Finite Group Databasebeta

← #100101

Subgroups — #100101

Order 512.

436 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 84 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#23357C211yesmaximal, characteristic
2#21646C211yesmaximal, characteristic
3#7605C211yesmaximal, characteristic, complemented
4#7604C211yesmaximal, characteristic, complemented
5#7603C211yesmaximal, characteristic
6#7019C211yesmaximal, characteristic, complemented
7#7019C211yesmaximal, characteristic, complemented
8C2×#486C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C4×#127C2211yescharacteristic
11#1990C411yescharacteristic
12#1957C411yescharacteristic
13#1936—22no—
14#1870—22no—
15#1870—22no—
16#1870—22no—
17#1839C2211yescharacteristic
18#1839C2211yescharacteristic
19#1839C2211yescharacteristic
20#1839C2211yescharacteristic
21#1834C411yescharacteristic
22#1762—22no—
23C4×#97C2211yescharacteristic
24#1272—22no—
25#1271—22no—
26#1270—22no—
27C22×M16—22no—
28C2×C4⋊Q8C2×C411yescharacteristic
29C2×#127C2×C411yescharacteristic
30C2×C4×Q8—21no—
31C2×C4×D4—21no—
32C2×C4×D4—21no—
33C2×C4×D4D412yes—
34#486—22no—
35#486—22no—
36C2×#102—22no—
37C2×#102—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×#97—21no—
42C2×#97—21no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45#391—21no—
46#390—21no—
47#390—21no—
48#390D412yes—
49#389—21no—
50#389—21no—
51#384C2×C411yescharacteristic
52#384C2×C411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55C43C2311yescharacteristic
56#340—22no—
57#340—22no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336D412yes—
69#336D412yes—
70C22×Q8—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4C22⋊C412yes—
74C23×C4—22no—
75C2×M16—82no—
76C2×M16—82no—
77C2×M16—82no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8—41no—
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8M1612yescomplemented
88#127—21no—
89#127—21no—
90#127—21no—
91#127—21no—
92C4×Q8—82no—
93C4×D4—42no—
94C4×D4—42no—
95C4×D4—81no—
96C4×D4—81no—
97C4×D4—81no—
98C4×D4—81no—
99C2×C4⋊C4—21no—
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—41no—
107C2×C4⋊C4C22⋊C412yes—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C42—41no—
112C2×C42C22×C411yescharacteristic
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42#4111yescharacteristic
118C2×C42#4111yescharacteristic
119#102—28no—
120#102—44no—
121C22⋊C8—82no—
122C22⋊C8—82no—
123C22⋊C8—82no—
124#97—42no—
125#97—42no—
126C4×C8—21no—
127C4×C8—21no—
128C4×C8—21no—
129C4×C8M1612yescomplemented
130#95—41no—
131#95—41no—
132C24—22no—
133C2×Q8—22no—
134C2×Q8—41no—
135C2×Q8—81no—
136C2×Q8—81no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—41no—
142C2×D4—41no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—81no—
146C2×D4—81no—
147C2×D4—81no—
148C2×D4—81no—
149C22×C4—22no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4#12011yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12311yescharacteristic
165M16—88no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—81no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—81no—
182C2×C8—81no—
183C2×C8—81no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—42no—
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—82no—
197C22⋊C4—82no—
198C42—22no—
199C42—41no—
200C42—41no—
201C42—41no—
202C42—41no—
203C42—41no—
204C42—82no—
205C42C2×M1611yescharacteristic
206C42C2×M1611yescharacteristic
207C42#13111yescharacteristic
208C42#13111yescharacteristic
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23#38611yescharacteristic
217Q8—82no—
218Q8—82no—
219D4—44no—
220D4—82no—
221D4—82no—
222D4—82no—
223D4—82no—
224D4—82no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
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—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#40711yescharacteristic
256C2×C4#40811yescharacteristic
257C2×C4#43511yescharacteristic
258C2×C4#43511yescharacteristic
259C2×C4#43611yescharacteristic
260C2×C4#43611yescharacteristic
261C2×C4#45512yes—
262C8—42no—
263C8—42no—
264C8—82no—
265C8—82no—
266C8—161no—
267C8—161no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22#187011yescharacteristic
276C22#188411yescharacteristic
277C22#188511yescharacteristic
278C22#190211yescharacteristic
279C22#190311yescharacteristic
280C22#190411yescharacteristic
281C22#190411yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—82no—
293C4—82no—
294C4—161no—
295C2—162no—
296C2#896111yescharacteristic
297C2#909111yescharacteristic
298C2#909311yescharacteristic
299C2#941511yescharacteristic
300C2#941711yescharacteristic
301C2#941911yescharacteristic
302C2#942511yescharacteristic
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. #7605C2semidirect
4. #7604C2semidirect
6. #7019C2semidirect
7. #7019C2semidirect
87. C4⋊Q8M16semidirect
129. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#233572
#216462
#76052
#76042
#76032
#70192