Finite Group Databasebeta

← #99390

Subgroups — #99390

Order 512.

653 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 653 conjugacy classes of subgroups — too many to draw. Only its 154 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#14467C211yesmaximal, characteristic
2#11267C212yesmaximal
3#11050C211yesmaximal, characteristic
4C2×#1272C211yesmaximal, characteristic
5#7025C212yesmaximal
6C22×#103—21no—
7C2×#391—21no—
8C2×#391C2211yescharacteristic
9C2×#384C2211yescharacteristic
10C2×#340—22no—
11C2×#340—22no—
12C2×#340—21no—
13C2×#336C2211yescharacteristic
14C2×#336C2212yes—
15#1444—22no—
16#1441—21no—
17#1436—22no—
18C4⋊#95C2212yes—
19#1272C414yescomplemented
20C23×Q8D411yescharacteristic
21C23×C8—21no—
22C23×C8—22no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D411yescharacteristic
29C22×C4⋊C4D412yes—
30C22×C42—21no—
31C22×C42C2311yescharacteristic
32C2×#103—42no—
33C2×#103—42no—
34C2×#103—42no—
35#391—22no—
36#391—42no—
37#391—42no—
38#391C2×C412yes—
39#391Q812yes—
40#384—42no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—21no—
46C2×#95—22no—
47#340—44no—
48#340—44no—
49#340—42no—
50#336—22no—
51#336—22no—
52#336C2×C414yes—
53C22×Q8—21no—
54C22×Q8—21no—
55C22×Q8—21no—
56C22×Q8—41no—
57C22×Q8—41no—
58C22×Q8—41no—
59C22×Q8—41no—
60C22×Q8C4⋊C412yes—
61C22×Q8SD1612yes—
62C22×Q8SD1612yes—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—41no—
68C23×C4C2×D411yescharacteristic
69C23×C4C2×D412yes—
70C22×C8—22no—
71C22×C8—24no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—42no—
83C22×C8—42no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—81no—
108C2×C4⋊C4—81no—
109C2×C4⋊C4—81no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4C4⋊C412yes—
112C2×C4⋊C4Q1614yes—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42—42no—
116C2×C42—81no—
117C2×C42—81no—
118C2×C42C22×C411yescharacteristic
119C2×C42C2×Q811yescharacteristic
120C2×C42#4112yes—
121#103—48no—
122#95—24no—
123#95—24no—
124#95—44no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—81no—
130#95—82no—
131C24#12011yescharacteristic
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—81no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8#10614yes—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—82no—
185C22×C4—81no—
186C22×C4—82no—
187C22×C4C2×C4⋊C411yescharacteristic
188C22×C4C4×D412yes—
189C22×C4C22⋊Q812yes—
190C22×C4#12311yescharacteristic
191C22×C4C2×SD1611yescharacteristic
192C22×C4C2×SD1611yescharacteristic
193C22×C4C2×Q1612yes—
194C22×C4#13512yes—
195C22×C4AGL(1,ℤ/8)11yescharacteristic
196C22×C4AGL(1,ℤ/8)12yes—
197C22×C4#13711yescharacteristic
198C22×C4#13712yes—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—44no—
202C2×C8—44no—
203C2×C8—42no—
204C2×C8—44no—
205C2×C8—82no—
206C2×C8—81no—
207C2×C8—82no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—81no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—81no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C42—41no—
230C42—41no—
231C42—42no—
232C42—84no—
233C23#38511yescharacteristic
234C23#44811yescharacteristic
235C23#44912yes—
236C23#45011yescharacteristic
237C23#45112yes—
238C23#45311yescharacteristic
239C23#45412yes—
240C23#45511yescharacteristic
241C23#45512yes—
242C23B(4,𝔽2)11yescharacteristic
243C23#45811yescharacteristic
244Q8—44no—
245Q8—82no—
246Q8—82no—
247Q8—82no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—161no—
301C2×C4—162no—
302C2×C4—161no—
303C2×C4C2×#10611yescharacteristic
304C2×C4#42811yescharacteristic
305C2×C4C4×Q1612yes—
306C2×C4#44012yes—
307C2×C4#47712yes—
308C2×C4#47912yes—
309C2×C4#48111yescharacteristic
310C2×C4#48311yescharacteristic
311C8—82no—
312C8—84no—
313C22#159611yescharacteristic
314C22#159712yes—
315C22#160111yescharacteristic
316C22#160212yes—
317C22#160811yescharacteristic
318C22#161212yes—
319C22#161311yescharacteristic
320C22#161712yes—
321C22#162411yescharacteristic
322C22#162612yes—
323C22#163311yescharacteristic
324C22#163712yes—
325C22#164212yes—
326C22#164412yes—
327C22#164512yes—
328C22#165711yescharacteristic
329C22#165911yescharacteristic
330C22#166212yes—
331C22#185311yescharacteristic
332C22#185611yescharacteristic
333C22#186012yes—
334C22#186312yes—
335C22#186711yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—44no—
345C4—84no—
346C4—82no—
347C4—162no—
348C4—162no—
349C2#774311yescharacteristic
350C2#775911yescharacteristic
351C2#777412yes—
352C2#781011yescharacteristic
353C2#781612yes—
354C2#782911yescharacteristic
355C2#885411yescharacteristic
356C2#886712yes—
357C2#889612yes—
358C2#891411yescharacteristic
359C2#892911yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #1272C4semidirect

Maximal subgroups

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

subgroupindex
#144672
#112672
#110502
C2×#12722
#70252