Finite Group Databasebeta

← #163796

Subgroups — #163796

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic
2#22641C212yesmaximal
3#7729C212yesmaximal, complemented
4#7713C212yesmaximal, complemented
5#7689C212yesmaximal, complemented
6#7673C212yesmaximal, complemented
7#7643C211yesmaximal, characteristic
8#7629C211yesmaximal, characteristic
9#7613C212yesmaximal
10#2863C2212yes—
11#2863C2212yes—
12#2645C412yes—
13#2645C412yes—
14#2514—22no—
15#2511—22no—
16#2493—22no—
17#2493—22no—
18#2317C411yescharacteristic
19#2316C411yescharacteristic
20#2310C411yescharacteristic
21#2310C411yescharacteristic
22#2296C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1940—22no—
26#1924—22no—
27#1923—22no—
28#1839C2212yes—
29#1839C2212yes—
30#1797C2212yes—
31#1771—22no—
32#1767C2212yes—
33C4×C4⋊C8C2212yes—
34#1749C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2211yescharacteristic
38#1272C2212yes—
39#1272C2211yescharacteristic
40#1272C2211yescharacteristic
41#1271C2212yes—
42#1271C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45C23×Q8C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117C2×C411yescharacteristic
49C2×#117C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×C4⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57C2×#97—22no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#401—22no—
61#401—22no—
62#400—22no—
63#400—22no—
64#400—22no—
65#400—22no—
66#400—22no—
67#400—22no—
68#391—21no—
69#391—22no—
70#391—21no—
71#391—22no—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#391C2311yescharacteristic
79#391C2311yescharacteristic
80#389—22no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2312yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384—21no—
96#383—22no—
97#383—22no—
98#382—21no—
99#381—21no—
100#381—21no—
101#381—22no—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109#376C2×C412yes—
110C2×#95—22no—
111C43C2312yes—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336C2312yes—
117#336C2312yes—
118#336C2312yes—
119#336C2312yes—
120C22×Q8—21no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8C22×C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D412yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C411yescharacteristic
152C2×C4⋊C4C22×C412yes—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—42no—
161C2×C42C22×C412yes—
162C2×C42C22×C412yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2411yescharacteristic
170C4⋊C8—44no—
171C4⋊C8—44no—
172C4⋊C8—44no—
173C22⋊C8—44no—
174C22⋊C8—82no—
175#97—44no—
176#97—82no—
177C4×C8—82no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95C22⋊C412yes—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200C24C2×C22⋊C411yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—42no—
210C2×Q8—88no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—82no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D412yes—
237C2×C8—24no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C4⋊C4—24no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—24no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—24no—
275C42—42no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42—42no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23#41312yes—
287C23C22×C22⋊C411yescharacteristic
288Q8—82no—
289Q8—82no—
290Q8—84no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—84no—
332C2×C4—162no—
333C2×C4#41012yes—
334C2×C4#56812yes—
335C2×C4#57712yes—
336C2×C4#57812yes—
337C8—44no—
338C8—82no—
339C8—82no—
340C8—82no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—42no—
345C22—42no—
346C22#287911yescharacteristic
347C22C2×#41011yescharacteristic
348C22C2×#41311yescharacteristic
349C22#289812yes—
350C22#290112yes—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—82no—
363C4—82no—
364C4—81no—
365C4—81no—
366C2—42no—
367C2#1739811yescharacteristic
368C2#1750612yes—
369C2#1761212yes—
370C2#1768611yescharacteristic
371C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7729C2semidirect
4. #7713C2semidirect
5. #7689C2semidirect
6. #7673C2semidirect

Maximal subgroups

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

subgroupindex
#353132
#226412
#77292
#77132
#76892
#76732
#76432
#76292
#76132