Finite Group Databasebeta

← #99947

Subgroups — #99947

Order 512.

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

normal subgroups only

This group has 469 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#22264C211yesmaximal, characteristic
2#22219C211yesmaximal, characteristic
3#9873C211yesmaximal, characteristic, complemented
4#7590C211yesmaximal, characteristic, complemented
5#7587C211yesmaximal, characteristic
6#7018C211yesmaximal, characteristic, complemented
7#7014C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C2×#436—21no—
10C4×C4⋊Q8C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#1990C411yescharacteristic
13#1957C411yescharacteristic
14#1954—21no—
15#1926—22no—
16#1870—22no—
17#1870—22no—
18#1864—22no—
19#1844C411yescharacteristic
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1744C2211yescharacteristic
23#1744C2211yescharacteristic
24#1744C2211yescharacteristic
25#1279—21no—
26#1279—21no—
27#1272—22no—
28#1270—22no—
29C22×M16—22no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×#127C2×C411yescharacteristic
32C2×C4×Q8—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4D412yes—
36#486—22no—
37#486—22no—
38#436—42no—
39#436—42no—
40C2×#106—22no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×#102—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×M16—44no—
81C2×M16—82no—
82C2×M16—82no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—41no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8M1612yescomplemented
94#127—21no—
95#127—21no—
96#127—21no—
97#127—21no—
98C4×Q8—82no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—82no—
104C4×D4—82no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C42—41no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124C2×C42#4111yescharacteristic
125#106—82no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#102—28no—
131#102—82no—
132C22⋊C8—44no—
133C22⋊C8—82no—
134C22⋊C8—82no—
135C4×C8—22no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—41no—
143C4×C8M1612yescomplemented
144C4×C8SD1612yes—
145#95—41no—
146#95—41no—
147C24—22no—
148C2×Q8—22no—
149C2×Q8—41no—
150C2×Q8—81no—
151C2×Q8—81no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—81no—
165C2×D4—81no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4C2×C22⋊C411yescharacteristic
176C22×C4C4×D411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4#12011yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12311yescharacteristic
182M16—88no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—81no—
215C4⋊C4—81no—
216C4⋊C4—82no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—41no—
225C42—41no—
226C42—41no—
227C42—82no—
228C42C2×M1611yescharacteristic
229C42C2×M1611yescharacteristic
230C42#13111yescharacteristic
231C42#13111yescharacteristic
232C42C2×SD1611yescharacteristic
233C42#13511yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42#13711yescharacteristic
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23#38611yescharacteristic
244Q8—82no—
245Q8—82no—
246D4—44no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—81no—
251D4—81no—
252D4—81no—
253D4—81no—
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—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4#40711yescharacteristic
282C2×C4#40811yescharacteristic
283C2×C4#43511yescharacteristic
284C2×C4#43511yescharacteristic
285C2×C4#43611yescharacteristic
286C2×C4#43611yescharacteristic
287C2×C4C4×SD1611yescharacteristic
288C2×C4#44011yescharacteristic
289C2×C4#44311yescharacteristic
290C2×C4#44411yescharacteristic
291C2×C4#45011yescharacteristic
292C2×C4#45211yescharacteristic
293C2×C4#46111yescharacteristic
294C2×C4#46311yescharacteristic
295C8—42no—
296C8—42no—
297C8—81no—
298C8—81no—
299C8—82no—
300C8—82no—
301C8—161no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22#187011yescharacteristic
310C22#187611yescharacteristic
311C22#187711yescharacteristic
312C22#190311yescharacteristic
313C22#190411yescharacteristic
314C22#190911yescharacteristic
315C22#191111yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—82no—
328C4—161no—
329C4#158311yescharacteristic
330C4#158811yescharacteristic
331C4#159011yescharacteristic
332C4#159211yescharacteristic
333C2—162no—
334C2#894911yescharacteristic
335C2#909311yescharacteristic
336C2#910311yescharacteristic
337C2#927911yescharacteristic
338C2#928311yescharacteristic
339C2#928711yescharacteristic
340C2#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. #7590C2semidirect
6. #7018C2semidirect
7. #7014C2semidirect
93. C4⋊Q8M16semidirect
143. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#222642
#222192
#98732
#75902
#75872
#70182
#70142