Finite Group Databasebeta

← #100747

Subgroups — #100747

Order 512.

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

normal subgroups only

This group has 415 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#29148C211yesmaximal, characteristic
2#23346C211yesmaximal, characteristic
3#9903C211yesmaximal, characteristic, complemented
4#9702C211yesmaximal, characteristic, complemented
5#8579C211yesmaximal, characteristic
6#7049C211yesmaximal, characteristic, complemented
7#7049C211yesmaximal, characteristic, complemented
8#2868C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2024C412yes—
11#2023C412yes—
12#1984—21no—
13#1959—21no—
14#1923—21no—
15#1923—21no—
16#1869—21no—
17#1850—21no—
18#1850—21no—
19#1836—21no—
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1803—21no—
23#1802—21no—
24#1767C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27C2×#333—21no—
28C2×#333—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⋊Q8C2×C412yes—
37C2×#124D411yescharacteristic
38C2×C4×Q8D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×#103—22no—
49C2×#102—22no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—21no—
54C2×#97—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57#391C2×C411yescharacteristic
58#390C2×C411yescharacteristic
59#388—21no—
60#388—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#382D411yescharacteristic
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C43C2311yescharacteristic
69#340—22no—
70#340—22no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336C2×C412yes—
80#333—42no—
81#333—42no—
82#329—42no—
83C22×Q8—22no—
84C22×Q8C22⋊C411yescharacteristic
85C22×D4C22⋊C411yescharacteristic
86C23×C4—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—81no—
95C22×C8—81no—
96C22×C8—81no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—82no—
102C4⋊Q8—82no—
103#124—21no—
104#124—21no—
105#124SD1612yes—
106#124Q1612yes—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×D4—41no—
110C4×D4—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C42—21no—
126C2×C42C22×C411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132C2×C42#4111yescharacteristic
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—81no—
144#103—82no—
145#102—82no—
146C22⋊C8—81no—
147C22⋊C8—81no—
148C22⋊C8—81no—
149#97—41no—
150#97—41no—
151#97—41no—
152#97—41no—
153#97—41no—
154#97—41no—
155C4×C8—81no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160C24—21no—
161C2×Q8—21no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—82no—
167C2×Q8—82no—
168C2×Q8#10312yes—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4#10312yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12311yescharacteristic
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—81no—
208C2×C8—81no—
209C2×C8—84no—
210C2×C8—81no—
211C2×C8—81no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—81no—
217C4⋊C4—82no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42C2×SD1611yescharacteristic
243C42C2×Q1611yescharacteristic
244C42#13511yescharacteristic
245C42#13511yescharacteristic
246C23—22no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#38611yescharacteristic
254Q8—42no—
255Q8—81no—
256Q8—162no—
257Q8—162no—
258D4—42no—
259D4—81no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4C2×#10311yescharacteristic
292C2×C4#41611yescharacteristic
293C2×C4#45311yescharacteristic
294C2×C4#45411yescharacteristic
295C2×C4#45511yescharacteristic
296C2×C4#45511yescharacteristic
297C2×C4#45511yescharacteristic
298C2×C4#45611yescharacteristic
299C2×C4#46511yescharacteristic
300C2×C4#46511yescharacteristic
301C2×C4#46711yescharacteristic
302C2×C4#46711yescharacteristic
303C8—82no—
304C8—82no—
305C8—82no—
306C8—81no—
307C8—81no—
308C8—81no—
309C8—81no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22#188211yescharacteristic
318C22#188311yescharacteristic
319C22#188511yescharacteristic
320C22#189211yescharacteristic
321C22#190211yescharacteristic
322C22#190311yescharacteristic
323C22#191411yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—162no—
337C4#165911yescharacteristic
338C4#166111yescharacteristic
339C4#166211yescharacteristic
340C4#166411yescharacteristic
341C2—82no—
342C2#899911yescharacteristic
343C2#931511yescharacteristic
344C2#931711yescharacteristic
345C2#933711yescharacteristic
346C2#934011yescharacteristic
347C2#939111yescharacteristic
348C2#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. #9903C2semidirect
4. #9702C2semidirect
6. #7049C2semidirect
7. #7049C2semidirect

Maximal subgroups

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

subgroupindex
#291482
#233462
#99032
#97022
#85792
#70492