Finite Group Databasebeta

← #104949

Subgroups — #104949

Order 512.

395 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 86 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#28279C211yesmaximal, characteristic
2#8809C211yesmaximal, characteristic, complemented
3#8590C211yesmaximal, characteristic, complemented
4#8588C211yesmaximal, characteristic
5#7591C211yesmaximal, characteristic, complemented
6#7587C211yesmaximal, characteristic
7#7040C211yesmaximal, characteristic, complemented
8C2×#474—21no—
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#2067—22no—
12#1982—21no—
13#1947—21no—
14#1842C2211yescharacteristic
15#1842C2211yescharacteristic
16#1831C2211yescharacteristic
17#1831C2211yescharacteristic
18#1802—21no—
19#1744C2211yescharacteristic
20C2×#334—22no—
21C2×#333—21no—
22C2×#331—21no—
23#1296—21no—
24#1296—21no—
25#1271—22no—
26#1270—22no—
27C2×C4⋊Q8D411yescharacteristic
28C2×#127D411yescharacteristic
29C2×C4×Q8—21no—
30C2×C4×D4—21no—
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33#474—42no—
34#474—42no—
35C2×#107—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D412yes—
44C2×#102—22no—
45C2×#102—22no—
46C2×C4×C8—21no—
47#391—21no—
48#390—21no—
49#390—21no—
50#390—21no—
51#389—21no—
52#389—21no—
53#384D411yescharacteristic
54#384D411yescharacteristic
55C4×C4⋊C4—21no—
56C4×C4⋊C4—21no—
57C43C2311yescharacteristic
58#340—22no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336D412yes—
65#334—44no—
66#333—42no—
67#331—42no—
68C22×Q8—21no—
69C22×D4—21no—
70C22×D4—21no—
71C22×D4—21no—
72C23×C4—41no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—41no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8SD1612yes—
82#127—21no—
83#127—21no—
84#127—21no—
85#127SD1612yes—
86C4×Q8—82no—
87C4×D4—41no—
88C4×D4—41no—
89C4×D4—41no—
90C4×D4—41no—
91C4×D4—82no—
92C4×D4—82no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C42—41no—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112C2×C42#4111yescharacteristic
113#107—44no—
114C4⋊C8—42no—
115C4⋊C8—42no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128#102—44no—
129#102—82no—
130C4×C8—22no—
131C4×C8—22no—
132#95—41no—
133#95—41no—
134C24—41no—
135C2×Q8—22no—
136C2×Q8—41no—
137C2×Q8—81no—
138C2×Q8—81no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4—81no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4#12011yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4#12311yescharacteristic
166C22×C4#12411yescharacteristic
167C2×C8—24no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—81no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—82no—
199C42—21no—
200C42—21no—
201C42—22no—
202C42—41no—
203C42—41no—
204C42—41no—
205C42—82no—
206C42C2×SD1611yescharacteristic
207C42C2×SD1611yescharacteristic
208C42#13511yescharacteristic
209C42#13511yescharacteristic
210C42AGL(1,ℤ/8)11yescharacteristic
211C42AGL(1,ℤ/8)11yescharacteristic
212C42#13711yescharacteristic
213C42#13711yescharacteristic
214C23—42no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23#39411yescharacteristic
222Q8—82no—
223Q8—82no—
224D4—42no—
225D4—42no—
226D4—82no—
227D4—82no—
228D4—82no—
229D4—82no—
230D4—81no—
231D4—81no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—162no—
255C2×C4—162no—
256C2×C4#44811yescharacteristic
257C2×C4#44911yescharacteristic
258C2×C4#45011yescharacteristic
259C2×C4#45211yescharacteristic
260C2×C4#45311yescharacteristic
261C2×C4#45611yescharacteristic
262C2×C4#46511yescharacteristic
263C2×C4#46511yescharacteristic
264C2×C4#46911yescharacteristic
265C2×C4#47011yescharacteristic
266C2×C4#47212yes—
267C2×C4#48111yescharacteristic
268C2×C4#48211yescharacteristic
269C2×C4#48311yescharacteristic
270C2×C4#48511yescharacteristic
271C8—44no—
272C8—82no—
273C8—161no—
274C8—161no—
275C8—161no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22—161no—
283C22#203411yescharacteristic
284C22#203511yescharacteristic
285C22#203911yescharacteristic
286C22#204511yescharacteristic
287C22#204811yescharacteristic
288C22#204911yescharacteristic
289C22#205311yescharacteristic
290C4—22no—
291C4—21no—
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—82no—
301C4—161no—
302C4—161no—
303C4#162211yescharacteristic
304C4#163111yescharacteristic
305C4#165111yescharacteristic
306C4#165211yescharacteristic
307C2—162no—
308C2#1043511yescharacteristic
309C2#1044111yescharacteristic
310C2#1044811yescharacteristic
311C2#1046711yescharacteristic
312C2#1046811yescharacteristic
313C2#1051211yescharacteristic
314C2#1052611yescharacteristic
subgroupG/H
center, socleC23#394
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8809C2semidirect
3. #8590C2semidirect
5. #7591C2semidirect
7. #7040C2semidirect

Maximal subgroups

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

subgroupindex
#282792
#88092
#85902
#85882
#75912
#75872
#70402