Finite Group Databasebeta

← #41013450

Subgroups — #41013450

Order 27008.

256 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20269984C211yesmaximal, characteristic
2C2×#20269912C211yesmaximal, characteristic, complemented
3#27213182C211yesmaximal, characteristic, complemented
4#27213163C211yesmaximal, characteristic
5#27212917C211yesmaximal, characteristic, complemented
6#27212916C211yesmaximal, characteristic, complemented
7#421×C211C211yesmaximal, characteristic
8#20269984—21no—
9#20269984—21no—
10#20269984C412yescomplemented
11#20269984C412yescomplemented
12C2×#16809946—21no—
13C2×#16809946C2211yescharacteristic
14C2×#16809946C2211yescharacteristic
15C22×C211⋊C8—21no—
16C22×C211⋊C8C2211yescharacteristic
17#20269936C2211yescharacteristic
18#20269912—21no—
19#20269912—21no—
20#20269910—21no—
21#20269908—21no—
22#20269904—22no—
23C2×#41×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25#117×C211C2211yescharacteristic
26#104×C211—22no—
27#16809946—22no—
28#16809946—22no—
29#16809946—21no—
30#16809946—21no—
31#16809946—21no—
32#16809946C2×C412yes—
33C2×C211⋊C8—22no—
34C2×C211⋊C8—22no—
35C2×C211⋊C8—21no—
36C2×C211⋊C8—21no—
37C2×C211⋊C8C2×C411yescharacteristic
38C2×C211⋊C8C2×C411yescharacteristic
39C2×C211⋊C8D411yescharacteristic
40C2×C211⋊C8D411yescharacteristic
41C2×C211⋊C8D411yescharacteristic
42C2×C211⋊C8D411yescharacteristic
43C4⋊Dic211—21no—
44C22⋊Dic211—21no—
45C4×Dic211—21no—
46#41×C211—21no—
47#41×C211—21no—
48#41×C211C2×C412yes—
49#41×C211D412yes—
50C2×Q8×C211Q811yescharacteristic
51C2×D4×C211—21no—
52C2×D4×C211Q811yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55M16×C211—21no—
56M16×C211—21no—
57C2×C8×C211—21no—
58C4⋊C4×C211—21no—
59C22⋊C4×C211—21no—
60C42×C211—21no—
61C2×Dic211—41no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64C211⋊C8C22⋊C411yescharacteristic
65C211⋊C8C22⋊C411yescharacteristic
66C211⋊C8C22⋊C411yescharacteristic
67C211⋊C8C22⋊C411yescharacteristic
68C23×C211—21no—
69C23×C211#4111yescharacteristic
70Q8×C211—21no—
71Q8×C211C4⋊C412yes—
72D4×C211—22no—
73D4×C211—22no—
74D4×C211—21no—
75D4×C211C4⋊C412yes—
76C2×C4×C211—21no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×Q811yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—41no—
87C22×C211—22no—
88C22×C211—41no—
89C22×C211—42no—
90C22×C211#11711yescharacteristic
91C22×C211#12111yescharacteristic
92C22×C211C22⋊Q811yescharacteristic
93C4×C211—22no—
94C4×C211—81no—
95C4×C211C2×C22⋊C411yescharacteristic
96C4×C211C2×C4⋊C411yescharacteristic
97C4×C211#12111yescharacteristic
98C4×C211C22⋊Q811yescharacteristic
99Dic211—81no—
100C2×C211—21no—
101C2×C211—21no—
102C2×C211—21no—
103C2×C211—42no—
104C2×C211#38011yescharacteristic
105C211#181411yescharacteristic, complemented, Sylow
106#1814—2111nomaximal, Sylow
107C2×#131—2111no—
108C2×#108—2111no—
109#427—2111no—
110#421—2111no—
111#421D21111yescharacteristic
112#344—2111no—
113#344—2111no—
114C2×#41C2×D21111yescharacteristic
115#131—4221no—
116#131—4221no—
117#131—2112no—
118#131—2112no—
119C2×M16—4221no—
120C2×M16—2111no—
121C2×M16—2111no—
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—22no—
132#104—4222no—
133#41—21no—
134#41—21no—
135#41C4×D21112yes—
136#41C22⋊D21112yes—
137C2×Q8Dic42211yescharacteristic
138C2×D4—21no—
139C2×D4Dic42211yescharacteristic
140C22×C4—21no—
141C22×C4C22×D21111yescharacteristic
142M16—21no—
143M16—21no—
144M16—4221no—
145M16—4221no—
146M16—4221no—
147M16—4222no—
148M16—4222no—
149M16—2112no—
150C2×C8—21no—
151C2×C8—4221no—
152C2×C8—4221no—
153C2×C8—4222no—
154C2×C8—4222no—
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—21no—
168C23#1680996111yescharacteristic
169Q8—21no—
170Q8Dic211⋊C412yes—
171D4—22no—
172D4—22no—
173D4—21no—
174D4Dic211⋊C412yes—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—8441no—
180C2×C4C2×C4×D21111yescharacteristic
181C2×C4D4×D21111yescharacteristic
182C2×C4D4×D21111yescharacteristic
183C2×C4C2×C22⋊D21111yescharacteristic
184C2×C4C2×Dic42211yescharacteristic
185C2×C4#1680996111yescharacteristic
186C8—41no—
187C8—8441no—
188C8—4222no—
189C8—2111no—
190C8—2111no—
191C8—2111no—
192C8—2111no—
193C22—22no—
194C22—41no—
195C22—42no—
196C22#2026993311yescharacteristic
197C22#2026994911yescharacteristic
198C22#2026995411yescharacteristic
199C4—22no—
200C4—81no—
201C4—16881no—
202C4C22⋊C4×D21111yescharacteristic
203C4C2×Dic211⋊C411yescharacteristic
204C4#2026994911yescharacteristic
205C4#2026995411yescharacteristic
206C2—21no—
207C2—21no—
208C2—21no—
209C2—42no—
210C2#2721300411yescharacteristic
subgroupG/H
centerC4#20269949
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#421×C211C2
socleC2×C211#380
prime radicalC2×#41×C211C22
cosocleC22×C4×C211C23
hypercenter#421D211
hypocenterC211#1814
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#20269912C2semidirect
3. #27213182C2semidirect
5. #27212917C2semidirect
6. #27212916C2semidirect
10. #20269984C4semidirect
11. #20269984C4semidirect
105. C211#1814semidirect

Maximal subgroups

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

subgroupindex
C2×#202699842
C2×#202699122
#272131822
#272131632
#272129172
#272129162
#421×C2112
#1814211