Finite Group Databasebeta

← #41014448

Subgroups — #41014448

Order 27008.

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

normal subgroups only

This group has 380 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
1C2×#20270002C211yesmaximal, characteristic, complemented
2#27213710C211yesmaximal, characteristic, complemented
3#27213163C211yesmaximal, characteristic, complemented
4#27213152C211yesmaximal, characteristic, complemented
5#27213120C211yesmaximal, characteristic, complemented
6#27212917C211yesmaximal, characteristic, complemented
7#575×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9#20270002—22no—
10#20270002—22no—
11C2×#16809949—21no—
12C2×#16809947—21no—
13C2×#16809946C2211yescharacteristic
14#20269973—21no—
15#20269966—21no—
16#20269951—22no—
17#20269945—22no—
18#20269936C2211yescharacteristic
19#20269936C2211yescharacteristic
20#20269904—22no—
21#20269899—22no—
22#20269892—22no—
23C2×#41×C211C2211yescharacteristic
24C2×#41×C211C2211yescharacteristic
25#137×C211—21no—
26#137×C211—21no—
27AGL(1,ℤ/8)×C211—21no—
28AGL(1,ℤ/8)×C211—21no—
29#135×C211—22no—
30C2×M16×C211C2211yescharacteristic
31C23×D211—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×D844—21no—
35C2×D844—21no—
36C2×D844—21no—
37C2×D844D412yes—
38C2×C22⋊D211—24no—
39#16809949—24no—
40#16809947—24no—
41#16809946—21no—
42#16809946D412yescomplemented
43C2×C211⋊C8—21no—
44C4⋊Dic211—21no—
45C4⋊Dic211—21no—
46C22⋊Dic211—21no—
47C22⋊Dic211D412yes—
48(C2×D211)⋊C4—24no—
49C4×Dic211—21no—
50C4×Dic211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54#41×C211—21no—
55#41×C211—21no—
56#41×C211—21no—
57#41×C211—21no—
58#41×C211—21no—
59C2×Q8×C211D411yescharacteristic
60C2×Q8×C211D411yescharacteristic, complemented
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211D411yescharacteristic
64C2×D4×C211D411yescharacteristic, complemented
65C22×C4×C211—21no—
66C22×C4×C211—21no—
67C22×C4×C211C2311yescharacteristic
68Q16×C211—22no—
69SD16×C211—22no—
70SD16×C211—22no—
71D8×C211—22no—
72M16×C211—21no—
73M16×C211—21no—
74C2×C8×C211D412yescomplemented
75C22×D211—22no—
76C22×D211—41no—
77C22×D211—41no—
78C22×D211—82no—
79D844—22no—
80D844—22no—
81D844—41no—
82D844—41no—
83C22⋊D211—84no—
84C2×Dic211—22no—
85C2×Dic211—41no—
86C211⋊C8—22no—
87C23×C211—21no—
88C23×C211—21no—
89C23×C211C2×D411yescharacteristic
90Q8×C211—21no—
91Q8×C211—21no—
92Q8×C211—21no—
93Q8×C211—21no—
94D4×C211—21no—
95D4×C211—21no—
96D4×C211—21no—
97D4×C211—21no—
98D4×C211—41no—
99D4×C211—41no—
100D4×C211—41no—
101D4×C211—41no—
102C2×C4×C211—21no—
103C2×C4×C211—21no—
104C2×C4×C211—41no—
105C2×C4×C211—41no—
106C2×C4×C211—41no—
107C2×C4×C211—41no—
108C2×C4×C211C2×D411yescharacteristic
109C2×C4×C211C2×D411yescharacteristic
110C2×C4×C211C2×D411yescharacteristic
111C2×C4×C211C2×D411yescharacteristic
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211#4111yescharacteristic
114C8×C211—22no—
115C22×C211—41no—
116C22×C211—42no—
117C22×C211—41no—
118C22×C211—41no—
119C22×C211—81no—
120C22×C211#12011yescharacteristic
121C22×C211#12111yescharacteristic
122C22×C211#12711yescharacteristic
123C2×D211—44no—
124C2×D211—42no—
125C2×D211—82no—
126C2×D211—82no—
127C4×C211—41no—
128C4×C211—41no—
129C4×C211#12011yescharacteristic
130C4×C211#12011yescharacteristic
131C4×C211#12111yescharacteristic
132C4×C211#12111yescharacteristic
133Dic211—42no—
