Finite Group Databasebeta

← #101455

Subgroups — #101455

Order 512.

455 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 96 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#29258C211yesmaximal, characteristic
2#22264C211yesmaximal, characteristic
3#9874C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7653C211yesmaximal, characteristic, complemented
6#7040C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×#436—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2094C412yes—
12#2023C412yes—
13#1982—21no—
14#1954—21no—
15#1948—21no—
16#1870—22no—
17#1870—22no—
18#1864—22no—
19#1842C2211yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1831C2211yescharacteristic
23#1744C2211yescharacteristic
24C2×#331—21no—
25#1296—21no—
26#1279—21no—
27#1279—21no—
28#1272—22no—
29C22×M16—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#127D411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4D412yes—
36#436—42no—
37#436—42no—
38C2×#106—22no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#102—22no—
46C2×#102—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—22no—
50C2×C4×C8D411yescharacteristic
51C2×C4×C8D411yescharacteristic
52#391—21no—
53#390—21no—
54#390—21no—
55#390C2×C412yes—
56#389—21no—
57#389—21no—
58#384C2×C411yescharacteristic
59#384D411yescharacteristic
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C43C2311yescharacteristic
63#340—22no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336C2×C412yes—
71#336D412yes—
72#331—42no—
73C22×Q8—21no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4C22⋊C412yes—
77C23×C4—22no—
78C2×M16—44no—
79C2×M16—82no—
80C2×M16—82no—
81C22×C8—21no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—21no—
85C22×C8—41no—
86C22×C8—41no—
87C4⋊Q8—42no—
88#127—21no—
89#127—21no—
90#127—21no—
91#127—21no—
92C4×Q8—81no—
93C4×Q8—81no—
94C4×D4—41no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—82no—
99C4×D4—82no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C22⋊C412yes—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C42—41no—
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C2×C42#4111yescharacteristic
119C2×C42#4111yescharacteristic
120#106—82no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—81no—
132#102—44no—
133#102—82no—
134C22⋊C8—44no—
135C22⋊C8—82no—
136C22⋊C8—82no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8SD1612yes—
141#95—41no—
142#95—41no—
143C24—22no—
144C2×Q8—22no—
145C2×Q8—41no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—81no—
161C2×D4—81no—
162C22×C4—22no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C4×D411yescharacteristic
173C22×C4C4×D411yescharacteristic
174C22×C4#12011yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4#12111yescharacteristic
177C22×C4#12311yescharacteristic
178M16—88no—
179C2×C8—22no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—82no—
213C22⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42—41no—
220C42—81no—
221C42—81no—
222C42C2×SD1611yescharacteristic
223C42#13511yescharacteristic
224C42AGL(1,ℤ/8)11yescharacteristic
225C42AGL(1,ℤ/8)11yescharacteristic
226C42#13711yescharacteristic
227C42#13711yescharacteristic
228C42#13711yescharacteristic
229C42#13711yescharacteristic
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23#38611yescharacteristic
238Q8—82no—
239Q8—82no—
240D4—44no—
241D4—82no—
242D4—82no—
243D4—82no—
244D4—81no—
245D4—81no—
246D4—81no—
247D4—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#41711yescharacteristic
276C2×C4#41911yescharacteristic
277C2×C4C4×SD1611yescharacteristic
278C2×C4#44011yescharacteristic
279C2×C4#45011yescharacteristic
280C2×C4#45211yescharacteristic
281C2×C4#46111yescharacteristic
282C2×C4#46311yescharacteristic
283C2×C4#46811yescharacteristic
284C2×C4#46811yescharacteristic
285C2×C4#47011yescharacteristic
286C2×C4#47011yescharacteristic
287C2×C4#47111yescharacteristic
288C2×C4#47211yescharacteristic
289C8—42no—
290C8—82no—
291C8—81no—
292C8—82no—
293C8—161no—
294C8—161no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22#187711yescharacteristic
303C22#188111yescharacteristic
304C22#189111yescharacteristic
305C22#189411yescharacteristic
306C22#190411yescharacteristic
307C22#190711yescharacteristic
308C22#191111yescharacteristic
309C4—21no—
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—161no—
322C4#169111yescharacteristic
323C4#169511yescharacteristic
324C4#169711yescharacteristic
325C4#170311yescharacteristic
326C2—162no—
327C2#904711yescharacteristic
328C2#922311yescharacteristic
329C2#929511yescharacteristic
330C2#930711yescharacteristic
331C2#937111yescharacteristic
332C2#948711yescharacteristic
333C2#952911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9874C2semidirect
5. #7653C2semidirect
6. #7040C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#292582
#222642
#98742
#85882
#76532
#70402
#70182