Finite Group Databasebeta

← #116458

Subgroups — #116458

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21300C211yesmaximal, characteristic
2C2×#1568C211yesmaximal, characteristic
3C2×#1537C212yesmaximal
4C2×#1307C212yesmaximal
5#7875C211yesmaximal, characteristic
6C2×#533C2211yescharacteristic
7C2×#432C2212yes—
8C2×#432C2212yes—
9C2×#423C2211yescharacteristic
10C22×C4×C8C2211yescharacteristic
11#1984—21no—
12#1984—21no—
13#1984—21no—
14#1984—21no—
15#1729C412yes—
16#1729C412yes—
17#1568—24no—
18C2×#335—21no—
19C2×#334—21no—
20#1537—24no—
21#1537—24no—
22C2×#330—24no—
23#1307—24no—
24#1277—21no—
25#1277—21no—
26C22×M16C2×C412yes—
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29#533—22no—
30#533—22no—
31#533—22no—
32#533—22no—
33C2×C4⋊Q8Q811yescharacteristic
34C2×C4⋊Q8Q811yescharacteristic
35C2×#125D412yes—
36C2×C22⋊Q8—21no—
37C2×C22⋊Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×#117C2×C412yes—
41C22×C42C2311yescharacteristic
42#432—22no—
43#432—22no—
44#432—22no—
45#432—22no—
46#432—22no—
47#432—22no—
48#432D414yes—
49#432D414yes—
50#423—22no—
51#423—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58C2×#97—22no—
59C2×#97—22no—
60C2×C4×C8—22no—
61C2×C4×C8—22no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66#336—22no—
67#336—22no—
68#336—22no—
69#336—22no—
70#335—24no—
71#334—24no—
72#330—216no—
73C22×Q8—21no—
74C22×Q8—21no—
75C23×C4C22×C411yescharacteristic
76C23×C4#4111yescharacteristic
77C23×C4#4111yescharacteristic
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—41no—
87C22×C8—41no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8C22⋊C411yescharacteristic
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98#125—24no—
99#125—24no—
100C22⋊Q8—42no—
101C22⋊Q8—42no—
102C22⋊Q8—42no—
103C22⋊Q8—42no—
104C4×Q8—41no—
105C4×Q8—42no—
106C4×Q8—42no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110#117—24no—
111#117—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C4⋊C412yes—
117C2×C4⋊C4C4⋊C412yes—
118C2×C22⋊C4—22no—
119C2×C42—22no—
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×Q811yescharacteristic
126C2×C42#4112yes—
127C2×C42#4112yes—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C4⋊C8—42no—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#97—24no—
140#97—24no—
141#97—42no—
142#97—42no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8—21no—
148C4×C8—21no—
149C24#11711yescharacteristic
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—41no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C2×C4⋊C411yescharacteristic
169C22×C4C4×D412yes—
170C22×C4C4×D412yes—
171C22×C4#12111yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12112yes—
174C22×C4C22⋊Q811yescharacteristic
175C22×C4C22⋊Q811yescharacteristic
176C22×C4#12312yes—
177C22×C4#12412yes—
178C22×C4#12612yes—
179M16—84no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—42no—
193C2×C8—44no—
194C2×C8—44no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C2×C8#10112yes—
202C2×C8#10112yes—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C22⋊C4—44no—
212C42—42no—
213C42—42no—
214C42#12012yes—
215C42#12011yescharacteristic
216C42#12112yes—
217C42#12112yes—
218C42C22⋊Q812yes—
219C42C22⋊Q812yes—
220C42#12312yes—
221C42#12311yescharacteristic
222C42#12411yescharacteristic
223C42#12711yescharacteristic
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23#38011yescharacteristic
231C23#38812yes—
232Q8—84no—
233Q8—82no—
234Q8—82no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#38512yes—
260C2×C4#38611yescharacteristic
261C2×C4#39011yescharacteristic
262C2×C4#39211yescharacteristic
263C2×C4#39311yescharacteristic
264C2×C4#39412yes—
265C2×C4#39612yes—
266C2×C4#39711yescharacteristic
267C2×C4#39911yescharacteristic
268C2×C4C2×#10111yescharacteristic
269C2×C4C2×#10111yescharacteristic
270C2×C4#41311yescharacteristic
271C2×C4#41311yescharacteristic
272C8—22no—
273C8—22no—
274C8—41no—
275C8—41no—
276C8—84no—
277C8—84no—
278C8—82no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#143911yescharacteristic
287C22#180111yescharacteristic
288C22#180111yescharacteristic
289C22#181411yescharacteristic
290C22#181411yescharacteristic
291C22#181411yescharacteristic
292C22#181411yescharacteristic
293C4—22no—
294C4—21no—
295C4—22no—
296C4—21no—
297C4—22no—
298C4—21no—
299C4—21no—
300C4—82no—
301C4—82no—
302C4#190111yescharacteristic
303C4#190111yescharacteristic
304C4#196611yescharacteristic
305C4#196711yescharacteristic
306C2—42no—
307C2#863111yescharacteristic
308C2#863311yescharacteristic
309C2#863511yescharacteristic
310C2#863511yescharacteristic
311C2#1104511yescharacteristic
312C2#1110411yescharacteristic
313C2#1110811yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#117
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#213002
C2×#15682
C2×#15372
C2×#13072
#78752