134Dic211—81no—
135C2×C211—21no—
136C2×C211—21no—
137C2×C211—21no—
138C2×C211—81no—
139C2×C211—81no—
140C2×C211#39211yescharacteristic
141D211—82no—
142C211#200811yescharacteristic, complemented, Sylow
143#2008—2111nomaximal, Sylow
144#575D21111yescharacteristic, complemented
145C2×AGL(1,ℤ/8)—2111no—
146#535—2111no—
147#421—2111no—
148#418—2111no—
149#410—2111no—
150#344—2111no—
151C2×#41C2×D21111yescharacteristic
152C2×#41C2×D21111yescharacteristic
153C22×D4—2111no—
154#137—21no—
155#137—21no—
156AGL(1,ℤ/8)—21no—
157AGL(1,ℤ/8)—21no—
158AGL(1,ℤ/8)—4222no—
159AGL(1,ℤ/8)—4222no—
160#135—22no—
161C2×SD16—4221no—
162C2×D8—4221no—
163C2×M16—2111no—
164C2×M16C2×D21111yescharacteristic
165#127—4221no—
166#124—4221no—
167#121—4222no—
168#120—4222no—
169#117—2111no—
170#117—2111no—
171#104—4222no—
172#102—4222no—
173#99—4222no—
174C24—4221no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183C2×Q8C22⋊D21111yescharacteristic
184C2×Q8C22⋊D21111yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—4224no—
188C2×D4—4221no—
189C2×D4—4221no—
190C2×D4—4221no—
191C2×D4—4221no—
192C2×D4—4221no—
193C2×D4—2112no—
194C2×D4C22⋊D21111yescharacteristic
195C2×D4C22⋊D21111yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—21no—
198C22×C4C22×D21111yescharacteristic
199Q16—22no—
200SD16—22no—
201SD16—22no—
202SD16—4224no—
203D8—22no—
204D8—4224no—
205M16—21no—
206M16—21no—
207M16—4221no—
208M16—2112no—
209C2×C8—4221no—
210C2×C8C22⋊D21112yescomplemented
211C4⋊C4—4221no—
212C4⋊C4—4221no—
213C22⋊C4—4224no—
214C22⋊C4—4221no—
215C22⋊C4—2112no—
216C42—4221no—
217C42—4221no—
218C23—21no—
219C23—21no—
220C23—8441no—
221C23—8441no—
222C23—16882no—
223C23—4222no—
224C23D4×D21111yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—8441no—
238D4—8441no—
239D4—16884no—
240D4—4222no—
241D4—4222no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—8441no—
249C2×C4—4222no—
250C2×C4D4×D21111yescharacteristic
251C2×C4D4×D21111yescharacteristic
252C2×C4C2×C22⋊D21111yescharacteristic
253C2×C4C2×C22⋊D21111yescharacteristic
254C2×C4C2×C22⋊D21111yescharacteristic
255C2×C4#1680996111yescharacteristic
256C8—22no—
257C8—4222no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22—8444no—
264C22—8442no—
265C22—16882no—
266C22—16882no—
267C22#2026994511yescharacteristic
268C22#2026994911yescharacteristic
269C22#2026997311yescharacteristic
270C4—41no—
271C4—41no—
272C4—8442no—
273C4—16881no—
274C4#2026994511yescharacteristic
275C4#2026994611yescharacteristic
276C4#2026994811yescharacteristic
277C4#2026994911yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—81no—
283C2—16882no—
284C2#2721305411yescharacteristic
subgroupG/H
centerC2#27213054
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#575×C211C2
socleC2×C211#392
hypercenter#575D211
hypocenterC211#2008
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#20270002C2semidirect
2. #27213710C2semidirect
3. #27213163C2semidirect
4. #27213152C2semidirect
5. #27213120C2semidirect
6. #27212917C2semidirect
7. #575×C211C2semidirect
42. #16809946D4semidirect
60. C2×Q8×C211D4semidirect
64. C2×D4×C211D4semidirect
74. C2×C8×C211D4semidirect
142. C211#2008semidirect
144. #575D211semidirect
184. C2×Q8C22⋊D211semidirect
195. C2×D4C22⋊D211semidirect
210. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202700022
#272137102
#272131632
#272131522
#272131202
#272129172
#575×C2112
#2008211