Finite Group Databasebeta

← #120844

Subgroups — #120844

Order 512.

534 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 116 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#21948C211yesmaximal, characteristic
2#8112C211yesmaximal, characteristic
3C2×#1306C211yesmaximal, characteristic
4#8024C211yesmaximal, characteristic
5#7946C211yesmaximal, characteristic
6#7943C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#381C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1810—22no—
18#1306—22no—
19#1306—22no—
20#1294—22no—
21#1284—22no—
22#1284—22no—
23#1282C414yescomplemented
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117Q812yes—
31C22×C22⋊C4—21no—
32C2×#108—24no—
33#407—21no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407D412yes—
44#407D412yes—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8C2×C412yes—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8D412yes—
56C2×C22⋊C8Q812yes—
57#381—21no—
58#381—21no—
59#381—21no—
60#381—21no—
61#381—21no—
62#381D412yes—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67#379—21no—
68#379—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4C2×C412yes—
73C2×#95—21no—
74C2×#95—21no—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—41no—
78C23×C4—82no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×Q811yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—82no—
100C2×M16—82no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—82no—
109#117—82no—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—82no—
122C2×C42—22no—
123C2×C42—22no—
124#108—84no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8C4⋊C414yes—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—41no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149C24—42no—
150C24—42no—
151C24—81no—
152C24#11711yescharacteristic
153C24C22⋊Q811yescharacteristic
154C24#12311yescharacteristic
155C24#12411yescharacteristic
156C24#12411yescharacteristic
157C24#12511yescharacteristic
158C24#12611yescharacteristic
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—82no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×Q811yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4#12311yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12411yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12511yescharacteristic
200C22×C4#12511yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4C4⋊Q811yescharacteristic
207M16—44no—
208M16—44no—
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213C2×C8—24no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C42—82no—
234C42—82no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—88no—
257C23#38211yescharacteristic
258C23#38711yescharacteristic
259C23#38811yescharacteristic
260C23#39311yescharacteristic
261C23#39711yescharacteristic
262C23#39811yescharacteristic
263C23#40011yescharacteristic
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—164no—
302C2×C4#38511yescharacteristic
303C2×C4#38711yescharacteristic
304C2×C4#38811yescharacteristic
305C2×C4#38911yescharacteristic
306C2×C4#39411yescharacteristic
307C2×C4#39811yescharacteristic
308C2×C4#40011yescharacteristic
309C2×C4#40011yescharacteristic
310C8—44no—
311C8—82no—
312C8—82no—
313C8—82no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—44no—
334C22—44no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22#144211yescharacteristic
341C22#183312yes—
342C22#203212yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—162no—
352C4—162no—
353C4—162no—
354C2—22no—
355C2—22no—
356C2—41no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2—41no—
361C2#1125911yescharacteristic
362C2#1127311yescharacteristic
363C2#1127911yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#81122
C2×#13062
#80242
#79462
#79432
#79252