Finite Group Databasebeta

← #116502

Subgroups — #116502

Order 512.

746 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 132 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#21938C211yesmaximal, characteristic, complemented
2#21741C211yesmaximal, characteristic
3#17592C212yesmaximal
4#8107C212yesmaximal, complemented
5C2×#1293C211yesmaximal, characteristic, complemented
6C2×#525C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10C2×#380C2212yes—
11#1870—22no—
12#1869—22no—
13#1853—24no—
14#1818C412yescomplemented
15#1807—22no—
16#1780C412yescomplemented
17#1520—22no—
18#1519—22no—
19#1513—24no—
20#1504—24no—
21#1305—24no—
22#1293—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×#117C2×C411yescharacteristic
41C22×C4⋊C4—22no—
42C22×C22⋊C4—22no—
43C2×#106—24no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407—22no—
49#407D414yes—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8C2×C412yes—
58C2×C22⋊C8D412yes—
59#380—22no—
60#380—22no—
61#380—22no—
62#380D414yes—
63C2×#95—22no—
64#340—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—42no—
79C23×C4—82no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C23×C4C2×Q811yescharacteristic
87C23×C4#4112yes—
88C23×C4#4112yes—
89C23×C4#4111yescharacteristic
90C2×M16—24no—
91C2×M16—42no—
92C2×M16—42no—
93C2×M16—82no—
94C2×M16—84no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—42no—
99C22×C8—82no—
100C22×C8C22⋊C414yes—
101#123—42no—
102#123—42no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107#121—22no—
108#121—22no—
109#121—22no—
110#121—22no—
111#120—42no—
112#120—42no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119#117—22no—
120#117—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—84no—
126C2×C4⋊C4C4⋊C411yescharacteristic
127C2×C4⋊C4C4⋊C411yescharacteristic
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—84no—
136C2×C42—21no—
137#106—84no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C22⋊C8—42no—
144C22⋊C8—41no—
145C22⋊C8—42no—
146C22⋊C8—41no—
147C22⋊C8—84no—
148#95—24no—
149#95—42no—
150C24—24no—
151C24—21no—
152C24—81no—
153C24—81no—
154C24#11711yescharacteristic
155C24#12112yes—
156C24#12111yescharacteristic
157C24C22⋊Q811yescharacteristic
158C24#12612yes—
159#41—84no—
160#41—84no—
161C2×Q8—22no—
162C2×Q8—81no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—41no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—44no—
169C2×D4—82no—
170C2×D4—82no—
171C2×D4—81no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—44no—
184C22×C4—81no—
185C22×C4—82no—
186C22×C4—81no—
187C22×C4—82no—
188C22×C4—81no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C4⋊C411yescharacteristic
194C22×C4C4×D412yes—
195C22×C4C4×D412yes—
196C22×C4#12011yescharacteristic
197C22×C4#12012yes—
198C22×C4#12112yes—
199C22×C4#12112yes—
200C22×C4#12111yescharacteristic
201C22×C4C22⋊Q812yes—
202C22×C4C22⋊Q812yes—
203C22×C4C22⋊Q811yescharacteristic
204C22×C4#12311yescharacteristic
205C22×C4#12312yes—
206C22×C4#12312yes—
207C22×C4#12411yescharacteristic
208C22×C4#12412yes—
209C22×C4#12711yescharacteristic
210M16—84no—
211M16—84no—
212M16—84no—
213C2×C8—24no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—44no—
219C2×C8—82no—
220C2×C8—84no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—48no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—41no—
227C4⋊C4—84no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—84no—
236C22⋊C4—84no—
237C42—41no—
238C42—41no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—42no—
245C23—41no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—88no—
259C23#38011yescharacteristic
260C23#38812yes—
261C23#39211yescharacteristic
262C23#39612yes—
263C23#39711yescharacteristic
264C23#47114yes—
265Q8—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—82no—
270D4—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—168no—
301C2×C4#38512yes—
302C2×C4#38611yescharacteristic
303C2×C4#39011yescharacteristic
304C2×C4#39311yescharacteristic
305C2×C4#39412yes—
306C2×C4#39911yescharacteristic
307C8—44no—
308C8—82no—
309C8—84no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—82no—
326C22—82no—
327C22—84no—
328C22—84no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22#143911yescharacteristic
333C22#181512yes—
334C22#201912yes—
335C22#207912yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—21no—
340C4—21no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—164no—
345C4#191312yes—
346C2—22no—
347C2—22no—
348C2—22no—
349C2—42no—
350C2—42no—
351C2—161no—
352C2#1104611yescharacteristic
353C2#1110611yescharacteristic
354C2#1110911yescharacteristic
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. #21938C2semidirect
4. #8107C2semidirect
5. C2×#1293C2semidirect
14. #1818C4semidirect
16. #1780C4semidirect

Maximal subgroups

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

subgroupindex
#219382
#217412
#175922
#81072
C2×#12932