Finite Group Databasebeta

← #100866

Subgroups — #100866

Order 512.

400 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 86 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#23347C211yesmaximal, characteristic
3#9903C211yesmaximal, characteristic, complemented
4#9705C211yesmaximal, characteristic, complemented
5#7635C211yesmaximal, characteristic
6#7042C211yesmaximal, characteristic, complemented
7#7042C211yesmaximal, characteristic, complemented
8#2868C2211yescharacteristic
9C4×#124C2211yescharacteristic
10#2024—21no—
11#2023—21no—
12#1984—21no—
13#1959C412yes—
14#1923—21no—
15#1923—21no—
16#1870—21no—
17#1851—21no—
18#1851—21no—
19#1836C412yes—
20#1831C2211yescharacteristic
21#1831C2211yescharacteristic
22#1803—21no—
23#1802—21no—
24#1767C2211yescharacteristic
25#1767C2211yescharacteristic
26#1767C2211yescharacteristic
27C2×#332—21no—
28C2×#332—21no—
29#1296—21no—
30#1296—21no—
31#1272—21no—
32#1270—21no—
33C22×M16—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×#124C2×C411yescharacteristic
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×#97C2×C412yes—
56C2×C4×C8—21no—
57#391D411yescharacteristic
58#390D411yescharacteristic
59#388—21no—
60#388—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384C2×C412yes—
65#382C2×C411yescharacteristic
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#336—21no—
80#332—42no—
81#332—42no—
82C22×Q8—41no—
83C22×Q8C22⋊C411yescharacteristic
84C22×D4C22⋊C411yescharacteristic
85C23×C4—21no—
86C2×M16—81no—
87C2×M16—81no—
88C2×M16—81no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—41no—
95C22×C8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—81no—
101C4⋊Q8—81no—
102C4⋊Q8—82no—
103#124—22no—
104#124—41no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×D4—41no—
108C4×D4—41no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C22⋊C411yescharacteristic
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C42—21no—
124C2×C42C22×C411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42#4111yescharacteristic
130C2×C42#4111yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—41no—
134C4⋊C8—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137C4⋊C8—41no—
138C4⋊C8—41no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—81no—
142#103—82no—
143#102—82no—
144C22⋊C8—81no—
145C22⋊C8—81no—
146C22⋊C8—81no—
147#97—42no—
148#97—81no—
149#97—81no—
150C4×C8—81no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95—21no—
155C24—21no—
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—81no—
160C2×Q8—81no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8#10112yes—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—41no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4C4×D411yescharacteristic
182C22×C4C4×D411yescharacteristic
183C22×C4#12011yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4#12311yescharacteristic
187M16—84no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
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—81no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8—81no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—81no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—81no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—41no—
242C42—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#38611yescharacteristic
251Q8—42no—
252Q8—81no—
253Q8—161no—
254Q8—161no—
255Q8—161no—
256Q8—161no—
257D4—42no—
258D4—81no—
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—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4C2×#10111yescharacteristic
291C2×C4#41311yescharacteristic
292C2×C4#45311yescharacteristic
293C2×C4#45411yescharacteristic
294C2×C4#45511yescharacteristic
295C2×C4#45511yescharacteristic
296C2×C4#45511yescharacteristic
297C2×C4#45611yescharacteristic
298C2×C4#47111yescharacteristic
299C2×C4#47111yescharacteristic
300C2×C4#47211yescharacteristic
301C2×C4#47311yescharacteristic
302C8—82no—
303C8—82no—
304C8—82no—
305C8—81no—
306C8—81no—
307C8—81no—
308C8—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#188211yescharacteristic
317C22#188311yescharacteristic
318C22#188511yescharacteristic
319C22#190111yescharacteristic
320C22#190711yescharacteristic
321C22#190811yescharacteristic
322C22#191011yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—42no—
333C4—81no—
334C4—81no—
335C4—162no—
336C4#165911yescharacteristic
337C4#166111yescharacteristic
338C4#166211yescharacteristic
339C4#166411yescharacteristic
340C2—161no—
341C2#899911yescharacteristic
342C2#935611yescharacteristic
343C2#935811yescharacteristic
344C2#937611yescharacteristic
345C2#937811yescharacteristic
346C2#943311yescharacteristic
347C2#945311yescharacteristic
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. #9705C2semidirect
6. #7042C2semidirect
7. #7042C2semidirect

Maximal subgroups

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

subgroupindex
#291482
#233472
#99032
#97052
#76352
#70422