Finite Group Databasebeta

← #190664

Subgroups — #190664

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34215C211yesmaximal, characteristic
2#21435C212yesmaximal
3#8718C212yesmaximal
4#8584C211yesmaximal, characteristic, complemented
5#8582C212yesmaximal, complemented
6#8580C211yesmaximal, characteristic, complemented
7#8338C211yesmaximal, characteristic
8#8338C211yesmaximal, characteristic
9#8227C212yesmaximal, complemented
10#8225C211yesmaximal, characteristic, complemented
11C4×#330C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13#2368—22no—
14#2368—22no—
15#2355—22no—
16#2355—22no—
17#2331C412yes—
18#2331C412yes—
19#2317C412yes—
20C4×#125C2211yescharacteristic
21C4×#125C2211yescharacteristic
22C4×#125C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#2274C412yes—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1923—22no—
29#1917—22no—
30#1916—22no—
31#1767C2212yes—
32#1767C2212yes—
33#1749C2212yes—
34#1749C2212yes—
35#1745C2212yes—
36C4×#97C2212yes—
37C2×#330C2211yescharacteristic
38C2×#330C2211yescharacteristic
39C2×#330C2211yescharacteristic
40C2×#330C2211yescharacteristic
41C2×#330C2212yes—
42C2×#330C2212yes—
43#1295C2212yes—
44#1295C2212yes—
45#1295C2212yes—
46#1295C2212yes—
47#533—24no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8C2×C412yes—
51C2×#125C2311yescharacteristic
52C2×#125C2311yescharacteristic
53C2×C22⋊Q8—22no—
54C2×C4×Q8—22no—
55C2×#117C2×C412yes—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×#97C2312yes—
66C2×#97C2312yes—
67C2×#97C2312yes—
68C2×#97C2312yes—
69C2×C4×C8—22no—
70#391—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#389C2×C412yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#385—22no—
87#385—22no—
88#384—22no—
89#382—22no—
90#382—22no—
91#382C2×C412yes—
92#382C2311yescharacteristic
93#382C2311yescharacteristic
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110#376C2×C411yescharacteristic
111#376C2×C411yescharacteristic
112C2×#95—22no—
113C43C2311yescharacteristic
114C43C2311yescharacteristic
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#330—22no—
122#330—22no—
123#330—22no—
124#330—22no—
125#330—24no—
126#330—24no—
127C22×Q8—22no—
128C23×C4—22no—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137#125—22no—
138#125—21no—
139#125—21no—
140#125#4112yes—
141#125#4112yes—
142#125#4114yes—
143C22⋊Q8—28no—
144C22⋊Q8—28no—
145C4×Q8—28no—
146C4×Q8—28no—
147#117—24no—
148#117—24no—
149#117—24no—
150#117—42no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—44no—
191C4⋊C8—44no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—44no—
203C4×C8—44no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219#95C22⋊C412yes—
220C24C2×C22⋊C411yescharacteristic
221C2×Q8—28no—
222C2×Q8—82no—
223C2×Q8—82no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D412yes—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—28no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4#10114yes—
278C4⋊C4#11714yes—
279C4⋊C4#11714yes—
280C22⋊C4—28no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C42—28no—
285C42—21no—
286C42—21no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—41no—
296C42—41no—
297C42—82no—
298C42—82no—
299C42—82no—
300C42—82no—
301C42C2×#4111yescharacteristic
302C42C2×#4111yescharacteristic
303C42C2×#4112yes—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—21no—
308C23#41312yes—
309C23C22×C22⋊C411yescharacteristic
310Q8—88no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—164no—
349C2×C4C2×#10112yes—
350C2×C4C2×#10112yes—
351C2×C4C2×#10112yes—
352C2×C4C2×#11711yescharacteristic
353C2×C4C2×#11711yescharacteristic
354C2×C4#52512yes—
355C2×C4#52512yes—
356C8—44no—
357C8—44no—
358C8—82no—
359C8—82no—
360C22—24no—
361C22—22no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22#228011yescharacteristic
366C22C22×#10111yescharacteristic
367C22C2×#41311yescharacteristic
368C22#290112yes—
369C22#290112yes—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—21no—
375C4—21no—
376C4—22no—
377C4—42no—
378C4—44no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—82no—
385C4—82no—
386C2—42no—
387C2#1768612yes—
388C2#1768911yescharacteristic
389C2#1769211yescharacteristic
390C2#2072711yescharacteristic
391C2#2074912yes—
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8584C2semidirect
5. #8582C2semidirect
6. #8580C2semidirect
9. #8227C2semidirect
10. #8225C2semidirect
11. C4×#330C2semidirect

Maximal subgroups

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

subgroupindex
#342152
#214352
#87182
#85842
#85822
#85802
#83382
#82272
#82252
C4×#3302