Finite Group Databasebeta

← #127658

Subgroups — #127658

Order 512.

722 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 146 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×#1822C211yesmaximal, characteristic
2C2×#1458C211yesmaximal, characteristic
3#13938C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6C2×#431—21no—
7C2×#407—21no—
8C2×#383C411yescharacteristic
9C2×C4×C4⋊C4C411yescharacteristic
10#1822—22no—
11#1822C414yescomplemented
12C2×#343—21no—
13C2×#343C2212yes—
14#1458—22no—
15#1458C414yescomplemented
16C2×#324C2212yes—
17C2×#323—21no—
18#1441C2212yes—
19#1439C412yes—
20#1437—21no—
21#1436—21no—
22#1434—21no—
23C22⋊#95—21no—
24C24×C4D411yescharacteristic
25C22×M16—22no—
26C2×#117—41no—
27C2×#117—41no—
28C2×#117—42no—
29C2×#117C2×C412yes—
30C2×#117C2×C412yes—
31C2×#117D412yes—
32C2×#117Q812yes—
33C22×C4⋊C4C2×C412yes—
34C22×C4⋊C4D411yescharacteristic
35C22×C4⋊C4C2311yescharacteristic
36C22×C22⋊C4—22no—
37C22×C22⋊C4C2×C411yescharacteristic
38C22×C22⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42D411yescharacteristic
42#431—24no—
43#431—42no—
44C2×C4⋊C8—41no—
45#407—24no—
46#407—42no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—41no—
49C2×C22⋊C8—41no—
50C2×C4×C8—41no—
51#383—42no—
52C4×C4⋊C4—41no—
53C4×C4⋊C4—41no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95C2×C412yes—
61#343—24no—
62#343—22no—
63#343C2×C414yes—
64#324—22no—
65#324C2×C414yes—
66#323—24no—
67C25C22⋊C411yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—42no—
75C23×C4C22⋊C412yes—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C4⋊C412yes—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—44no—
86C2×M16—44no—
87C22×C8—42no—
88C22×C8—42no—
89#117—42no—
90#117—42no—
91#117—42no—
92#117—42no—
93#117—42no—
94#117—42no—
95#117—42no—
96#117—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4C22⋊C412yes—
104C2×C4⋊C4C4⋊C412yes—
105C2×C4⋊C4C22×C411yescharacteristic
106C2×C4⋊C4C22×C411yescharacteristic
107C2×C4⋊C4C2×D411yescharacteristic
108C2×C4⋊C4C2×Q811yescharacteristic
109C2×C22⋊C4—41no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—44no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—82no—
116C2×C22⋊C4C4212yes—
117C2×C22⋊C4C22⋊C412yes—
118C2×C22⋊C4C4⋊C412yes—
119C2×C22⋊C4C4⋊C412yes—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—22no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—42no—
127C2×C42—82no—
128C2×C42C4212yes—
129C2×C42C22⋊C412yes—
130C2×C42C4⋊C412yes—
131C2×C42C4⋊C412yes—
132C4⋊C8—44no—
133C22⋊C8—24no—
134C22⋊C8—42no—
135C22⋊C8—44no—
136C22⋊C8—44no—
137C4×C8—44no—
138#95—42no—
139#95—44no—
140#95—42no—
141#95—41no—
142#95—41no—
143#95—42no—
144#95—44no—
145#95—82no—
146C24—21no—
147C24—22no—
148C24—21no—
149C24—22no—
150C24—42no—
151C24—41no—
152C24—41no—
153C24#9512yes—
154C24C2×C22⋊C411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#9512yes—
200C22×C4#9512yes—
201C22×C4#9512yes—
202C22×C4#9912yes—
203C22×C4#9911yescharacteristic
204C22×C4#9911yescharacteristic
205C22×C4C2×C4211yescharacteristic
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C4⋊C411yescharacteristic
209C22×C4C2×C4⋊C411yescharacteristic
210C22×C4C2×C4⋊C411yescharacteristic
211C22×C4#13514yes—
212M16—44no—
213M16—44no—
214M16—84no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—84no—
218C2×C8—84no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—88no—
232C42—22no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—84no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—44no—
250C23—44no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—82no—
259C23—82no—
260C23C2×#9511yescharacteristic
261C23C2×#9911yescharacteristic
262C23C2×#9911yescharacteristic
263C23#41612yes—
264C23#42112yes—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4#34212yes—
317C2×C4#34212yes—
318C2×C4#41612yes—
319C2×C4#42712yes—
320C8—84no—
321C8—84no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—44no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22—84no—
336C22—82no—
337C22—82no—
338C22#151612yes—
339C22#151612yes—
340C22#173311yescharacteristic
341C22#173511yescharacteristic
342C22C2×#34211yescharacteristic
343C22#212012yes—
344C22#213411yescharacteristic
345C22#213511yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—84no—
357C4—162no—
358C2—22no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2#802411yescharacteristic
363C2#1184712yes—
364C2#1184912yes—
365C2#1257311yescharacteristic
366C2#1258711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×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
11. #1822C4semidirect
15. #1458C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18222
C2×#14582
#139382
#72512