Finite Group Databasebeta

← #116504

Subgroups — #116504

Order 512.

710 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 124 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, complemented
2#21741C211yesmaximal, characteristic
3#17582C212yesmaximal
4#8115C212yesmaximal, complemented
5#7982C211yesmaximal, characteristic, complemented
6C2×#525C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10#2280C2212yes—
11#1870—22no—
12#1869—22no—
13#1855—24no—
14#1818C412yescomplemented
15#1810—22no—
16#1780C412yescomplemented
17#1516—24no—
18#1513—24no—
19#1508—22no—
20#1507—22no—
21#1306—24no—
22#1289—22no—
23C22×#41C2×C411yescharacteristic
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—22no—
27C23×C8D412yes—
28#525—22no—
29#525—22no—
30#525—22no—
31#525—22no—
32C2×#123—21no—
33C2×C22⋊Q8D411yescharacteristic
34C2×C22⋊Q8Q811yescharacteristic
35C2×#121D411yescharacteristic
36C2×#121Q811yescharacteristic
37C2×#120—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×#117—22no—
41C2×#117C2×C411yescharacteristic
42C2×#108—24no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407D414yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8D412yes—
58#381D414yes—
59#380—22no—
60#380—22no—
61#379—22no—
62C4×C22⋊C4—22no—
63C4×C22⋊C4—22no—
64#341—24no—
65#336—24no—
66#328—28no—
67#327—28no—
68C25#4111yescharacteristic
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—82no—
72C22×Q8C4⋊C411yescharacteristic
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4C4⋊C411yescharacteristic
77C23×C4—21no—
78C23×C4—82no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D412yes—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4C2×Q811yescharacteristic
86C23×C4#4112yes—
87C23×C4#4112yes—
88C23×C4#4111yescharacteristic
89C2×M16—24no—
90C2×M16—42no—
91C2×M16—42no—
92C2×M16—82no—
93C2×M16—84no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—42no—
98C22×C8—82no—
99C22×C8C22⋊C414yes—
100#123—42no—
101#123—42no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106#121—22no—
107#121—22no—
108#121—22no—
109#121—22no—
110#120—42no—
111#120—42no—
112C4×D4—41no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118#117—22no—
119#117—22no—
120#117—84no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C4⋊C411yescharacteristic
125C2×C4⋊C4C4⋊C411yescharacteristic
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C42—21no—
135C2×C42—42no—
136#108—84no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—42no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C22⋊C8—84no—
147#95—24no—
148#95—24no—
149C24—21no—
150C24—42no—
151C24—81no—
152C24—81no—
153C24#11711yescharacteristic
154C24#12111yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24#12312yes—
157C24#12412yes—
158#41—84no—
159#41—84no—
160C2×Q8—22no—
161C2×Q8—81no—
162C2×D4—22no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—82no—
169C2×D4—82no—
170C2×D4—81no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—44no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—44no—
183C22×C4—81no—
184C22×C4—82no—
185C22×C4—81no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4C2×C4⋊C411yescharacteristic
190C22×C4C4×D412yes—
191C22×C4C4×D412yes—
192C22×C4#12011yescharacteristic
193C22×C4#12012yes—
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4#12111yescharacteristic
198C22×C4C22⋊Q812yes—
199C22×C4C22⋊Q812yes—
200C22×C4C22⋊Q811yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12312yes—
203C22×C4#12411yescharacteristic
204C22×C4#12612yes—
205C22×C4#12711yescharacteristic
206M16—48no—
207M16—84no—
208M16—84no—
209C2×C8—24no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—42no—
214C2×C8—44no—
215C2×C8—82no—
216C2×C8—84no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—84no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—84no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C42—41no—
234C42—41no—
235C42—84no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—41no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—41no—
255C23—88no—
256C23#38011yescharacteristic
257C23#38812yes—
258C23#39311yescharacteristic
259C23#39412yes—
260C23#39911yescharacteristic
261Q8—82no—
262D4—82no—
263D4—82no—
264D4—82no—
265D4—84no—
266D4—84no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4#38512yes—
297C2×C4#38611yescharacteristic
298C2×C4#39011yescharacteristic
299C2×C4#39211yescharacteristic
300C2×C4#39612yes—
301C2×C4#39711yescharacteristic
302C8—44no—
303C8—82no—
304C8—84no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—44no—
320C22—82no—
321C22—82no—
322C22—84no—
323C22—84no—
324C22—84no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#143911yescharacteristic
329C22#181412yes—
330C4—21no—
331C4—21no—
332C4—22no—
333C4—22no—
334C4—21no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—164no—
339C4#191312yes—
340C2—22no—
341C2—42no—
342C2—41no—
343C2—41no—
344C2—42no—
345C2—161no—
346C2#1104611yescharacteristic
347C2#1110811yescharacteristic
348C2#1111011yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC2×#525C22
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
1. #21948C2semidirect
4. #8115C2semidirect
5. #7982C2semidirect
14. #1818C4semidirect
16. #1780C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#217412
#175822
#81152
#79822