Finite Group Databasebeta

← #22560460

Subgroups — #22560460

Order 8576.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#14524008C211yesmaximal, characteristic
2C2×#14523968C211yesmaximal, characteristic
3C2×#14523966C211yesmaximal, characteristic
4#16863292C211yesmaximal, characteristic
5#16863158C211yesmaximal, characteristic
6#16863150C211yesmaximal, characteristic
7C2×#107×C67C211yesmaximal, characteristic
8C22×Dic134C2211yescharacteristic
9C22×C67⋊C8C2211yescharacteristic
10#14524008—22no—
11#14524008—22no—
12C2×C4⋊Dic67C2211yescharacteristic
13C2×Dic67⋊C4—21no—
14C2×Dic67⋊C4—21no—
15C2×C4×Dic67—21no—
16C2×C4×Dic67C2211yescharacteristic
17#14523968C414yescomplemented
18#14523966—22no—
19C67⋊#95—21no—
20C22×C8×C67C2211yescharacteristic
21C2×C4⋊C4×C67C2211yescharacteristic
22C2×C4⋊C4×C67C2211yescharacteristic
23#107×C67—22no—
24C2×Dic134—21no—
25C2×Dic134—21no—
26C2×Dic134—21no—
27C2×Dic134C2×C412yes—
28C2×Dic134D412yes—
29C2×Dic134Q812yes—
30C22×Dic67—21no—
31C22×Dic67—21no—
32C22×Dic67D411yescharacteristic
33C22×Dic67D411yescharacteristic
34C2×C67⋊C8—21no—
35C2×C67⋊C8—21no—
36C2×C67⋊C8—21no—
37C4⋊Dic67—21no—
38C4⋊Dic67C2×C412yes—
39Dic67⋊C4—24no—
40Dic67⋊C4—42no—
41C4×Dic67—24no—
42C4×Dic67—21no—
43C4×Dic67—21no—
44C22×C4×C67—21no—
45C22×C4×C67—21no—
46C22×C4×C67C2311yescharacteristic
47C2×C8×C67—21no—
48C2×C8×C67—21no—
49C2×C8×C67C2×C412yes—
50C4⋊C4×C67—21no—
51C4⋊C4×C67—21no—
52C4⋊C4×C67—21no—
53C4⋊C4×C67D412yes—
54C23×C67C2×D411yescharacteristic
55C2×C4×C67—22no—
56C2×C4×C67—41no—
57C2×C4×C67—41no—
58C2×C4×C67—41no—
59C2×C4×C67—41no—
60C2×C4×C67—41no—
61C2×C4×C67C22×C411yescharacteristic
62C2×C4×C67C2×D411yescharacteristic
63C2×C4×C67C2×D411yescharacteristic
64C2×C4×C67C2×Q811yescharacteristic
65C2×C4×C67#4111yescharacteristic
66C2×C4×C67#4111yescharacteristic
67C2×Dic67—22no—
68C2×Dic67—21no—
69C2×Dic67—22no—
70C2×Dic67—22no—
71C2×Dic67—21no—
72C2×Dic67—22no—
73C2×Dic67—21no—
74C2×Dic67—21no—
75C2×Dic67—21no—
76C2×Dic67—41no—
77C2×Dic67—41no—
78C2×Dic67Q1612yes—
79Dic134—22no—
80Dic134—22no—
81Dic134—22no—
82Dic134C4⋊C414yescomplemented
83C8×C67—22no—
84C67⋊C8—41no—
85C22×C67C4×D411yescharacteristic
86C22×C67#12011yescharacteristic
87C22×C67C22⋊Q811yescharacteristic
88C22×C67C2×Q1611yescharacteristic
89C22×C67#13511yescharacteristic
90C22×C67AGL(1,ℤ/8)11yescharacteristic
91C22×C67#13711yescharacteristic
92C4×C67—42no—
93C4×C67—81no—
94C4×C67C2×C4⋊C411yescharacteristic
95C4×C67C4×D411yescharacteristic
96C4×C67C22⋊Q811yescharacteristic
97C4×C67#12311yescharacteristic
98Dic67—24no—
99Dic67—22no—
100Dic67—42no—
101Dic67—41no—
102C2×C67#38511yescharacteristic
103C2×C67C4×Q1611yescharacteristic
104C2×C67#44011yescharacteristic
105C2×C67#44911yescharacteristic
