Finite Group Databasebeta

← #100707

Subgroups — #100707

Order 512.

402 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 88 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#29870C211yesmaximal, characteristic
2#22702C211yesmaximal, characteristic
3#7723C211yesmaximal, characteristic, complemented
4#7680C211yesmaximal, characteristic, complemented
5#7580C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×#478—21no—
9C2×#435—21no—
10#2859C2211yescharacteristic
11#2293C2211yescharacteristic
12#2085—21no—
13#2084C411yescharacteristic
14#2083C411yescharacteristic
15#2073—21no—
16#2071C411yescharacteristic
17#2071C411yescharacteristic
18#1989—21no—
19#1940—21no—
20#1851—21no—
21#1850—21no—
22#1839C2211yescharacteristic
23#1831C2211yescharacteristic
24#1831C2211yescharacteristic
25#1798—21no—
26#1767C2211yescharacteristic
27#1744C2211yescharacteristic
28#1729—21no—
29#1271—21no—
30#1271—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C22×M16—21no—
36C23×C8—21no—
37C2×#125—21no—
38C2×C4×D4D411yescharacteristic
39C2×#117—21no—
40#478—42no—
41#478—42no—
42#435—42no—
43#435—42no—
44C2×#107—21no—
45C2×#106—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×#97—21no—
56C2×C4×C8—21no—
57#390C2×C411yescharacteristic
58#390D411yescharacteristic
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389C2×C411yescharacteristic
64#389D411yescharacteristic
65#388—21no—
66#388—21no—
67#384C2×C411yescharacteristic
68#384C2×C411yescharacteristic
69#382—21no—
70C4×C4⋊C4D411yescharacteristic
71C4×C22⋊C4—21no—
72C43C2311yescharacteristic
73#340—21no—
74#340—21no—
75#340—21no—
76#340—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90#336D411yescharacteristic
91#336D411yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C23×C4—21no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—81no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—42no—
104C22×C8—42no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—81no—
108#125—41no—
109#125—41no—
110#125—41no—
111#125—41no—
112C4×D4—21no—
113C4×D4—21no—
114C4×D4D812yes—
115C4×D4SD1612yes—
116#117—81no—
117#117—81no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C22⋊C411yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C42—21no—
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42#4111yescharacteristic
144#107—42no—
145#106—42no—
146C4⋊C8—42no—
147C4⋊C8—41no—
148C4⋊C8—41no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—81no—
152C4⋊C8—81no—
153#102—42no—
154#102—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#97—41no—
158#97—41no—
159C4×C8—22no—
160C4×C8—41no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—41no—
165#95—41no—
166C24—21no—
167C2×D4—21no—
168C2×D4—41no—
169C2×D4—41no—
170C2×D4#10212yes—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
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×C4C2×C22⋊C411yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12311yescharacteristic
189M16—84no—
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
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—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—84no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C2×C8—81no—
214C2×C8—81no—
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4#10212yes—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—41no—
250C42C2×D811yescharacteristic
251C42C2×SD1611yescharacteristic
252C42AGL(1,ℤ/8)11yescharacteristic
253C42#13711yescharacteristic
254C23—22no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#38611yescharacteristic
262D4—42no—
263D4—81no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
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—81no—
299C2×C4—81no—
300C2×C4C2×#10211yescharacteristic
301C2×C4#41711yescharacteristic
302C2×C4#44311yescharacteristic
303C2×C4#44411yescharacteristic
304C2×C4#44711yescharacteristic
305C2×C4#44811yescharacteristic
306C2×C4#48011yescharacteristic
307C2×C4#48111yescharacteristic
308C8—42no—
309C8—81no—
310C8—81no—
311C8—81no—
312C8—81no—
313C8—81no—
314C8—82no—
315C8—161no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22#187411yescharacteristic
324C22#187611yescharacteristic
325C22#188111yescharacteristic
326C22#190411yescharacteristic
327C22#190611yescharacteristic
328C22#190911yescharacteristic
329C22#191511yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—161no—
345C4—161no—
346C2—82no—
347C2#899511yescharacteristic
348C2#917911yescharacteristic
349C2#919511yescharacteristic
350C2#926611yescharacteristic
351C2#927911yescharacteristic
352C2#933511yescharacteristic
353C2#936911yescharacteristic
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. #7723C2semidirect
4. #7680C2semidirect
6. #7041C2semidirect
7. #7018C2semidirect

Maximal subgroups

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

subgroupindex
#298702
#227022
#77232
#76802
#75802
#70412
#70182