Finite Group Databasebeta

← #116871

Subgroups — #116871

Order 512.

720 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 132 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#21939C211yesmaximal, characteristic
2#20705C212yesmaximal
3#20685C211yesmaximal, characteristic
4#8115C212yesmaximal
5#8008C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2211yescharacteristic
9C22×C22⋊C8C2211yescharacteristic
10#2280C2212yes—
11#1854—24no—
12#1826—24no—
13#1826—24no—
14#1818C414yescomplemented
15#1817—22no—
16#1808—21no—
17#1807—21no—
18#1733—24no—
19#1729—21no—
20#1729—21no—
21#1306—24no—
22#1293—22no—
23C24×C4C2311yescharacteristic
24C22×M16—21no—
25C22×M16—22no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#117—21no—
29C2×#117—21no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117C2×C412yes—
35C2×#117D414yes—
36C2×#117D412yes—
37C22×C4⋊C4Q811yescharacteristic
38C22×C4⋊C4Q811yescharacteristic
39C22×C22⋊C4—21no—
40C22×C42—21no—
41C2×#107—22no—
42C2×#106—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407D414yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8D412yes—
57#381—22no—
58#380—22no—
59#380—22no—
60#379—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63#340—22no—
64#340—22no—
65#339—216no—
66#336—24no—
67#336—24no—
68C25#4111yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—42no—
72C23×C4—42no—
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D412yes—
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D412yes—
79C23×C4C2×Q811yescharacteristic
80C23×C4#4112yes—
81C23×C4#4112yes—
82C23×C4#4111yescharacteristic
83C2×M16—24no—
84C2×M16—42no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—44no—
90C2×M16—44no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—22no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8C22⋊C412yes—
100C22×C8C22⋊C412yes—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—24no—
105#117—41no—
106#117—41no—
107#117—42no—
108#117—41no—
109#117—42no—
110#117—41no—
111#117—44no—
112#117—44no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4C4⋊C412yes—
121C2×C4⋊C4C4⋊C412yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—42no—
133C2×C42—42no—
134#107—82no—
135#106—82no—
136C22⋊C8—24no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—24no—
140C22⋊C8—24no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143C22⋊C8—42no—
144#95—42no—
145#95—42no—
146C24—42no—
147C24—42no—
148C24—81no—
149C24#11711yescharacteristic
150C24#12111yescharacteristic
151C24#12112yes—
152C24C22⋊Q811yescharacteristic
153C24#12612yes—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
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—44no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—44no—
175C22×C4—82no—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C2×C4⋊C411yescharacteristic
178C22×C4C4×D412yes—
179C22×C4C4×D412yes—
180C22×C4#12012yes—
181C22×C4#12011yescharacteristic
182C22×C4#12112yes—
183C22×C4#12112yes—
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q812yes—
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q812yes—
188C22×C4#12311yescharacteristic
189C22×C4#12312yes—
190C22×C4#12312yes—
191C22×C4#12411yescharacteristic
192C22×C4#12412yes—
193C22×C4#12711yescharacteristic
194M16—48no—
195M16—82no—
196M16—82no—
197M16—82no—
198M16—82no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—44no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—84no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—84no—
224C22⋊C4—84no—
225C22⋊C4—84no—
226C42—48no—
227C42—42no—
228C42—42no—
229C42—44no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—24no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—42no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—42no—
245C23—88no—
246C23#38011yescharacteristic
247C23#38812yes—
248C23#39211yescharacteristic
249C23#39612yes—
250C23#39711yescharacteristic
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4#38512yes—
282C2×C4#38611yescharacteristic
283C2×C4#39011yescharacteristic
284C2×C4#39311yescharacteristic
285C2×C4#39412yes—
286C2×C4#39911yescharacteristic
287C2×C4#45812yes—
288C2×C4#45812yes—
289C8—42no—
290C8—42no—
291C8—82no—
292C8—84no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—44no—
307C22—44no—
308C22—84no—
309C22—82no—
310C22—84no—
311C22#143911yescharacteristic
312C22#181412yes—
313C22#202212yes—
314C4—21no—
315C4—22no—
316C4—22no—
317C4—21no—
318C4—82no—
319C4—82no—
320C4—164no—
321C4#189011yescharacteristic
322C4#189011yescharacteristic
323C4#203012yes—
324C2—22no—
325C2—22no—
326C2—42no—
327C2—41no—
328C2—42no—
329C2#1107011yescharacteristic
330C2#1110411yescharacteristic
331C2#1110911yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#219392
#207052
#206852
#81152
#80082