211—22no—
26C22×C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422D412yes—
33C22×Dic211—21no—
34C22×Dic211—21no—
35C22×Dic211D412yes—
36#16809952—28no—
37C2×C211⋊C8—21no—
38C2×C211⋊C8—21no—
39C2×C211⋊C8D412yes—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42Dic211⋊C4—28no—
43C4×Dic211—21no—
44C4×Dic211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211—21no—
50C2×Q8×C211—21no—
51C2×Q8×C211—21no—
52C2×Q8×C211—21no—
53C2×Q8×C211—21no—
54C2×Q8×C211—21no—
55C2×Q8×C211D412yes—
56C2×Q8×C211D412yes—
57C22×C4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211C2311yescharacteristic
60Q16×C211—28no—
61C2×C8×C211—21no—
62C2×C8×C211—21no—
63C2×C8×C211D412yes—
64Dic422—22no—
65Dic422—22no—
66Dic422—41no—
67Dic422—41no—
68C2×Dic211—22no—
69C2×Dic211—22no—
70C2×Dic211—22no—
71C2×Dic211—41no—
72C2×Dic211—41no—
73C2×Dic211—41no—
74C2×Dic211—42no—
75C2×Dic211Q1614yescomplemented
76C211⋊C8—22no—
77C23×C211C2×D411yescharacteristic
78Q8×C211—22no—
79Q8×C211—22no—
80Q8×C211—22no—
81Q8×C211—22no—
82Q8×C211—41no—
83Q8×C211—41no—
84Q8×C211—41no—
85Q8×C211—41no—
86C2×C4×C211—22no—
87C2×C4×C211—22no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211C2×D411yescharacteristic
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C8×C211—22no—
99C22×C211#12011yescharacteristic
100C22×C211#12111yescharacteristic
101C22×C211#12711yescharacteristic
102C22×C211C2×Q1612yes—
103C22×C211#13712yes—
104C4×C211—42no—
105C4×C211—42no—
106C4×C211#12011yescharacteristic
107C4×C211#12011yescharacteristic
108C4×C211#12111yescharacteristic
109C4×C211#12111yescharacteristic
110Dic211—24no—
111Dic211—42no—
112Dic211—81no—
113C2×C211#39211yescharacteristic
114C2×C211#45112yes—
115C2×C211#46212yes—
116C2×C211#49411yescharacteristic
117C2×C211#49711yescharacteristic
118C211#201611yescharacteristic, complemented, Sylow
119#2016—2111nomaximal, Sylow
120C22×Q16—2111no—
121C22×Q16D21111yescharacteristic
122C2×C4⋊Q8—2111no—
123C2×#103—2111no—
124C2×#103—2111no—
125#391—2111no—
126#336—2111no—
127C22×Q8—2111no—
128C22×Q8C2×D21111yescharacteristic
129C22×Q8C2×D21111yescharacteristic
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—4222no—
134C2×Q16—4222no—
135C2×Q16—4222no—
136C22×C8—2111no—
137C22×C8C2×D21111yescharacteristic
138C4⋊Q8—4222no—
139C4⋊Q8—4222no—
140C2×C4⋊C4—4222no—
141C2×C4⋊C4—2111no—
142C2×C42—2111no—
143#103—4222no—
144#103—4222no—
145#95—4222no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—4221no—
157C2×Q8—4221no—
158C2×Q8—4221no—
159C2×Q8—4221no—
160C2×Q8—4221no—
161C2×Q8—2112no—
162C2×Q8C22⋊D21112yes—
163C2×Q8C22⋊D21112yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—4221no—
167C22×C4—4221no—
168C22×C4—2112no—
169C22×C4C22×D21111yescharacteristic
170Q16—28no—
171Q16—4228no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—4221no—
175C2×C8—4221no—
176C2×C8—2112no—
177C2×C8C22⋊D21112yes—
178C4⋊C4—4228no—
179C4⋊C4—4221no—
180C4⋊C4—4221no—
181C42—4221no—
182C42—4221no—
183C23D4×D21111yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—41no—
192Q8—8441no—
193Q8—8441no—
194Q8—4222no—
195Q8—4222no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—8441no—
205C2×C4—8442no—
206C2×C4—4222no—
207C2×C4—4222no—
208C2×C4—4222no—
209C2×C4—2114no—
210C2×C4D4×D21111yescharacteristic
211C2×C4D4×D21111yescharacteristic
212C2×C4C2×C22⋊D21111yescharacteristic
213C2×C4C2×C22⋊D21111yescharacteristic
214C2×C4C2×C22⋊D21111yescharacteristic
215C2×C4#1680996111yescharacteristic
216C8—22no—
217C8—4222no—
218C22#2026994511yescharacteristic
219C22#2026994911yescharacteristic
220C22#2026997311yescharacteristic
221C22Q16×D21112yes—
222C22#2027000912yes—
223C4—42no—
224C4—42no—
225C4—8442no—
226C4—16881no—
227C4—4224no—
228C4#2026994511yescharacteristic
229C4#2026994611yescharacteristic
230C4#2026994811yescharacteristic
231C4#2026994911yescharacteristic
232C2#2721305411yescharacteristic
233C2#2721330012yes—
234C2#2721335412yes—
235C2#2721352411yescharacteristic
236C2#2721353911yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC22×Q16×C211C2
socle, prime radicalC23×C211C2×D4
hypercenterC22×Q16D211
hypocenterC211#2016
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
75. C2×Dic211Q16semidirect
118. C211#2016semidirect

Maximal subgroups

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

subgroupindex
C22×#168099522
C2×#202699752
C2×#202699022
C2×#202699012
#272130532
#272129002
C22×Q16×C2112
#2016211