Finite Group Databasebeta

← #41013814

Subgroups — #41013814

Order 27008.

342 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#27213913C211yesmaximal, characteristic, complemented
2#27213653C211yesmaximal, characteristic
3#27213109C212yesmaximal, complemented
4#27212918C212yesmaximal, complemented
5#413×C211C211yesmaximal, characteristic, complemented
6C22×C4×D211C2211yescharacteristic
7#20269984—24no—
8C2×#16809946—22no—
9C2×#16809946C2212yes—
10#20269958C412yes—
11#20269953—21no—
12#20269948—21no—
13#20269945C412yes—
14C4×C22⋊D211—22no—
15#20269936C2211yescharacteristic
16(C2×D211)⋊C8—24no—
17C2×#41×C211C2211yescharacteristic
18C2×M16×C211C2212yes—
19#101×C211—21no—
20#100×C211—21no—
21C23×D211C2×C411yescharacteristic
22C2×C22⋊D211—22no—
23C22×Dic211C2×C411yescharacteristic
24C2×C4×D211—22no—
25C2×C4×D211—21no—
26C2×C4×D211—21no—
27C2×C4×D211—22no—
28#16809946—24no—
29#16809946—22no—
30#16809946—22no—
31C2×C211⋊C8—24no—
32C2×C211⋊C8D414yescomplemented
33C4⋊Dic211—21no—
34Dic211⋊C4—22no—
35C22⋊Dic211—22no—
36C22⋊Dic211C2×C412yes—
37(C2×D211)⋊C4—22no—
38C4×Dic211—21no—
39#41×C211—22no—
40#41×C211—22no—
41C2×Q8×C211D411yescharacteristic
42C2×D4×C211C2×C412yes—
43C2×D4×C211D411yescharacteristic
44C22×C4×C211D412yes—
45C22×C4×C211C2311yescharacteristic
46M16×C211—22no—
47M16×C211—22no—
48C2×C8×C211—22no—
49C22×D211—21no—
50C22×D211—22no—
51C22×D211—42no—
52C22⋊D211—44no—
53C2×Dic211—22no—
54C2×Dic211—21no—
55C2×Dic211—22no—
56C4×D211—41no—
57C4×D211—42no—
58C4×D211—41no—
59C211⋊C8—24no—
60C23×C211C22⋊C412yes—
61C23×C211C22×C411yescharacteristic
62Q8×C211—22no—
63D4×C211—24no—
64D4×C211—22no—
65C2×C4×C211—22no—
66C2×C4×C211—22no—
67C2×C4×C211C22⋊C412yes—
68C2×C4×C211C2×D412yes—
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C2×C4×C211#4112yes—
72C8×C211—42no—
73C22×C211—22no—
74C22×C211—41no—
75C22×C211—42no—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211C4×D412yes—
78C2×D211—22no—
79C2×D211—24no—
80C2×D211—41no—
81C2×D211—42no—
82C2×D211—41no—
83C4×C211—22no—
84C4×C211#12011yescharacteristic
85C4×C211#12112yes—
86C4×C211#12311yescharacteristic
87Dic211—42no—
88Dic211—41no—
89C2×C211—21no—
90C2×C211—22no—
91C2×C211—42no—
92C2×C211#38611yescharacteristic
93D211—42no—
94C211#189111yescharacteristic, complemented, Sylow
95#1891—2111nomaximal, Sylow
96#568—2111no—
97#525—2111no—
98#413D21111yescharacteristic, complemented
99#407—2112no—
100#344—2112no—
101C2×#41C2×D21111yescharacteristic
102C23×C4—2111no—
103#131—4224no—
104C2×M16—4222no—
105C2×M16—2112no—
106C2×M16C2×D21112yes—
107#123—2112no—
108C22⋊Q8—4221no—
109#121—4221no—
110#120—2112no—
111C4×D4—4222no—
112#117—2111no—
113#101—21no—
114#100—21no—
115C22⋊C8—4224no—
116C24—2111no—
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D21111yescharacteristic
120C2×D4—4222no—
121C2×D4C4×D21112yes—
122C2×D4C22⋊D21111yescharacteristic
123C22×C4—4222no—
124C22×C4—4221no—
125C22×C4—4221no—
126C22×C4—4222no—
127C22×C4—2111no—
128C22×C4D84412yes—
129C22×C4C22×D21111yescharacteristic
130M16—22no—
131M16—22no—
132M16—4224no—
133M16—4222no—
134M16—4222no—
135C2×C8—22no—
136C2×C8—4224no—
137C2×C8—2114no—
138C4⋊C4—4222no—
139C4⋊C4—4221no—
140C22⋊C4—4222no—
141C22⋊C4—4222no—
142C22⋊C4—2112no—
143C42—4221no—
144C23—8442no—
145C23—4221no—
146C23—4222no—
147C23(C2×D211)⋊C412yes—
148C23C2×C4×D21111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—8444no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—8441no—
156C2×C4—8442no—
157C2×C4—8441no—
158C2×C4—4222no—
159C2×C4—4221no—
160C2×C4—4222no—
161C2×C4(C2×D211)⋊C412yes—
162C2×C4D4×D21112yes—
163C2×C4C2×C22⋊D21111yescharacteristic
164C2×C4C2×D84411yescharacteristic
165C2×C4#1680996112yes—
166C8—42no—
167C8—4224no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—8441no—
172C22—8442no—
173C22—8441no—
174C22—4222no—
175C22—4224no—
176C22C2×(C2×D211)⋊C411yescharacteristic
177C22(C22⋊D211)⋊C412yes—
178C4—22no—
179C4—8442no—
180C4—8441no—
181C4#2026994711yescharacteristic
182C4#2026994912yes—
183C4#2026996111yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—8442no—
188C2#2721302311yescharacteristic
subgroupG/H
centerC4#20269947
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#413×C211C2
socleC2×C211#386
prime radical#27213653C2
cosocleC22×C4×C211C23
hypercenter#413D211
hypocenterC211#1891
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #27213913C2semidirect
3. #27213109C2semidirect
4. #27212918C2semidirect
5. #413×C211C2semidirect
32. C2×C211⋊C8D4semidirect
94. C211#1891semidirect
98. #413D211semidirect

Maximal subgroups

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

subgroupindex
#272139132
#272136532
#272131092
#272129182
#413×C2112
#1891211