Finite Group Databasebeta

← #100710

Subgroups — #100710

Order 512.

363 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 80 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#22838C211yesmaximal, characteristic
3#7723C211yesmaximal, characteristic, complemented
4#7680C211yesmaximal, characteristic, complemented
5#7580C211yesmaximal, characteristic
6#7041C211yesmaximal, characteristic, complemented
7#7018C211yesmaximal, characteristic, complemented
8C2×#478—21no—
9#2859C2211yescharacteristic
10#2293C2211yescharacteristic
11#2085—21no—
12#2084C411yescharacteristic
13#2083C411yescharacteristic
14#2073—21no—
15#2071C411yescharacteristic
16#2071C411yescharacteristic
17#1989—21no—
18#1955—21no—
19#1924—21no—
20#1923—21no—
21#1839C2211yescharacteristic
22#1831C2211yescharacteristic
23#1831C2211yescharacteristic
24#1818—21no—
25#1771—21no—
26#1767C2211yescharacteristic
27#1750—21no—
28#1744C2211yescharacteristic
29#1271—21no—
30#1271—21no—
31#1271—21no—
32#1271—21no—
33#1271—21no—
34#1270—21no—
35C2×#125—21no—
36C2×C4×D4D411yescharacteristic
37C2×#117—21no—
38#478—42no—
39#478—42no—
40C2×#107—21no—
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×#102—21no—
46C2×#102—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×#97—21no—
52C2×#97—21no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55#390C2×C411yescharacteristic
56#390D411yescharacteristic
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389C2×C411yescharacteristic
62#389D411yescharacteristic
63#388—21no—
64#388—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#382—21no—
68C4×C4⋊C4D411yescharacteristic
69C4×C22⋊C4—21no—
70C43C2311yescharacteristic
71#340—21no—
72#340—21no—
73#340—21no—
74#340—21no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336C2×C411yescharacteristic
85#336C2×C411yescharacteristic
86#336D411yescharacteristic
87#336D411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C23×C4—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—41no—
99C22×C8—41no—
100#125—41no—
101#125—41no—
102#125—41no—
103#125—41no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108#117—81no—
109#117—81no—
110C2×C4⋊C4—21no—
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—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4C22⋊C411yescharacteristic
121C2×C4⋊C4C22⋊C411yescharacteristic
122C2×C4⋊C4C22⋊C411yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C42—21no—
129C2×C42C22×C411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42#4111yescharacteristic
135C2×C42#4111yescharacteristic
136#107—42no—
137#106—42no—
138C4⋊C8—42no—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143#102—42no—
144#102—42no—
145C22⋊C8—42no—
146C22⋊C8—42no—
147C22⋊C8—81no—
148C22⋊C8—81no—
149#97—41no—
150#97—41no—
151#97—81no—
152C4×C8—22no—
153C4×C8—41no—
154C4×C8—81no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—41no—
159#95—41no—
160C24—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—41no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4C2×C22⋊C411yescharacteristic
177C22×C4C4×D411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12311yescharacteristic
183C2×C8—22no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
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—81no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C42—21no—
234C42—21no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42—41no—
242C42#13511yescharacteristic
243C42#13511yescharacteristic
244C42AGL(1,ℤ/8)11yescharacteristic
245C42AGL(1,ℤ/8)11yescharacteristic
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#38611yescharacteristic
254D4—42no—
255D4—81no—
256C2×C4—21no—
257C2×C4—21no—
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—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4#41611yescharacteristic
293C2×C4#41811yescharacteristic
294C2×C4#44311yescharacteristic
295C2×C4#44411yescharacteristic
296C2×C4#44911yescharacteristic
297C2×C4#44911yescharacteristic
298C2×C4#48211yescharacteristic
299C2×C4#48211yescharacteristic
300C8—42no—
301C8—81no—
302C8—81no—
303C8—81no—
304C8—81no—
305C8—81no—
306C8—82no—
307C8—161no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#187511yescharacteristic
316C22#187611yescharacteristic
317C22#188111yescharacteristic
318C22#190411yescharacteristic
319C22#190611yescharacteristic
320C22#190911yescharacteristic
321C22#191511yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—161no—
337C4—161no—
338C2—161no—
339C2#899611yescharacteristic
340C2#918111yescharacteristic
341C2#919611yescharacteristic
342C2#926611yescharacteristic
343C2#927911yescharacteristic
344C2#933511yescharacteristic
345C2#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
#228382
#77232
#76802
#75802
#70412
#70182