Finite Group Databasebeta

← #103774

Subgroups — #103774

Order 512.

487 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 110 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#29218C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#9865C211yesmaximal, characteristic
5#9694C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic
7#7031C211yesmaximal, characteristic
8C2×#462—21no—
9C2×#462—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13#2100—22no—
14#2058—22no—
15#2056—22no—
16#1990—21no—
17#1990—21no—
18#1987—22no—
19#1948—22no—
20#1844—21no—
21#1842C2211yescharacteristic
22#1834—21no—
23#1803—21no—
24#1803—21no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29#1279—22no—
30#1271—22no—
31C22×Q16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39#462—42no—
40#462—42no—
41#462—42no—
42#462—42no—
43C2×#106—24no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D412yes—
49C2×C4⋊C8D412yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×C4×C8—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#391—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384D411yescharacteristic
65#384D411yescharacteristic
66#384D412yes—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—24no—
74#340—24no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#332—44no—
80C22×Q8—21no—
81C22×Q8—21no—
82C22×Q8—41no—
83C2×Q16—44no—
84C2×Q16—44no—
85C2×Q16—82no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C4⋊Q8—21no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—82no—
101C4⋊Q8SD1612yes—
102C4⋊Q8SD1612yes—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#106—84no—
139C4⋊C8—24no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—42no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149#103—44no—
150#103—44no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—42no—
154C4×C8—41no—
155C4×C8—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—82no—
173C2×Q8—82no—
174C2×Q8—81no—
175C2×Q8—81no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4#12011yescharacteristic
182C22×C4#12011yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12711yescharacteristic
188Q16—88no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—81no—
200C2×C8—81no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—82no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C42—21no—
229C42—21no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—81no—
234C42—81no—
235C42—81no—
236C42—81no—
237C42C2×SD1611yescharacteristic
238C42C2×SD1611yescharacteristic
239C42#13511yescharacteristic
240C42#13511yescharacteristic
241C42#13512yes—
242C42AGL(1,ℤ/8)11yescharacteristic
243C42AGL(1,ℤ/8)11yescharacteristic
244C42#13711yescharacteristic
245C42#13711yescharacteristic
246C42#13712yes—
247C23#39211yescharacteristic
248Q8—42no—
249Q8—42no—
250Q8—42no—
251Q8—42no—
252Q8—81no—
253Q8—81no—
254Q8—81no—
255Q8—81no—
256Q8—162no—
257Q8—162no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4#44811yescharacteristic
283C2×C4#44811yescharacteristic
284C2×C4#44911yescharacteristic
285C2×C4#44911yescharacteristic
286C2×C4#45011yescharacteristic
287C2×C4#45011yescharacteristic
288C2×C4#45211yescharacteristic
289C2×C4#45211yescharacteristic
290C2×C4#45212yes—
291C2×C4#45311yescharacteristic
292C2×C4#45411yescharacteristic
293C2×C4#45612yes—
294C2×C4#46412yes—
295C2×C4#46511yescharacteristic
296C2×C4#46511yescharacteristic
297C2×C4#46811yescharacteristic
298C2×C4#46811yescharacteristic
299C2×C4#47312yes—
300C2×C4#49511yescharacteristic
301C2×C4#49711yescharacteristic
302C8—44no—
303C8—82no—
304C8—82no—
305C8—82no—
306C22#200011yescharacteristic
307C22#200011yescharacteristic
308C22#200511yescharacteristic
309C22#200611yescharacteristic
310C22#200811yescharacteristic
311C22#201511yescharacteristic
312C22#202011yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—82no—
321C4—82no—
322C4—161no—
323C4—161no—
324C4—162no—
325C4#162911yescharacteristic
326C4#163411yescharacteristic
327C4#163511yescharacteristic
328C4#163511yescharacteristic
329C4#165212yes—
330C4#165312yes—
331C4#168012yes—
332C4#170012yes—
333C2#1006911yescharacteristic
334C2#1011711yescharacteristic
335C2#1018111yescharacteristic
336C2#1019511yescharacteristic
337C2#1023511yescharacteristic
338C2#1024511yescharacteristic
339C2#1027211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#292182
#292062
#282552
#98652
#96942
#75702
#70312