Finite Group Databasebeta

← #41014612

Subgroups — #41014612

Order 27008.

278 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213189C211yesmaximal, characteristic, complemented
2C2×#20269904C211yesmaximal, characteristic
3#27213147C211yesmaximal, characteristic
4C2×#20269896C211yesmaximal, characteristic, complemented
5#27213053C211yesmaximal, characteristic, complemented
6#27212905C211yesmaximal, characteristic, complemented
7#536×C211C211yesmaximal, characteristic
8C2×#16809946C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×C4×Dic211C2211yescharacteristic
12#20269910—21no—
13#20269908—21no—
14#20269904—22no—
15#20269901—21no—
16#20269898—21no—
17#20269896—22no—
18C211⋊#95—22no—
19C2×#41×C211C2211yescharacteristic
20C22×Q8×C211C2211yescharacteristic
21C4⋊Q8×C211—21no—
22#124×C211—21no—
23#123×C211—22no—
24C22⋊Q8×C211—21no—
25C22⋊Q8×C211—21no—
26#117×C211C2211yescharacteristic
27C22×Dic211—21no—
28C22×Dic211—21no—
29#16809946—21no—
30#16809946—21no—
31#16809946—21no—
32#16809946D412yes—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4×Dic211—21no—
38C4×Dic211—21no—
39#41×C211—21no—
40#41×C211—21no—
41#41×C211—21no—
42#41×C211—21no—
43C2×Q8×C211—21no—
44C2×Q8×C211—21no—
45C2×Q8×C211—21no—
46C2×Q8×C211—21no—
47C2×Q8×C211—21no—
48C2×Q8×C211—21no—
49C2×Q8×C211D411yescharacteristic
50C2×D4×C211—21no—
51C2×D4×C211D411yescharacteristic
52C22×C4×C211—21no—
53C22×C4×C211D412yes—
54C22×C4×C211C2311yescharacteristic
55C4⋊C4×C211—22no—
56C4⋊C4×C211—22no—
57C4⋊C4×C211D411yescharacteristic
58C4⋊C4×C211D411yescharacteristic
59C22⋊C4×C211—22no—
60C22⋊C4×C211—22no—
61C22⋊C4×C211—21no—
62C42×C211—21no—
63C2×Dic211—22no—
64C2×Dic211—41no—
65C2×Dic211—41no—
66C2×Dic211—42no—
67C2×Dic211—41no—
68C211⋊C8—22no—
69C211⋊C8—41no—
70C23×C211—21no—
71C23×C211C2×D411yescharacteristic
72Q8×C211—21no—
73Q8×C211—21no—
74Q8×C211—41no—
75Q8×C211—41no—
76Q8×C211—41no—
77Q8×C211—41no—
78D4×C211—21no—
79D4×C211—21no—
80D4×C211—41no—
81D4×C211—41no—
82C2×C4×C211—21no—
83C2×C4×C211—21no—
84C2×C4×C211—22no—
85C2×C4×C211—21no—
86C2×C4×C211—22no—
87C2×C4×C211—22no—
88C2×C4×C211—41no—
89C2×C4×C211—41no—
90C2×C4×C211C2×D411yescharacteristic
91C2×C4×C211C2×D411yescharacteristic
92C2×C4×C211C2×D411yescharacteristic
93C2×C4×C211#4111yescharacteristic
94C2×C4×C211#4111yescharacteristic
95C2×C4×C211#4111yescharacteristic
96C22×C211—21no—
97C22×C211—21no—
98C22×C211—41no—
99C22×C211—41no—
100C22×C211—41no—
101C22×C211#12011yescharacteristic
102C22×C211#12111yescharacteristic
103C22×C211#12311yescharacteristic
104C4×C211—42no—
105C4×C211—41no—
106C4×C211—41no—
107C4×C211—41no—
108C4×C211#12111yescharacteristic
109C4×C211#12111yescharacteristic
110C4×C211#12311yescharacteristic
111C4×C211#12411yescharacteristic
112Dic211—42no—
113Dic211—81no—
114C2×C211—21no—
115C2×C211—21no—
116C2×C211—81no—
117C2×C211#39411yescharacteristic
118C2×C211#45411yescharacteristic
119C2×C211#45611yescharacteristic
120C211#203811yescharacteristic, complemented, Sylow
121#2038—2111nomaximal, Sylow
122#536D21111yescharacteristic
123#428—2111no—
124C2×#104—2111no—
125#417—2111no—
126C2×#101—2111no—
127#391—2111no—
128#339—2111no—
129C2×#41C2×D21111yescharacteristic
130C22×Q8C2×D21111yescharacteristic
131C2×M16—2111no—
132C2×M16—2111no—
133C4⋊Q8—21no—
134#124—21no—
135#123—22no—
136C22⋊Q8—21no—
137C22⋊Q8—21no—
138#117C2×D21111yescharacteristic
139C2×C4⋊C4—2111no—
140C2×C42—2111no—
141#107—4221no—
142#106—4221no—
143#104—4222no—
144#103—4221no—
145#102—4221no—
146#101—4222no—
147#95—4222no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8C22⋊D21111yescharacteristic
159C2×D4—21no—
160C2×D4C22⋊D21111yescharacteristic
161C22×C4—21no—
162C22×C4—4221no—
163C22×C4—4221no—
164C22×C4C22⋊D21112yes—
165C22×C4C22×D21111yescharacteristic
166M16—4221no—
167M16—4221no—
168M16—4221no—
169M16—2112no—
170C2×C8—4221no—
171C2×C8—4221no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—4221no—
175C4⋊C4—4221no—
176C4⋊C4C22⋊D21111yescharacteristic
177C4⋊C4C22⋊D21111yescharacteristic
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—21no—
181C42—21no—
182C42—4221no—
183C42—4221no—
184C23—21no—
185C23C2×C22⋊D21111yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—41no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—8441no—
205C2×C4—8441no—
206C2×C4—8442no—
207C2×C4—8441no—
208C2×C4—4222no—
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×C22⋊D21111yescharacteristic
211C2×C4C2×C22⋊D21111yescharacteristic
212C2×C4#1680996111yescharacteristic
213C2×C4#1680996111yescharacteristic
214C2×C4#1680996111yescharacteristic
215C8—8441no—
216C8—4222no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22#2026994611yescharacteristic
223C22#2026994911yescharacteristic
224C22#2026995811yescharacteristic
225C4—42no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—8442no—
230C4—16881no—
231C4#2026994811yescharacteristic
232C4#2026994911yescharacteristic
233C4#2026995811yescharacteristic
234C4#2026996411yescharacteristic
235C2—21no—
236C2—21no—
237C2—81no—
238C2#2721306011yescharacteristic
239C2#2721331911yescharacteristic
240C2#2721332911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#536×C211C2
socleC22×C211#120
prime radicalC2×#41×C211C22
hypercenter#536D211
hypocenterC211#2038
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #27213189C2semidirect
4. C2×#20269896C2semidirect
5. #27213053C2semidirect
6. #27212905C2semidirect
120. C211#2038semidirect

Maximal subgroups

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

subgroupindex
#272131892
C2×#202699042
#272131472
C2×#202698962
#272130532
#272129052
#536×C2112
#2038211