Finite Group Databasebeta

← #352707

Subgroups — #352707

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic, complemented
2#33289C211yesmaximal, characteristic, complemented
3#33277C211yesmaximal, characteristic
4#33238C211yesmaximal, characteristic
5#30220C211yesmaximal, characteristic, complemented
6#30203C211yesmaximal, characteristic
7#9840C212yesmaximal, complemented
8#9838C212yesmaximal
9#9822C212yesmaximal
10#9821C212yesmaximal, complemented
11C8×#97C211yesmaximal, characteristic, complemented
12C2×#487C2212yes—
13C2×#486C2212yes—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2858C2211yescharacteristic
18#2071—22no—
19#2043—22no—
20#2043—22no—
21#2023—22no—
22#1993—22no—
23#1992—22no—
24#1990—22no—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2211yescharacteristic
28#1990C2212yes—
29#1990C2211yescharacteristic
30#1959—22no—
31#1959—22no—
32#1957C2211yescharacteristic
33#1957C2212yes—
34#1957C2211yescharacteristic
35#1956C2212yes—
36#1956C2212yes—
37#1934—22no—
38#1934—22no—
39#1834—22no—
40#1834C2212yes—
41#1834C2212yes—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×C82C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2312yes—
54C2×#127C2311yescharacteristic
55C2×#125—21no—
56C2×#124—21no—
57C2×#124—21no—
58#487—24no—
59#487—24no—
60#486—24no—
61#486—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#102—24no—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—24no—
69C2×#97D412yes—
70C2×#97D412yes—
71C2×C4×C8D412yes—
72C2×C4×C8D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2312yes—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#384C2312yes—
95#382—21no—
96C43C2311yescharacteristic
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#336D412yes—
105#336D412yes—
106#322—24no—
107C82—24no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—41no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D414yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—82no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129#127—22no—
130#127—22no—
131#125—44no—
132#124—44no—
133#124—44no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2411yescharacteristic
167#103—48no—
168#103—84no—
169#103—84no—
170#102—44no—
171#102—48no—
172#102—44no—
173#102—84no—
174#97—24no—
175#97—24no—
176C4×C8—24no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—42no—
182C4×C8#4114yes—
183C4×C8#4114yes—
184#95—41no—
185#95—41no—
186C24—41no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—82no—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#12711yescharacteristic
213C22×C4#12711yescharacteristic
214C22×C4#12711yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—24no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8#12418yes—
230C2×C8#13714yes—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—82no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42#13514yes—
265C42#13514yes—
266C42C2×#4112yes—
267C42C2×#4112yes—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—81no—
275C23C2×#12711yescharacteristic
276Q8—84no—
277Q8—84no—
278Q8—84no—
279Q8—164no—
280D4—84no—
281D4—82no—
282D4—84no—
283D4—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4#49512yes—
308C2×C4#49512yes—
309C2×C4#49512yes—
310C2×C4#49512yes—
311C2×C4C2×#12412yes—
312C2×C4#53211yescharacteristic
313C2×C4#53212yes—
314C2×C4#53211yescharacteristic
315C2×C4C2×#13512yes—
316C2×C4C2×#13512yes—
317C2×C4C2×#13712yes—
318C2×C4#57512yes—
319C2×C4#57612yes—
320C2×C4#57712yes—
321C8—28no—
322C8—24no—
323C8—42no—
324C8—44no—
325C22—161no—
326C22—161no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22#285811yescharacteristic
333C22C2×#49511yescharacteristic
334C22C2×#49511yescharacteristic
335C22#315211yescharacteristic
336C22#315311yescharacteristic
337C22#315411yescharacteristic
338C22#315511yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—162no—
344C4—161no—
345C4—161no—
346C4—161no—
347C4—162no—
348C4#313114yes—
349C4#317212yes—
350C4#317914yes—
351C4#318112yes—
352C2—321no—
353C2#1962611yescharacteristic
354C2#1962711yescharacteristic
355C2#1963811yescharacteristic
356C2#1964011yescharacteristic
357C2#3322711yescharacteristic
358C2#3324311yescharacteristic
359C2#3325011yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33297C2semidirect
2. #33289C2semidirect
5. #30220C2semidirect
7. #9840C2semidirect
10. #9821C2semidirect
11. C8×#97C2semidirect

Maximal subgroups

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

subgroupindex
#332972
#332892
#332772
#332382
#302202
#302032
#98402
#98382
#98222
#98212
C8×#972