Finite Group Databasebeta

← #121958

Subgroups — #121958

Order 512.

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

normal subgroups only

This group has 544 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#21933C211yesmaximal, characteristic
2#8009C211yesmaximal, characteristic
3#7946C211yesmaximal, characteristic
4#7943C211yesmaximal, characteristic
5#7943C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7913C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#381C2211yescharacteristic
15#1851—22no—
16#1850—22no—
17#1798C414yescomplemented
18#1293—22no—
19#1284—22no—
20#1284—22no—
21#1284—22no—
22#1282—22no—
23#1280—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117C2×C412yes—
31C22×C22⋊C4—21no—
32C2×C4⋊C8—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#407—21no—
44#407—21no—
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#381—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67#379—21no—
68#379—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4D412yes—
71C4×C22⋊C4D412yes—
72C4×C22⋊C4D412yes—
73C2×#95—21no—
74C2×#95Q812yes—
75#336—24no—
76C25#4111yescharacteristic
77C23×C4—22no—
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—41no—
94C2×M16—41no—
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—22no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—82no—
122C2×C42—22no—
123C2×C42C22⋊C414yes—
124C4⋊C8—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—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—41no—
147#95—41no—
148#95—41no—
149C24—42no—
150C24—42no—
151C24—81no—
152C24#11711yescharacteristic
153C24C22⋊Q811yescharacteristic
154C24#12311yescharacteristic
155C24#12311yescharacteristic
156C24#12311yescharacteristic
157C24#12511yescharacteristic
158C24#12611yescharacteristic
159C22×C4—24no—
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—44no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4#11711yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4C4×Q811yescharacteristic
184C22×C4C4×Q811yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4#12311yescharacteristic
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#12611yescharacteristic
203C22×C4#12611yescharacteristic
204C22×C4#12611yescharacteristic
205C22×C4#12711yescharacteristic
206C22×C4C4⋊Q811yescharacteristic
207M16—82no—
208M16—82no—
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213C2×C8—44no—
214C2×C8—44no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C42—44no—
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#38111yescharacteristic
258C23#38711yescharacteristic
259C23#38711yescharacteristic
260C23#39711yescharacteristic
261C23#39811yescharacteristic
262C23#39911yescharacteristic
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—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—164no—
302C2×C4#38711yescharacteristic
303C2×C4#38711yescharacteristic
304C2×C4#39011yescharacteristic
305C2×C4#39111yescharacteristic
306C2×C4#39711yescharacteristic
307C2×C4#39811yescharacteristic
308C2×C4#39911yescharacteristic
309C2×C4#40011yescharacteristic
310C8—82no—
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#144511yescharacteristic
341C22#182812yes—
342C22#209812yes—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—84no—
352C4—162no—
353C4—162no—
354C2—22no—
355C2—22no—
356C2—41no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2—41no—
361C2#1134911yescharacteristic
362C2#1135111yescharacteristic
363C2#1136911yescharacteristic
subgroupG/H
center, socleC22#1445
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
17. #1798C4semidirect

Maximal subgroups

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

subgroupindex
#219332
#80092
#79462
#79432
#79252
#79132