Finite Group Databasebeta

← #121844

Subgroups — #121844

Order 512.

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

normal subgroups only

This group has 507 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#21949C211yesmaximal, characteristic
2#8112C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8057C211yesmaximal, characteristic
5#8009C211yesmaximal, characteristic
6C2×#1293C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2280C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811C414yescomplemented
18#1306—22no—
19#1305—22no—
20#1298—22no—
21#1293—22no—
22#1293—22no—
23#1282—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C22×M16—21no—
29C22×M16C2×C412yes—
30C23×C8—21no—
31C2×#117—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#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407—21no—
48#407—21no—
49#407C2×C412yes—
50#407C2×C412yes—
51#407D412yes—
52#407D412yes—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8D412yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8Q812yes—
69#381—21no—
70#380—21no—
71#380—21no—
72#379Q812yes—
73C4×C22⋊C4—21no—
74C4×C22⋊C4—21no—
75#341—24no—
76C25C2×Q811yescharacteristic
77C23×C4—82no—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×Q811yescharacteristic
84C23×C4#4111yescharacteristic
85C23×C4#4111yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—22no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—82no—
103C2×M16—82no—
104C2×M16—82no—
105C2×M16—82no—
106C2×M16—82no—
107C22×C8—22no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—82no—
115C22×C8C22⋊C414yes—
116#117—82no—
117C2×C4⋊C4—41no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C42—41no—
122#108—84no—
123C22⋊C8—22no—
124C22⋊C8—22no—
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—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147#95—41no—
148#95—41no—
149C24—42no—
150C24—81no—
151C24—81no—
152C24C4×Q811yescharacteristic
153C24C22⋊Q811yescharacteristic
154C24C22⋊Q811yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24#12511yescharacteristic
157C24#12511yescharacteristic
158C24C4⋊Q811yescharacteristic
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—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4#11711yescharacteristic
176C22×C4#11711yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×Q811yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4C22⋊Q811yescharacteristic
183C22×C4C22⋊Q811yescharacteristic
184C22×C4C22⋊Q811yescharacteristic
185C22×C4#12311yescharacteristic
186C22×C4#12311yescharacteristic
187C22×C4#12311yescharacteristic
188C22×C4#12311yescharacteristic
189C22×C4#12311yescharacteristic
190C22×C4#12311yescharacteristic
191C22×C4#12311yescharacteristic
192C22×C4#12311yescharacteristic
193C22×C4#12411yescharacteristic
194C22×C4#12411yescharacteristic
195C22×C4#12511yescharacteristic
196C22×C4#12511yescharacteristic
197C22×C4#12611yescharacteristic
198C22×C4#12611yescharacteristic
199C22×C4#12611yescharacteristic
200C22×C4#12611yescharacteristic
201C22×C4#12711yescharacteristic
202M16—44no—
203M16—44no—
204M16—82no—
205M16—82no—
206M16—82no—
207M16—82no—
208M16—82no—
209M16—82no—
210M16—82no—
211M16—82no—
212C2×C8—24no—
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C4⋊C4—82no—
227C22⋊C4—44no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—82no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—88no—
253C23#38711yescharacteristic
254C23#38711yescharacteristic
255C23#39111yescharacteristic
256C23#39811yescharacteristic
257C23#39811yescharacteristic
258C23#39911yescharacteristic
259C23#40011yescharacteristic
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
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—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4#38111yescharacteristic
292C2×C4#38711yescharacteristic
293C2×C4#38711yescharacteristic
294C2×C4#39011yescharacteristic
295C2×C4#39711yescharacteristic
296C2×C4#39711yescharacteristic
297C2×C4#39911yescharacteristic
298C2×C4#40011yescharacteristic
299C8—44no—
300C8—82no—
301C8—82no—
302C8—82no—
303C8—82no—
304C8—82no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—44no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22#144511yescharacteristic
333C22#210312yes—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—162no—
342C4#182812yes—
343C2—22no—
344C2—41no—
345C2—41no—
346C2—41no—
347C2—41no—
348C2—41no—
349C2—41no—
350C2#1135111yescharacteristic
351C2#1136311yescharacteristic
352C2#1136811yescharacteristic
subgroupG/H
center, socleC22#1445
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×Q8
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. #1811C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#81122
#81082
#80572
#80092
C2×#12932
#79252