Finite Group Databasebeta

← #41013451

Subgroups — #41013451

Order 27008.

256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20269985C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213182C211yesmaximal, characteristic, complemented
4#27213164C211yesmaximal, characteristic
5#27213162C211yesmaximal, characteristic
6#27212918C211yesmaximal, characteristic, complemented
7#344×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9#20269985—21no—
10#20269985—21no—
11#20269985C412yescomplemented
12#20269985C412yescomplemented
13C2×#16809946C2211yescharacteristic
14C2×#16809945—21no—
15C22×C211⋊C8C2211yescharacteristic
16C2×C8×D211—21no—
17#20269936C2211yescharacteristic
18#20269912—21no—
19#20269912—21no—
20#20269910—21no—
21#20269908—21no—
22#20269905—22no—
23#20269903—22no—
24C2×M16×C211C2211yescharacteristic
25C2×M16×C211C2211yescharacteristic
26#117×C211C2211yescharacteristic
27#16809962—21no—
28#16809962—21no—
29#16809962C2×C412yes—
30#16809962D412yes—
31C2×Dic422Q811yescharacteristic
32C2×D844Q811yescharacteristic
33C2×C22⋊D211—21no—
34C2×C4×D211—21no—
35#16809946—21no—
36#16809946—21no—
37#16809945—22no—
38#16809945—22no—
39C2×C211⋊C8—21no—
40C2×C211⋊C8C2×C411yescharacteristic
41C2×C211⋊C8C2×C411yescharacteristic
42C2×C211⋊C8D411yescharacteristic
43C2×C211⋊C8D411yescharacteristic
44C2×C211⋊C8D411yescharacteristic
45C2×C211⋊C8D411yescharacteristic
46C8×D211—22no—
47C8×D211—22no—
48C4⋊Dic211—21no—
49C22⋊Dic211—21no—
50C4×Dic211—21no—
51C22×C4×C211C2311yescharacteristic
52M16×C211—21no—
53M16×C211—21no—
54M16×C211—21no—
55M16×C211C2×C412yes—
56C2×C8×C211—21no—
57C2×C8×C211—21no—
58C4⋊C4×C211—21no—
59C22⋊C4×C211—21no—
60C42×C211—21no—
61C22×D211—21no—
62Dic422—21no—
63Dic422C4⋊C412yes—
64D844—21no—
65D844C4⋊C412yes—
66C22⋊D211—22no—
67C22⋊D211—22no—
68C2×Dic211—21no—
69C2×Dic211—41no—
70C4×D211—22no—
71C4×D211—22no—
72C211⋊C8—41no—
73C211⋊C8C22⋊C411yescharacteristic
74C211⋊C8C22⋊C411yescharacteristic
75C211⋊C8C22⋊C411yescharacteristic
76C211⋊C8C22⋊C411yescharacteristic
77C23×C211#4111yescharacteristic
78C2×C4×C211—41no—
79C2×C4×C211C22×C411yescharacteristic
80C2×C4×C211C2×D411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×Q811yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C8×C211—22no—
86C8×C211—41no—
87C22×C211—41no—
88C22×C211#11711yescharacteristic
89C22×C211#12111yescharacteristic
90C22×C211C22⋊Q811yescharacteristic
91C2×D211—22no—
92C2×D211—42no—
93C4×C211—81no—
94C4×C211C2×C22⋊C411yescharacteristic
95C4×C211C2×C4⋊C411yescharacteristic
96C4×C211#12111yescharacteristic
97C4×C211C22⋊Q811yescharacteristic
98Dic211—22no—
99Dic211—81no—
100C2×C211—21no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211#38011yescharacteristic
104D211—42no—
105C211#181411yescharacteristic, complemented, Sylow
106#1814—2111nomaximal, Sylow
107C2×#131—2111no—
108C2×#108—2111no—
109#427—2111no—
110#421—2111no—
111#421—2111no—
112#344—2111no—
113#344D21111yescharacteristic, complemented
114C2×#41—2111no—
115#131—4221no—
116#131—4221no—
117#131—2112no—
118#131—2112no—
119C2×M16—4221no—
120C2×M16—2111no—
121C2×M16C2×D21111yescharacteristic
122C2×M16C2×D21111yescharacteristic
123C22×C8—4221no—
124C22×C8—2111no—
125#117—2111no—
126#117C2×D21111yescharacteristic
127#108—4221no—
128#108—4221no—
129#107—4221no—
130#106—4221no—
131#104—4222no—
132#104—4222no—
133#41—4221no—
134#41—4221no—
135#41—2112no—
136#41—2112no—
137C2×Q8—2111no—
138C2×D4—4221no—
139C2×D4—2111no—
140C22×C4—4221no—
141C22×C4C22×D21111yescharacteristic
142M16—21no—
143M16—21no—
144M16—21no—
145M16—4222no—
146M16—4222no—
147M16—4221no—
148M16—4221no—
149M16C4×D21112yes—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—4222no—
153C2×C8—4222no—
154C2×C8—4221no—
155C2×C8—2111no—
156C2×C8—2111no—
157C2×C8—2111no—
158C2×C8—2111no—
159C2×C8—2111no—
160C2×C8—2111no—
161C4⋊C4—21no—
162C4⋊C4—4221no—
163C22⋊C4—21no—
164C22⋊C4—4221no—
165C42—21no—
166C42—4221no—
167C23—4221no—
168C23#1680996211yescharacteristic
169Q8—4221no—
170Q8—2112no—
171D4—4222no—
172D4—4222no—
173D4—4221no—
174D4—2112no—
175C2×C4—41no—
176C2×C4—8441no—
177C2×C4—4221no—
178C2×C4—4222no—
179C2×C4—4222no—
180C2×C4C2×C4×D21111yescharacteristic
181C2×C4D4×D21111yescharacteristic
182C2×C4D4×D21111yescharacteristic
183C2×C4D4×D21111yescharacteristic
184C2×C4Q8×D21111yescharacteristic
185C2×C4#1680996211yescharacteristic
186C8—22no—
187C8—41no—
188C8—8441no—
189C8—2111no—
190C8—2111no—
191C8—2111no—
192C8—2111no—
193C22—41no—
194C22—8442no—
195C22—4222no—
196C22#2026993411yescharacteristic
197C22#2026995011yescharacteristic
198C22#2026995511yescharacteristic
199C4—81no—
200C4—16881no—
201C4—4222no—
202C4C22⋊C4×D21111yescharacteristic
203C4C4⋊C4×D21111yescharacteristic
204C4#2026995011yescharacteristic
205C4#2026995511yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—8442no—
210C2#2721300511yescharacteristic
subgroupG/H
centerC4#20269950
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#344×C211C2
socleC2×C211#380
prime radicalC2×#16809962C22
cosocleC22×C4×C211C23
hypercenter#344D211
hypocenterC211#1814
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
3. #27213182C2semidirect
6. #27212918C2semidirect
7. #344×C211C2semidirect
11. #20269985C4semidirect
12. #20269985C4semidirect
105. C211#1814semidirect
113. #344D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699852
C2×#202699122
#272131822
#272131642
#272131622
#272129182
#344×C2112
#1814211