Finite Group Databasebeta

← #22561349

Subgroups — #22561349

Order 8576.

290 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14524042C211yesmaximal, characteristic
2C2×#14523968C211yesmaximal, characteristic
3C2×#14523966C211yesmaximal, characteristic
4#16863301C211yesmaximal, characteristic
5#16863158C211yesmaximal, characteristic
6#16863150C211yesmaximal, characteristic
7#340×C67C211yesmaximal, characteristic
8C22×Dic134—21no—
9C22×Dic134C2211yescharacteristic
10C22×C67⋊C8C2211yescharacteristic
11#14524042—22no—
12#14524042—22no—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×Dic67⋊C4—21no—
15C2×C4×Dic67C2211yescharacteristic
16#14523968—22no—
17#14523966—22no—
18C67⋊#95—21no—
19C67⋊#95—21no—
20C22×C8×C67C2211yescharacteristic
21C2×C4⋊C4×C67C2211yescharacteristic
22C2×C4⋊C4×C67C2211yescharacteristic
23C2×Dic134—21no—
24C2×Dic134—21no—
25C2×Dic134—21no—
26C2×Dic134—21no—
27C2×Dic134—24no—
28C2×Dic134—21no—
29C2×Dic134—42no—
30C2×Dic134—42no—
31C2×Dic134Q812yescomplemented
32C22×Dic67—21no—
33C22×Dic67—21no—
34C22×Dic67D411yescharacteristic
35C22×Dic67D411yescharacteristic
36C2×C67⋊C8—21no—
37C2×C67⋊C8—21no—
38C2×C67⋊C8—21no—
39C4⋊Dic67—21no—
40C4⋊Dic67D412yes—
41Dic67⋊C4—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×C67—21no—
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×C67C2×D411yescharacteristic
62C2×C4×C67C2×D411yescharacteristic
63C2×C4×C67C2×Q811yescharacteristic
64C2×C4×C67#4111yescharacteristic
65C2×C4×C67#4111yescharacteristic
66C2×C4×C67#4111yescharacteristic
67C2×Dic67—21no—
68C2×Dic67—21no—
69C2×Dic67—21no—
70C2×Dic67—21no—
71C2×Dic67—22no—
72C2×Dic67—22no—
73C2×Dic67—41no—
74C2×Dic67—41no—
75C2×Dic67—41no—
76C2×Dic67—41no—
77C2×Dic67SD1612yes—
78C2×Dic67Q1612yes—
79Dic134—22no—
80Dic134—22no—
81Dic134—48no—
82Dic134—41no—
83Dic134—41no—
84C8×C67—41no—
85C67⋊C8—41no—
86C22×C67#12011yescharacteristic
87C22×C67C22⋊Q811yescharacteristic
88C22×C67#12411yescharacteristic
89C22×C67C2×SD1611yescharacteristic
90C22×C67C2×Q1611yescharacteristic
91C22×C67AGL(1,ℤ/8)11yescharacteristic
92C22×C67#13711yescharacteristic
93C4×C67—42no—
94C4×C67—81no—
95C4×C67C22⋊Q811yescharacteristic
96C4×C67C22⋊Q811yescharacteristic
97C4×C67#12411yescharacteristic
98C4×C67#12411yescharacteristic
99Dic67—22no—
100Dic67—22no—
101Dic67—42no—
102Dic67—42no—
103C2×C67#39311yescharacteristic
104C2×C67#45011yescharacteristic
105C2×C67#45111yescharacteristic
106C2×C67#47511yescharacteristic
107C2×C67#47711yescharacteristic
108C2×C67#48711yescharacteristic
109C2×C67#48911yescharacteristic
110#2024—671nomaximal, Sylow
111C67#202411yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—671no—
113C2×#103—671no—
114C2×#103—671no—
115#391—671no—
116#340—671no—
117#340D6711yescharacteristic
118#336—671no—
119C22×Q8—671no—
120C22×Q8—1341no—
121C22×C8—671no—
122C22×C8C2×D6711yescharacteristic
123C4⋊Q8—1342no—
124C4⋊Q8—1342no—
125C2×C4⋊C4—671no—
126C2×C4⋊C4—1341no—
127C2×C4⋊C4C2×D6711yescharacteristic
128C2×C4⋊C4C2×D6711yescharacteristic
129C2×C42—671no—
130#103—1342no—
131#103—1342no—
132#95—1341no—
133#95—1341no—
134C2×Q8—2682no—
135C2×Q8—2682no—
136C2×Q8—672no—
137C2×Q8—1344no—
138C2×Q8—1341no—
139C2×Q8—1341no—
140C2×Q8—1341no—
141C2×Q8—1341no—
142C2×Q8—1341no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—671no—
146C22×C4—671no—
147C22×C4—1341no—
148C22×C4—1341no—
149C22×C4C22×D6711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1341no—
154C2×C8—1341no—
155C2×C8—1341no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—672no—
160C4⋊C4—1344no—
161C4⋊C4—1341no—
162C4⋊C4D26812yes—
163C42—1341no—
164C42—1341no—
165C23D4×D6711yescharacteristic
166Q8—2688no—
167Q8—2681no—
168Q8—2681no—
169Q8—1342no—
170Q8—1342no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—2681no—
178C2×C4—2681no—
179C2×C4—2681no—
180C2×C4—2681no—
181C2×C4—672no—
182C2×C4—672no—
183C2×C4—1342no—
184C2×C4—1342no—
185C2×C4—1341no—
186C2×C4—1341no—
187C2×C4—1341no—
188C2×C4—1341no—
189C2×C4D4×D6711yescharacteristic
190C2×C4C2×D26811yescharacteristic
191C2×C4Q8×D6711yescharacteristic
192C2×C4#1176162711yescharacteristic
193C2×C4#1176162811yescharacteristic
194C2×C4#1176162811yescharacteristic
195C8—41no—
196C8—2681no—
197C22#1452401311yescharacteristic
198C22#1452402111yescharacteristic
199C22#1452402911yescharacteristic
200C22SD16×D6711yescharacteristic
201C22Q16×D6711yescharacteristic
202C22#1452406711yescharacteristic
203C22#1452407211yescharacteristic
204C4—42no—
205C4—81no—
206C4—2682no—
207C4—2682no—
208C4—1342no—
209C4—1342no—
210C4#1452402111yescharacteristic
211C4#1452402311yescharacteristic
212C4#1452402911yescharacteristic
213C4#1452403111yescharacteristic
214C2#1686330911yescharacteristic
215C2#1686354811yescharacteristic
216C2#1686355311yescharacteristic
217C2#1686368011yescharacteristic
218C2#1686369011yescharacteristic
219C2#1686374511yescharacteristic
220C2#1686375411yescharacteristic
subgroupG/H
centerC23D4×D67
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
Fitting#340×C67C2
socleC23×C67C2×D4
hypercenter#340D67
hypocenterC67#2024
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
31. C2×Dic134Q8semidirect
111. C67#2024semidirect