106C2×C67#45111yescharacteristic
107C2×C67#47711yescharacteristic
108C2×C67#47911yescharacteristic
109#1863—671nomaximal, Sylow
110C67#186311yescharacteristic, complemented, Sylow
111C2×C4×Q8—671no—
112C2×#107D6711yescharacteristic
113C2×#103—671no—
114C2×#103—671no—
115#389—671no—
116#340—671no—
117#336—671no—
118C22×Q8—671no—
119C22×C8—671no—
120C22×C8C2×D6711yescharacteristic
121C4×Q8—1342no—
122C4×Q8—1342no—
123C2×C4⋊C4—671no—
124C2×C4⋊C4—1341no—
125C2×C4⋊C4—1341no—
126C2×C4⋊C4C2×D6711yescharacteristic
127C2×C4⋊C4C2×D6711yescharacteristic
128C2×C42—671no—
129C2×C42—1341no—
130#107—22no—
131#103—674no—
132#103—1342no—
133#95—1341no—
134C2×Q8—672no—
135C2×Q8—672no—
136C2×Q8—672no—
137C2×Q8—1341no—
138C2×Q8—1341no—
139C2×Q8—1341no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—671no—
143C22×C4—671no—
144C22×C4—1341no—
145C22×C4—1341no—
146C22×C4C22×D6711yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—1341no—
150C2×C8—1341no—
151C2×C8—1341no—
152C2×C8C4×D6712yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2682no—
157C4⋊C4—672no—
158C4⋊C4—1344no—
159C4⋊C4—1341no—
160C4⋊C4C22⋊D6712yes—
161C42—1344no—
162C42—1341no—
163C42—1341no—
164C23D4×D6711yescharacteristic
165Q8—674no—
166Q8—1342no—
167Q8—1342no—
168Q8—1342no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—2681no—
176C2×C4—2681no—
177C2×C4—672no—
178C2×C4—1342no—
179C2×C4—1341no—
180C2×C4—1341no—
181C2×C4—1341no—
182C2×C4—1342no—
183C2×C4—1341no—
184C2×C4—1342no—
185C2×C4—1342no—
186C2×C4—1341no—
187C2×C4C2×C4×D6711yescharacteristic
188C2×C4D4×D6711yescharacteristic
189C2×C4C2×C22⋊D6711yescharacteristic
190C2×C4Q8×D6711yescharacteristic
191C2×C4#1176162811yescharacteristic
192C2×C4#1176162911yescharacteristic
193C8—22no—
194C8—2681no—
195C22#1452400511yescharacteristic
196C22#1452401111yescharacteristic
197C22#1452402111yescharacteristic
198C22Q16×D6711yescharacteristic
199C22#1452406211yescharacteristic
200C22#1452406711yescharacteristic
201C22#1452407511yescharacteristic
202C4—42no—
203C4—81no—
204C4—2682no—
205C4—2681no—
206C4—1344no—
207C4—1342no—
208C4C4⋊C4×D6711yescharacteristic
209C4C4×C22⋊D6711yescharacteristic
210C4#1452401911yescharacteristic
211C4#1452402811yescharacteristic
212C2#1686326711yescharacteristic
213C2#1686349011yescharacteristic
214C2#1686349511yescharacteristic
215C2#1686353811yescharacteristic
216C2#1686355011yescharacteristic
217C2#1686369011yescharacteristic
218C2#1686370211yescharacteristic
subgroupG/H
centerC23D4×D67
commutatorC2×C4×C67C22×C4
FrattiniC22×C4C22×D67
FittingC2×#107×C67C2
socleC23×C67C2×D4
hypercenterC2×#107D67
hypocenterC67#1863
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
17. #14523968C4semidirect
82. Dic134C4⋊C4semidirect
110. C67#1863semidirect