Finite Group Databasebeta

← #41014041

Subgroups — #41014041

Order 27008.

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

normal subgroups only

This group has 254 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×#20269911C211yesmaximal, characteristic
2C2×#20269901C211yesmaximal, characteristic
3#27213053C211yesmaximal, characteristic
4C4⋊C4×Dic211C211yesmaximal, characteristic
5#27212908C211yesmaximal, characteristic
6#27212900C211yesmaximal, characteristic
7C2×#103×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C2×C4⋊Dic211C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211—21no—
12C2×C4×Dic211C2211yescharacteristic
13#20269911—22no—
14#20269901—22no—
15C211⋊#95—21no—
16C211⋊#95—21no—
17C211⋊#95—21no—
18C22×Q8×C211C2211yescharacteristic
19C22×C8×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21#103×C211C414yescomplemented
22C22×Dic211—21no—
23C22×Dic211—21no—
24C22×Dic211D411yescharacteristic
25C22×Dic211D411yescharacteristic
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211D412yes—
33C4×Dic211—21no—
34C4×Dic211—21no—
35C4×Dic211—42no—
36C2×Q8×C211—21no—
37C2×Q8×C211—21no—
38C2×Q8×C211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×Q8×C211C2×C412yes—
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C2×C8×C211C2×C412yes—
48C4⋊C4×C211—21no—
49C4⋊C4×C211C2×C412yes—
50C2×Dic211—21no—
51C2×Dic211—21no—
52C2×Dic211—22no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—41no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C2×Dic211—41no—
60C2×Dic211—41no—
61C2×Dic211Q1612yes—
62C211⋊C8—41no—
63C23×C211C2×D411yescharacteristic
64Q8×C211—22no—
65Q8×C211—22no—
66Q8×C211—41no—
67Q8×C211—41no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211C22×C411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4111yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C2×C4×C211#4111yescharacteristic
80C8×C211—22no—
81C22×C211C4×D411yescharacteristic
82C22×C211#12111yescharacteristic
83C22×C211#12311yescharacteristic
84C22×C211C2×Q1611yescharacteristic
85C22×C211#13511yescharacteristic
86C22×C211AGL(1,ℤ/8)11yescharacteristic
87C22×C211#13711yescharacteristic
88C4×C211—42no—
89C4×C211—42no—
90C4×C211#11711yescharacteristic
91C4×C211C4×D411yescharacteristic
92C4×C211#12411yescharacteristic
93C4×C211#12611yescharacteristic
94Dic211—22no—
95Dic211—42no—
96Dic211—41no—
97Dic211—81no—
98C2×C211#38811yescharacteristic
99C2×C211C4×Q1611yescharacteristic
100C2×C211#44011yescharacteristic
101C2×C211#46211yescharacteristic
102C2×C211#46311yescharacteristic
103C2×C211#48411yescharacteristic
104C2×C211#48511yescharacteristic
105C211#192811yescharacteristic, complemented, Sylow
106#1928—2111nomaximal, Sylow
107C2×#107—2111no—
108C2×#103—2111no—
109C2×#103D21111yescharacteristic
110#391—2111no—
111C4×C4⋊C4—2111no—
112#340—2111no—
113#336—2111no—
114C22×Q8C2×D21111yescharacteristic
115C22×C8—2111no—
116C22×C8C2×D21111yescharacteristic
117C2×C4⋊C4—2111no—
118C2×C4⋊C4—2111no—
119C2×C4⋊C4C2×D21111yescharacteristic
120C2×C42—4221no—
121C2×C42—2111no—
122#107—4222no—
123#103—4222no—
124#103Dic21114yescomplemented
125#95—4221no—
126#95—4221no—
127#95—4221no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C2×Dic21112yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—4221no—
137C22×C4—4221no—
138C22×C4—2111no—
139C22×C4—2111no—
140C22×C4C22×D21111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—4221no—
146C2×C8C2×Dic21112yes—
147C4⋊C4—21no—
148C4⋊C4—4221no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—2112no—
152C4⋊C4C2×Dic21112yes—
153C42—8442no—
154C42—4221no—
155C42—4221no—
156C23D4×D21111yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8441no—
168C2×C4—8441no—
169C2×C4—8441no—
170C2×C4—8441no—
171C2×C4—8441no—
172C2×C4—4222no—
173C2×C4—4221no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4—2112no—
179C2×C4C22×Dic21111yescharacteristic
180C2×C4D4×D21111yescharacteristic
181C2×C4#1680996111yescharacteristic
182C2×C4#1680996111yescharacteristic
183C2×C4#1680996211yescharacteristic
184C2×C4#1680996211yescharacteristic
185C8—22no—
186C8—8441no—
187C22D4×Dic21111yescharacteristic
188C22#2026995011yescharacteristic
189C22#2026995911yescharacteristic
190C22Q16×D21111yescharacteristic
191C22#2027000011yescharacteristic
192C22#2027000411yescharacteristic
193C22#2027000611yescharacteristic
194C4—42no—
195C4—42no—
196C4—8442no—
197C4—8441no—
198C4—16881no—
199C4—4222no—
200C4#2026993611yescharacteristic
201C4D4×Dic21111yescharacteristic
202C4#2026996311yescharacteristic
203C4#2026997011yescharacteristic
204C2#2721303911yescharacteristic
205C2Q16×Dic21111yescharacteristic
206C2#2721324711yescharacteristic
207C2#2721335511yescharacteristic
208C2#2721336411yescharacteristic
209C2#2721348011yescharacteristic
210C2#2721348611yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#103×C211C2
socle, prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC2×#103D211
hypocenterC211#1928
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
21. #103×C211C4semidirect
105. C211#1928semidirect
124. #103Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699112
C2×#202699012
#272130532
C4⋊C4×Dic2112
#272129082
#272129002
C2×#103×C2112
#1928211