Finite Group Databasebeta

← #22573012

Subgroups — #22573012

Order 8576.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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#16864125C211yesmaximal, characteristic
2#16863766C212yesmaximal
3#16863748C211yesmaximal, characteristic
4#16863700C212yesmaximal
5#16863678C211yesmaximal, characteristic
6#16863606C211yesmaximal, characteristic
7#16863606C212yesmaximal
8#16863527C211yesmaximal, characteristic
9#16863501C212yesmaximal
10C4×#11761618C211yesmaximal, characteristic
11#557×C67C211yesmaximal, characteristic
12C2×#11761618C2211yescharacteristic
13C2×#11761618C2212yes—
14#14524043C2211yescharacteristic
15#14524043C2212yes—
16#14524034—21no—
17#14524034—22no—
18Q8×Dic67—21no—
19C4×Dic134C2211yescharacteristic
20C4×Dic134C2212yes—
21#14523976C2211yescharacteristic
22#14523974C2212yes—
23C4⋊(C67⋊C8)C2211yescharacteristic
24C4⋊(C67⋊C8)C2212yes—
25#14523967C2211yescharacteristic
26#14523966C2211yescharacteristic
27#14523966C2212yes—
28#14523966C2212yes—
29#14523966C2212yes—
30#14523966C2212yes—
31#14523954C2212yes—
32C4×C67⋊C8C2211yescharacteristic
33C4⋊Q8×C67—21no—
34C4⋊Q8×C67C2211yescharacteristic
35#125×C67—22no—
36#125×C67C2212yes—
37C4×Q8×C67—21no—
38C4×Q8×C67C2211yescharacteristic
39C4×Q8×C67C2212yes—
40C4×Q8×C67C2211yescharacteristic
41C2×Dic134C2311yescharacteristic
42C2×Dic134C2312yes—
43#11761618—22no—
44#11761618—24no—
45C2×C67⋊C8C2311yescharacteristic
46C2×C67⋊C8C2311yescharacteristic
47C2×C67⋊C8C2312yes—
48C4⋊Dic67—22no—
49C4⋊Dic67—21no—
50C4⋊Dic67—22no—
51C4⋊Dic67—21no—
52C4⋊Dic67C2311yescharacteristic
53Dic67⋊C4—22no—
54Dic67⋊C4—21no—
55C4×Dic67—22no—
56C4×Dic67—21no—
57C2×Q8×C67—21no—
58C2×Q8×C67D411yescharacteristic
59C2×Q8×C67C2311yescharacteristic
60C4⋊C4×C67—22no—
61C4⋊C4×C67—21no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67—22no—
65C4⋊C4×C67D411yescharacteristic
66C4⋊C4×C67D412yes—
67C4⋊C4×C67C2311yescharacteristic
68C4⋊C4×C67C2312yes—
69C42×C67—22no—
70C42×C67—21no—
71C42×C67C2312yes—
72C42×C67C2311yescharacteristic
73Q8×C67—22no—
74Q8×C67—21no—
75Q8×C67—42no—
76Q8×C67#4112yes—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—22no—
80C2×C4×C67C2×D412yes—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2×D412yes—
83C2×C4×C67C2×D411yescharacteristic
84C2×C4×C67C2411yescharacteristic
85C2×Dic67—21no—
86C2×Dic67—22no—
87C2×Dic67—21no—
88Dic134—21no—
89Dic134—22no—
90Dic134—21no—
91Dic134—22no—
92C67⋊C8—21no—
93C67⋊C8—22no—
94C67⋊C8—21no—
95C22×C67C22×D411yescharacteristic
96C4×C67—22no—
97C4×C67—22no—
98C4×C67—22no—
99C4×C67—21no—
100C4×C67—42no—
101C4×C67—41no—
102C4×C67#13712yes—
103C4×C67C2×#4111yescharacteristic
104C4×C672−1+411yescharacteristic
105Dic67—41no—
106Dic67—42no—
107Dic67—41no—
108C2×C67#54811yescharacteristic
109C2×C67C2×#13711yescharacteristic
110C2×C67#57811yescharacteristic
111#3374—671nomaximal, Sylow
112C67#337411yescharacteristic, complemented, Sylow
113#557—671no—
114#557D6711yescharacteristic
115#491—672no—
116#487—671no—
117#479—672no—
118#475—671no—
119#462—672no—
120#462—671no—
121#446—671no—
122#441—672no—
123C4×Q16—671no—
124C2×Q16—671no—
125C2×Q16—672no—
126C4⋊Q8—21no—
127C4⋊Q8—671no—
128C4⋊Q8—672no—
129C4⋊Q8C2×D6711yescharacteristic
130#125—22no—
131#125—1341no—
132#125—1342no—
133#125C2×D6712yes—
134C4×Q8—21no—
135C4×Q8—671no—
136C4×Q8—672no—
137C4×Q8—1341no—
138C4×Q8C2×D6711yescharacteristic
139C4×Q8C2×D6712yes—
140C4×Q8C2×D6711yescharacteristic
141#107—671no—
142#106—672no—
143C4⋊C8—671no—
144C4⋊C8—672no—
145#103—671no—
146#103—671no—
147#103—672no—
148#103—672no—
149#103—672no—
150#103—672no—
151#97—672no—
152C4×C8—671no—
153C2×Q8—21no—
154C2×Q8—671no—
155C2×Q8—672no—
156C2×Q8C22⋊D6711yescharacteristic
157C2×Q8C22×D6711yescharacteristic
158Q16—1342no—
159Q16—1344no—
160C2×C8—671no—
161C2×C8—671no—
162C2×C8—672no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—671no—
169C4⋊C4—1342no—
170C4⋊C4—1341no—
171C4⋊C4—1342no—
172C4⋊C4—1341no—
173C4⋊C4—1342no—
174C4⋊C4—1341no—
175C4⋊C4C22⋊D6711yescharacteristic
176C4⋊C4C22⋊D6712yes—
177C4⋊C4C22×D6712yes—
178C4⋊C4C22×D6711yescharacteristic
179C42—22no—
180C42—21no—
181C42—1342no—
182C42—1341no—
183C42C22×D6711yescharacteristic
184C42C22×D6712yes—
185Q8—22no—
186Q8—21no—
187Q8—42no—
188Q8—1341no—
189Q8—1342no—
190Q8—1341no—
191Q8—1342no—
192Q8#1176162912yes—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—1341no—
197C2×C4—1342no—
198C2×C4—1341no—
199C2×C4C2×C22⋊D6712yes—
200C2×C4C2×C22⋊D6712yes—
201C2×C4C2×C22⋊D6711yescharacteristic
202C2×C4C2×C22⋊D6711yescharacteristic
203C2×C4C23×D6711yescharacteristic
204C8—1341no—
205C8—1342no—
206C8—1341no—
207C22C22×C22⋊D6711yescharacteristic
208C4—22no—
209C4—22no—
210C4—22no—
211C4—21no—
212C4—42no—
213C4—41no—
214C4—2682no—
215C4—2681no—
216C4—2681no—
217C4#1452407312yes—
218C4C2×#1176162911yescharacteristic
219C4#1452409111yescharacteristic
220C2#1686405711yescharacteristic
221C2C2×#1452407311yescharacteristic
222C2#1686421111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#557×C67C2
socleC22×C67C22×D4
hypercenter#557D67
hypocenterC67#3374
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
112. C67#3374semidirect