Finite Group Databasebeta

← #134236

Subgroups — #134236

Order 512.

522 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 114 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#20383C211yesmaximal, characteristic
2#11431C212yesmaximal
3#8601C212yesmaximal, complemented
4#7188C212yesmaximal, complemented
5C22×#131C411yescharacteristic
6C2×#408C411yescharacteristic
7C2×#405C2211yescharacteristic
8C22×C4×C8C411yescharacteristic
9#2297C2211yescharacteristic
10#1753—22no—
11C8×C22⋊C4—22no—
12#1729C2211yescharacteristic
13#1728C411yescharacteristic
14#1564—22no—
15#1458C2214yes—
16#1319—22no—
17#1318—22no—
18#1275—22no—
19C22×#41C2×C411yescharacteristic
20C2×#131—22no—
21C2×#131—41no—
22C2×#131—41no—
23C2×#131—42no—
24C2×#131C814yescomplemented
25C22×M16—21no—
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C23×C8—21no—
29C23×C8C2×C411yescharacteristic
30C23×C8C2×C411yescharacteristic
31C2×#117C2311yescharacteristic
32C22×C42C2×C411yescharacteristic
33#431—24no—
34#408—22no—
35#408—22no—
36C2×C22⋊C8—22no—
37#405—41no—
38#405—41no—
39C2×#97D412yes—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—42no—
43C2×C4×C8D412yes—
44#391D411yescharacteristic
45#391Q811yescharacteristic
46#390D411yescharacteristic
47#390Q811yescharacteristic
48#386—22no—
49C4×C22⋊C4—22no—
50#336—22no—
51#336—22no—
52#335—42no—
53#334—42no—
54#324—24no—
55#323—24no—
56C2×#41—21no—
57C2×#41—21no—
58C2×#41—41no—
59C2×#41—41no—
60C2×#41—41no—
61C2×#41—41no—
62C2×#41C2×C812yes—
63C2×#41M1612yes—
64C22×Q8C4⋊C411yescharacteristic
65C22×D4—21no—
66C22×D4C4⋊C411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C22×C411yescharacteristic
70#131—44no—
71#131—44no—
72#131—44no—
73#131—44no—
74C2×M16—21no—
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—24no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—42no—
81C2×M16—42no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16C2×C812yes—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—24no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—42no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8C22⋊C412yes—
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C2×C812yes—
101C22×C8M1612yes—
102C22×C8M1612yes—
103#117—41no—
104#117—41no—
105C2×C4⋊C4C2×D411yescharacteristic
106C2×C4⋊C4C2×Q811yescharacteristic
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4#4112yes—
110C2×C42—21no—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C4⋊C411yescharacteristic
114C2×C42C4⋊C411yescharacteristic
115C2×C42C2×D412yes—
116C4⋊C8—44no—
117C22⋊C8—28no—
118C22⋊C8—44no—
119#97—42no—
120C4×C8—22no—
121C4×C8—22no—
122C4×C8—44no—
123C4×C8—42no—
124C4×C8—42no—
125#95—22no—
126#95—22no—
127C24—21no—
128C24#11711yescharacteristic
129#41—42no—
130#41—42no—
131#41—42no—
132#41—42no—
133#41—42no—
134#41—42no—
135#41—42no—
136#41—42no—
137C2×Q8—21no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8C4⋊C812yes—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—21no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C4⋊C812yes—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4C2×C22⋊C411yescharacteristic
166C22×C4C2×C4⋊C411yescharacteristic
167C22×C4#12112yes—
168C22×C4C22⋊Q812yes—
169C22×C4C22×C811yescharacteristic
170C22×C4C2×M1611yescharacteristic
171C22×C4C2×M1611yescharacteristic
172C22×C4C2×M1611yescharacteristic
173M16—48no—
174M16—42no—
175M16—42no—
176M16—42no—
177M16—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—44no—
195C2×C8—44no—
196C2×C8—42no—
197C2×C8—84no—
198C2×C8C22⋊C812yes—
199C2×C8C22⋊C812yes—
200C2×C8#9911yescharacteristic
201C2×C8#9911yescharacteristic
202C2×C8#10012yes—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C42—22no—
209C42—22no—
210C42—41no—
211C42—41no—
212C42—41no—
213C42—41no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—41no—
218C23—41no—
219C23—42no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23#38011yescharacteristic
224C23#43111yescharacteristic
225C23#43211yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230D4—44no—
231D4—44no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—48no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4C2×C22⋊C811yescharacteristic
261C2×C4#40711yescharacteristic
262C2×C4C2×#9911yescharacteristic
263C2×C4#41011yescharacteristic
264C2×C4C2×#10011yescharacteristic
265C2×C4#41311yescharacteristic
266C2×C4C2×C4⋊C811yescharacteristic
267C2×C4#42311yescharacteristic
268C8—22no—
269C8—42no—
270C8—42no—
271C8—42no—
272C8—164no—
273C8#32312yes—
274C8#32312yes—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—44no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—82no—
287C22—81no—
288C22—81no—
289C22#179711yescharacteristic
290C22#179911yescharacteristic
291C22#180011yescharacteristic
292C4—21no—
293C4—21no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—42no—
298C4—42no—
299C4C2×#32311yescharacteristic
300C4#145811yescharacteristic
301C4#146511yescharacteristic
302C4#146611yescharacteristic
303C2—21no—
304C2—21no—
305C2—41no—
306C2—41no—
307C2—82no—
308C2#855711yescharacteristic
309C2#1280711yescharacteristic
310C2#1280911yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8601C2semidirect
4. #7188C2semidirect
24. C2×#131C8semidirect

Maximal subgroups

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

subgroupindex
#203832
#114312
#86012
#71882