Finite Group Databasebeta

← #100749

Subgroups — #100749

Order 512.

425 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 96 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#28726C211yesmaximal, characteristic
2#23346C211yesmaximal, characteristic
3#9905C211yesmaximal, characteristic, complemented
4#9705C211yesmaximal, characteristic, complemented
5#8579C211yesmaximal, characteristic
6#7049C211yesmaximal, characteristic, complemented
7#7049C211yesmaximal, characteristic, complemented
8C2×#489—21no—
9#2868C2211yescharacteristic
10C4×#124C2211yescharacteristic
11#2072C412yes—
12#2071C412yes—
13#1993—21no—
14#1984—21no—
15#1923—21no—
16#1923—21no—
17#1869—21no—
18#1850—21no—
19#1850—21no—
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1803—21no—
23#1802—21no—
24#1767C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27C2×#332—21no—
28C2×#332—21no—
29C2×#329—21no—
30#1296—21no—
31#1296—21no—
32#1295—21no—
33C23×C8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×#124D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40#489—42no—
41#489—42no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49C2×C4⋊C8D411yescharacteristic
50C2×#103—22no—
51C2×#102—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×#97—21no—
57C2×#97—21no—
58C2×C4×C8—21no—
59#391C2×C411yescharacteristic
60#390C2×C411yescharacteristic
61#388—21no—
62#388—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384C2×C412yes—
67#382D411yescharacteristic
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C43C2311yescharacteristic
71#340—22no—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336C2×C412yes—
82#332—42no—
83#332—42no—
84#329—42no—
85C22×Q8—41no—
86C22×Q8C22⋊C411yescharacteristic
87C22×D4C22⋊C411yescharacteristic
88C23×C4—21no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—81no—
97C22×C8—81no—
98C22×C8—81no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106C4⋊Q8—41no—
107C4⋊Q8—82no—
108#124—21no—
109#124—21no—
110#124D812yes—
111#124SD1612yes—
112C4×Q8—41no—
113C4×Q8—41no—
114C4×D4—41no—
115C4×D4—41no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4C22⋊C411yescharacteristic
124C2×C4⋊C4C22⋊C411yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C42—21no—
131C2×C42C22×C411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42#4111yescharacteristic
137C2×C42#4111yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146C4⋊C8—41no—
147C4⋊C8—41no—
148C4⋊C8—81no—
149#103—44no—
150#102—44no—
151C22⋊C8—81no—
152C22⋊C8—81no—
153C22⋊C8—81no—
154#97—41no—
155#97—41no—
156#97—41no—
157#97—41no—
158#97—41no—
159#97—41no—
160C4×C8—81no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165C24—21no—
166C2×Q8—21no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×Q8#10212yes—
176C2×D4—21no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4#10212yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4C4×D411yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12311yescharacteristic
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—81no—
215C2×C8—81no—
216C2×C8—84no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—81no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42C2×D811yescharacteristic
251C42C2×SD1611yescharacteristic
252C42#13511yescharacteristic
253C42#13511yescharacteristic
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#38611yescharacteristic
262Q8—42no—
263Q8—81no—
264Q8—161no—
265Q8—161no—
266Q8—161no—
267Q8—161no—
268D4—42no—
269D4—81no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4C2×#10211yescharacteristic
302C2×C4#41611yescharacteristic
303C2×C4#45311yescharacteristic
304C2×C4#45411yescharacteristic
305C2×C4#45511yescharacteristic
306C2×C4#45511yescharacteristic
307C2×C4#45511yescharacteristic
308C2×C4#45611yescharacteristic
309C2×C4#46511yescharacteristic
310C2×C4#46511yescharacteristic
311C2×C4#46611yescharacteristic
312C2×C4#46611yescharacteristic
313C8—82no—
314C8—82no—
315C8—82no—
316C8—81no—
317C8—81no—
318C8—81no—
319C8—81no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#188211yescharacteristic
328C22#188311yescharacteristic
329C22#188511yescharacteristic
330C22#189311yescharacteristic
331C22#190211yescharacteristic
332C22#190311yescharacteristic
333C22#191511yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—162no—
347C4#165911yescharacteristic
348C4#166111yescharacteristic
349C4#166211yescharacteristic
350C4#166411yescharacteristic
351C2—82no—
352C2#899911yescharacteristic
353C2#931611yescharacteristic
354C2#931811yescharacteristic
355C2#933811yescharacteristic
356C2#933911yescharacteristic
357C2#939311yescharacteristic
358C2#941911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9905C2semidirect
4. #9705C2semidirect
6. #7049C2semidirect
7. #7049C2semidirect

Maximal subgroups

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

subgroupindex
#287262
#233462
#99052
#97052
#85792
#70492