Finite Group Databasebeta

← #101296

Subgroups — #101296

Order 512.

557 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 557 conjugacy classes of subgroups — too many to draw. Only its 94 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#23732C211yesmaximal, characteristic
2#14501C211yesmaximal, characteristic
3#8717C212yesmaximal
4#7595C211yesmaximal, characteristic
5#7037C212yesmaximal
6C2×#428—21no—
7C2×#419—21no—
8C2×#391—21no—
9C2×#391C2211yescharacteristic
10#2285C2211yescharacteristic
11#2097—22no—
12#2096C412yescomplemented
13#2095C412yescomplemented
14#1817C2212yes—
15#1728C2211yescharacteristic
16#1442—22no—
17#1441—21no—
18#1436—22no—
19#1434C2212yes—
20#1294—22no—
21#1294—22no—
22#1272—22no—
23C23×Q8D411yescharacteristic
24C22×M16—21no—
25C2×#117—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4D411yescharacteristic
29C22×C42—21no—
30C22×C42C2311yescharacteristic
31#428—42no—
32#428—42no—
33C2×#107—22no—
34C2×#106—22no—
35#419—42no—
36#419—42no—
37C2×#103—22no—
38C2×#103—22no—
39C2×C22⋊C8—22no—
40#391—22no—
41#391—42no—
42#391—42no—
43#391C2×C412yes—
44#391D412yes—
45#384C2×C411yescharacteristic
46#384C2×C411yescharacteristic
47#382—21no—
48#380—22no—
49C4×C4⋊C4—22no—
50C2×#95—22no—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—22no—
54C2×#95—22no—
55C2×#95—21no—
56C2×#95D412yes—
57C2×#95D412yes—
58#336—21no—
59#336—22no—
60#336—22no—
61#336C2×C412yes—
62C22×Q8—21no—
63C22×Q8—21no—
64C22×Q8—21no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8C22⋊C412yes—
70C22×Q8SD1612yes—
71C22×Q8Q1612yes—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D412yes—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C22×C8—22no—
85C22×C8—42no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—44no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—81no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4C22⋊C412yes—
103C2×C22⋊C4—41no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42—41no—
108C2×C42—81no—
109C2×C42—81no—
110C2×C42C22×C411yescharacteristic
111C2×C42C2×D411yescharacteristic
112C2×C42#4112yes—
113#107—44no—
114#106—44no—
115#103—44no—
116#103—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#95—24no—
120#95—24no—
121#95—42no—
122#95—42no—
123#95—44no—
124#95—44no—
125#95—44no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—81no—
130C24#12011yescharacteristic
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—81no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—81no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8#10314yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D412yes—
185C22×C4#12112yes—
186C22×C4#12311yescharacteristic
187C22×C4C2×SD1611yescharacteristic
188C22×C4C2×Q1611yescharacteristic
189C22×C4AGL(1,ℤ/8)11yescharacteristic
190C22×C4#13711yescharacteristic
191M16—42no—
192M16—42no—
193M16—42no—
194M16—42no—
195C2×C8—24no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—82no—
199C2×C8—82no—
200C2×C8—82no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—84no—
210C4⋊C4—84no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C42—41no—
214C42—41no—
215C42—42no—
216C42—81no—
217C42—81no—
218C42—84no—
219C23—22no—
220C23—22no—
221C23#38611yescharacteristic
222C23#45011yescharacteristic
223C23#45111yescharacteristic
224C23#45311yescharacteristic
225C23#45311yescharacteristic
226C23#45811yescharacteristic
227C23#45811yescharacteristic
228Q8—44no—
229Q8—82no—
230Q8—82no—
231Q8—82no—
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—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4C2×#10311yescharacteristic
283C2×C4#41711yescharacteristic
284C2×C4#48311yescharacteristic
285C2×C4#48411yescharacteristic
286C8—44no—
287C8—162no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—21no—
295C22—21no—
296C22#161311yescharacteristic
297C22#161311yescharacteristic
298C22#161411yescharacteristic
299C22#161411yescharacteristic
300C22#187211yescharacteristic
301C22#188711yescharacteristic
302C22#189011yescharacteristic
303C22#190512yes—
304C22#190612yes—
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—42no—
311C4—44no—
312C4—84no—
313C4—82no—
314C4—162no—
315C4—161no—
316C4—161no—
317C2—21no—
318C2—21no—
319C2—21no—
320C2—21no—
321C2#902111yescharacteristic
322C2#917612yes—
323C2#942911yescharacteristic
324C2#944111yescharacteristic
325C2#952612yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2096C4semidirect
13. #2095C4semidirect

Maximal subgroups

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

subgroupindex
#237322
#145012
#87172
#75952
#70372