Finite Group Databasebeta

← #135307

Subgroups — #135307

Order 512.

662 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 100 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#14340C211yesmaximal, characteristic
2#10728C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6C2×#407—21no—
7C2×#384C411yescharacteristic
8C2×#382C411yescharacteristic
9C2×#380—21no—
10#1817—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324C2212yes—
14C2×#323—21no—
15#1441—21no—
16#1441C2212yes—
17#1439—21no—
18#1439C412yes—
19C4⋊#95—21no—
20C24×C4D411yescharacteristic
21C22×M16—22no—
22C2×#117—24no—
23C2×#117—42no—
24C2×#117—42no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4C2311yescharacteristic
31C22×C22⋊C4—22no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36#407—24no—
37#407—42no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—42no—
40#384—42no—
41#382—42no—
42#380—42no—
43#380—42no—
44C2×#95—21no—
45C2×#95—22no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95C2×C412yes—
49#343—24no—
50#343—42no—
51#336—42no—
52#324—24no—
53#323—42no—
54C25C22⋊C411yescharacteristic
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—22no—
61C23×C4—41no—
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4D812yes—
67C23×C4SD1612yes—
68C23×C4C22×C411yescharacteristic
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—42no—
78C22×C8—42no—
79#117—416no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—82no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4#4114yes—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—82no—
104C2×C22⋊C4—82no—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—22no—
110C2×C42—41no—
111C22⋊C8—48no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114#95—42no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—82no—
119#95—82no—
120C24—21no—
121C24—21no—
122C24—21no—
123C24—21no—
124C24—21no—
125C24—21no—
126C24—41no—
127C24—41no—
128C24—41no—
129C24—41no—
130C24#10212yes—
131C24C2×C22⋊C411yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4#9912yes—
182C22×C4#9912yes—
183C22×C4#10212yes—
184C22×C4#10412yes—
185C22×C4#11712yes—
186C22×C4#12312yes—
187C22×C4#12312yes—
188C22×C4C2×D811yescharacteristic
189C22×C4C2×SD1611yescharacteristic
190C22×C4#13712yes—
191M16—44no—
192M16—44no—
193M16—82no—
194M16—82no—
195C2×C8—44no—
196C2×C8—82no—
197C2×C8—82no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—88no—
210C42—28no—
211C42—44no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23—81no—
237C23—81no—
238C23—81no—
239C23#38111yescharacteristic
240C23C2×#9911yescharacteristic
241C23C2×#9911yescharacteristic
242C23C2×#10211yescharacteristic
243C23#41911yescharacteristic
244C23C2×#10411yescharacteristic
245C23#42111yescharacteristic
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—48no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4#32714yes—
297C2×C4#48012yes—
298C2×C4#48312yes—
299C8—84no—
300C8—162no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—44no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×#32711yescharacteristic
325C22#150811yescharacteristic
326C22#151111yescharacteristic
327C22#151911yescharacteristic
328C22#181911yescharacteristic
329C22#182311yescharacteristic
330C22#182611yescharacteristic
331C22#212211yescharacteristic
332C22#212311yescharacteristic
333C22#213411yescharacteristic
334C22#213511yescharacteristic
335C4—22no—
336C4—24no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—81no—
342C4—162no—
343C4—162no—
344C2—21no—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—82no—
349C2#871211yescharacteristic
350C2#1169911yescharacteristic
351C2#1170011yescharacteristic
352C2#1181811yescharacteristic
353C2#1181911yescharacteristic
354C2#1285511yescharacteristic
355C2#1286311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#143402
#107282
#83002
#72512