Finite Group Databasebeta

← #207492

Subgroups — #207492

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34629C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic
3#20547C211yesmaximal, characteristic
4#9867C212yesmaximal, complemented
5#9677C212yesmaximal, complemented
6#8938C212yesmaximal
7#8937C212yesmaximal
8#8178C212yesmaximal, complemented
9#8176C212yesmaximal, complemented
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2655—22no—
13#2652—22no—
14#2647C411yescharacteristic
15#2645C412yes—
16#2642C411yescharacteristic
17#2629C411yescharacteristic
18#2625C412yes—
19#2621C411yescharacteristic
20#2402—22no—
21#2401—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25#2295C2212yes—
26C4×#117C2211yescharacteristic
27#1923—22no—
28#1916—22no—
29#1770—22no—
30#1767C2212yes—
31#1767C2212yes—
32C4×C4⋊C8C2212yes—
33#1752—22no—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C23×C8—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125C2×C411yescharacteristic
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8C2312yes—
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C22×C42C2311yescharacteristic
53#431—24no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—24no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—22no—
61C2×#97—22no—
62C2×#97—22no—
63C2×C4×C8—21no—
64C2×C4×C8—22no—
65C2×C4×C8C2312yes—
66C2×C4×C8C2312yes—
67#401—22no—
68#400—22no—
69#400—22no—
70#400—22no—
71#400—22no—
72#400—22no—
73#400—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#397—22no—
80#397—22no—
81#397—22no—
82#396—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2×C412yes—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#380C2×C412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4C2312yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110#376—22no—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43C2311yescharacteristic
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120Q8⋊C8—24no—
121Q8⋊C8—24no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124C22×Q8C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—24no—
132C22×C8—22no—
133C22×C8—82no—
134C4⋊Q8—42no—
135#125—42no—
136C22⋊Q8—28no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141#117—24no—
142#117—42no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C22⋊C412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—24no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—44no—
181C4⋊C8#4114yes—
182C4⋊C8#4114yes—
183C22⋊C8—24no—
184C22⋊C8—44no—
185#97—82no—
186#97—82no—
187C4×C8—24no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—44no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24C2×C22⋊C411yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#13114yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—84no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—24no—
270C22⋊C4—42no—
271C22⋊C4—44no—
272C22⋊C4—82no—
273C42—22no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42C2×#4112yes—
292C42C2×#4112yes—
293C23—21no—
294C23—22no—
295C23—22no—
296C23—21no—
297C23#40812yes—
298C23C22×C22⋊C411yescharacteristic
299Q8—44no—
300Q8—44no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—28no—
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—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4#40812yes—
338C2×C4C4×#4112yes—
339C2×C4#52511yescharacteristic
340C2×C4#52511yescharacteristic
341C2×C4#53212yes—
342C2×C4C2×#13112yes—
343C2×C4#56812yes—
344C2×C4#57712yes—
345C2×C4#57812yes—
346C8—44no—
347C8—44no—
348C8—44no—
349C8—82no—
350C22—24no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22#229711yescharacteristic
356C22C2×#40811yescharacteristic
357C22#287911yescharacteristic
358C22#289812yes—
359C22#290112yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4#296814yes—
373C4#312012yes—
374C4#312311yescharacteristic
375C4#312711yescharacteristic
376C2—42no—
377C2#1748412yes—
378C2#1750612yes—
379C2#2137711yescharacteristic
380C2#2138411yescharacteristic
381C2#2139811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9677C2semidirect
8. #8178C2semidirect
9. #8176C2semidirect

Maximal subgroups

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

subgroupindex
#346292
#214302
#205472
#98672
#96772
#89382
#89372
#81782
#81762