Finite Group Databasebeta

← #99951

Subgroups — #99951

Order 512.

485 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 100 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#22268C211yesmaximal, characteristic
2#22264C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic, complemented
4#7591C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#492C411yescharacteristic
9C2×#436—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#1957C411yescharacteristic
13#1956C411yescharacteristic
14#1954—21no—
15#1880—22no—
16#1877—22no—
17#1870—22no—
18#1870—22no—
19#1834C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25#1279—21no—
26#1279—21no—
27#1271—22no—
28#1270—22no—
29C22×SD16—22no—
30C22×M16—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×#127C2×C411yescharacteristic
33C2×C4×Q8—21no—
34C2×C4×D4—21no—
35C2×C4×D4—21no—
36C2×C4×D4D412yes—
37#492—22no—
38#492—22no—
39#436—42no—
40#436—42no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#103—22no—
44C2×#102—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C4×C8—21no—
49C2×C4×C8C2×C411yescharacteristic
50C2×C4×C8C2×C411yescharacteristic
51C2×C4×C8D411yescharacteristic
52C2×C4×C8D411yescharacteristic
53#391—21no—
54#390—21no—
55#390—21no—
56#390D412yes—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61C4×C4⋊C4—21no—
62C4×C4⋊C4—21no—
63C43C2311yescharacteristic
64#340—22no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336D412yes—
75C22×Q8—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4C22⋊C412yes—
79C23×C4—22no—
80C2×SD16—28no—
81C2×SD16—82no—
82C2×SD16—82no—
83C2×M16—44no—
84C2×M16—82no—
85C2×M16—82no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8M1612yescomplemented
97#127—21no—
98#127—21no—
99#127—21no—
100#127—21no—
101C4×Q8—82no—
102C4×D4—41no—
103C4×D4—41no—
104C4×D4—41no—
105C4×D4—41no—
106C4×D4—82no—
107C4×D4—82no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C42—41no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42#4111yescharacteristic
127C2×C42#4111yescharacteristic
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132#103—82no—
133#102—82no—
134C22⋊C8—44no—
135C22⋊C8—82no—
136C22⋊C8—82no—
137C4×C8—22no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—41no—
145C4×C8M1612yescomplemented
146C4×C8SD1612yescomplemented
147#95—41no—
148#95—41no—
149C24—22no—
150C2×Q8—22no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—41no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4—81no—
167C2×D4—81no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184SD16—88no—
185M16—88no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—41no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—82no—
223C22⋊C4—82no—
224C42—21no—
225C42—21no—
226C42—41no—
227C42—41no—
228C42—41no—
229C42—41no—
230C42—82no—
231C42C2×M1611yescharacteristic
232C42C2×M1611yescharacteristic
233C42#13111yescharacteristic
234C42#13111yescharacteristic
235C42C2×SD1611yescharacteristic
236C42#13511yescharacteristic
237C42AGL(1,ℤ/8)11yescharacteristic
238C42#13711yescharacteristic
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23#38611yescharacteristic
247Q8—44no—
248Q8—82no—
249D4—44no—
250D4—82no—
251D4—82no—
252D4—82no—
253D4—81no—
254D4—81no—
255D4—81no—
256D4—81no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4#40711yescharacteristic
285C2×C4#40811yescharacteristic
286C2×C4#43511yescharacteristic
287C2×C4#43511yescharacteristic
288C2×C4#43611yescharacteristic
289C2×C4#43611yescharacteristic
290C2×C4C4×SD1611yescharacteristic
291C2×C4#44011yescharacteristic
292C2×C4#44311yescharacteristic
293C2×C4#44411yescharacteristic
294C2×C4#45011yescharacteristic
295C2×C4#45211yescharacteristic
296C2×C4#46111yescharacteristic
297C2×C4#46311yescharacteristic
298C8—42no—
299C8—42no—
300C8—81no—
301C8—81no—
302C8—82no—
303C8—82no—
304C8—161no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22#187011yescharacteristic
313C22#187611yescharacteristic
314C22#187711yescharacteristic
315C22#190211yescharacteristic
316C22#190411yescharacteristic
317C22#191011yescharacteristic
318C22#191111yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—161no—
332C4#158311yescharacteristic
333C4#158811yescharacteristic
334C4#159011yescharacteristic
335C4#159211yescharacteristic
336C2—162no—
337C2#894911yescharacteristic
338C2#909111yescharacteristic
339C2#910511yescharacteristic
340C2#928111yescharacteristic
341C2#928511yescharacteristic
342C2#928911yescharacteristic
343C2#929511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9873C2semidirect
4. #7591C2semidirect
6. #7018C2semidirect
7. #7014C2semidirect
96. C4⋊Q8M16semidirect
145. C4×C8M16semidirect
146. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#222682
#222642
#98732
#75912
#75872
#70182
#70142