Finite Group Databasebeta

← #105170

Subgroups — #105170

Order 512.

675 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 144 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#17072C211yesmaximal, characteristic
2#11323C211yesmaximal, characteristic
3#11267C212yesmaximal
4#7025C212yesmaximal
5#7025C211yesmaximal, characteristic
6C22×#106—21no—
7C2×#395—21no—
8C2×#395—22no—
9C2×#384C2211yescharacteristic
10C2×#340—22no—
11C2×#340—22no—
12C2×#340—21no—
13C2×#336C2212yes—
14C2×#336C2211yescharacteristic
15#1441—22no—
16#1441—21no—
17C4⋊#95C2212yes—
18C4⋊#95C2211yescharacteristic
19C23×Q8—21no—
20C23×C8—21no—
21C23×C8—22no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4D411yescharacteristic
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C42C2311yescharacteristic
31C2×#106—42no—
32C2×#106—42no—
33C2×#106—42no—
34#395—42no—
35#395—44no—
36#395—42no—
37#395—44no—
38#384—42no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—21no—
45#340—44no—
46#340—42no—
47#340—44no—
48#336—21no—
49#336—22no—
50#336Q812yescomplemented
51#336Q814yescomplemented
52C22×Q8—42no—
53C22×Q8—44no—
54C22×Q8—81no—
55C22×Q8—82no—
56C22×Q8—81no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—41no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D412yes—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—24no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—41no—
71C22×C8—41no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—81no—
106C2×C4⋊C4—82no—
107C2×C4⋊C4D812yes—
108C2×C4⋊C4SD1614yes—
109C2×C4⋊C4SD1614yes—
110C2×C4⋊C4Q1612yes—
111C2×C42—41no—
112C2×C42—41no—
113C2×C42C2×Q811yescharacteristic
114C2×C42C2×Q812yes—
115C2×C42#4111yescharacteristic
116#106—48no—
117#95—42no—
118#95—44no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—81no—
126C24#12011yescharacteristic
127C2×Q8—84no—
128C2×Q8—88no—
129C2×Q8—84no—
130C2×Q8—88no—
131C2×Q8—84no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—81no—
166C22×C4—82no—
167C22×C4—81no—
168C22×C4—82no—
169C22×C4C22⋊Q811yescharacteristic
170C22×C4C22⋊Q812yes—
171C22×C4C4⋊Q812yes—
172C22×C4C4⋊Q811yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×SD1612yes—
175C22×C4C2×SD1612yes—
176C22×C4C2×Q1611yescharacteristic
177C22×C4AGL(1,ℤ/8)12yes—
178C22×C4AGL(1,ℤ/8)11yescharacteristic
179C22×C4#13711yescharacteristic
180C22×C4#13712yes—
181C2×C8—24no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—44no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C2×C8—82no—
195C4⋊C4—42no—
196C4⋊C4—44no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—81no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—88no—
213C42—41no—
214C42—42no—
215C42—41no—
216C23#39511yescharacteristic
217C23#44711yescharacteristic
218C23#44812yes—
219C23#45012yes—
220C23#45111yescharacteristic
221C23#45311yescharacteristic
222C23#45412yes—
223C23#45512yes—
224C23#45611yescharacteristic
225C23B(4,𝔽2)11yescharacteristic
226C23#45811yescharacteristic
227Q8—816no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—161no—
279C2×C4—162no—
280C2×C4—161no—
281C2×C4#47411yescharacteristic
282C2×C4#47512yes—
283C2×C4#47612yes—
284C2×C4#47711yescharacteristic
285C2×C4#49812yes—
286C2×C4#50011yescharacteristic
287C2×C4#50111yescharacteristic
288C2×C4#50112yes—
289C8—44no—
290C8—84no—
291C22#159511yescharacteristic
292C22#159912yes—
293C22#160112yes—
294C22#160511yescharacteristic
295C22#160812yes—
296C22#161011yescharacteristic
297C22#161411yescharacteristic
298C22#161612yes—
299C22#162311yescharacteristic
300C22#162712yes—
301C22#162911yescharacteristic
302C22#163712yes—
303C22#163912yes—
304C22#164512yes—
305C22#165112yes—
306C22#165711yescharacteristic
307C22#165911yescharacteristic
308C22#166412yes—
309C22#205711yescharacteristic
310C22#205812yes—
311C22#206011yescharacteristic
312C22#206112yes—
313C22#206211yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—82no—
323C4—84no—
324C4—84no—
325C4—82no—
326C4—162no—
327C2#774111yescharacteristic
328C2#775011yescharacteristic
329C2#776612yes—
330C2#779112yes—
331C2#781211yescharacteristic
332C2#782611yescharacteristic
333C2#1056711yescharacteristic
334C2#1056812yes—
335C2#1057411yescharacteristic
336C2#1057512yes—
337C2#1058411yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC22×C42C23
LCSG ▷ C22×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
50. #336Q8semidirect
51. #336Q8semidirect

Maximal subgroups

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

subgroupindex
#170722
#113232
#112672
#70252