Finite Group Databasebeta

← #195724

Subgroups — #195724

Order 512.

1144 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34365C211yesmaximal, characteristic
3#21892C211yesmaximal, characteristic
4#21837C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9066C214yesmaximal, complemented
7#9060C214yesmaximal, complemented
8#7899C211yesmaximal, characteristic
9#7894C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2381—22no—
22#2369C2214yes—
23#2344—22no—
24#2344—22no—
25#2331—22no—
26#2331—22no—
27#2310—22no—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30#1991—22no—
31#1988C2212yes—
32#1987C2212yes—
33#1987C2212yes—
34#1981C2212yes—
35#1894—22no—
36#1825—22no—
37#1805—22no—
38#1548—24no—
39C2×#333C2214yes—
40#1279C2214yes—
41#1279C2214yes—
42C23×Q8D412yes—
43C22×Q16—24no—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×#117—24no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#494—24no—
70#494C2×C418yes—
71#487—24no—
72#487C2×C418yes—
73#462—216no—
74#451—216no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431C2×C414yes—
79#423—22no—
80#423—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—24no—
86C2×#103—24no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#391—28no—
92#391—28no—
93#389—24no—
94#389—24no—
95#387—28no—
96#387—28no—
97#383—21no—
98#383—21no—
99#382D412yes—
100#382D412yes—
101#381—24no—
102#381—21no—
103#381—21no—
104#380D412yes—
105#380D412yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2312yes—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112#376—21no—
113#376—21no—
114#340—24no—
115#340—24no—
116#340—24no—
117#333—28no—
118C22×Q8—22no—
119C22×Q8—24no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—84no—
123C22×Q8C22⋊C418yes—
124C22×Q8C2×D414yes—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×Q16—216no—
130C2×Q16—84no—
131C2×Q16—84no—
132C2×M16—42no—
133C2×M16—82no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C22⋊C418yes—
151C4⋊Q8C22×C412yes—
152C4⋊Q8C22×C412yes—
153C4⋊Q8C22×C414yes—
154C4⋊Q8C2×D414yes—
155#125—24no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×Q8—42no—
165#117—24no—
166#117—42no—
167#117—88no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—41no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—28no—
191C2×C42—41no—
192C2×C42C22×C412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—42no—
206#103—216no—
207#103—84no—
208C22⋊C8—24no—
209C22⋊C8—42no—
210C4×C8—22no—
211C4×C8—42no—
212C4×C8C22×C414yes—
213#95—28no—
214#95—28no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C24#12011yescharacteristic
224C2×Q8—24no—
225C2×Q8—28no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—84no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×Q8C2×C22⋊C418yes—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—28no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—48no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C4×D412yes—
252C22×C4C4×D412yes—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12112yes—
257C22×C4#12112yes—
258C22×C4#12312yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262Q16—816no—
263M16—84no—
264C2×C8—24no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—44no—
270C2×C8—44no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—88no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—42no—
295C42—44no—
296C42—42no—
297C42—44no—
298C42—88no—
299C42C2×C22⋊C412yes—
300C42C4×D412yes—
301C42C4×D412yes—
302C42#12112yes—
303C42#12112yes—
304C42#12312yes—
305C42C23×C411yescharacteristic
306C42C22×D411yescharacteristic
307C42C2×#4111yescharacteristic
308C42C2×#4111yescharacteristic
309C23—22no—
310C23—22no—
311C23—22no—
312C23#38612yes—
313C23C2×#12011yescharacteristic
314Q8—48no—
315Q8—44no—
316Q8—44no—
317Q8—84no—
318Q8—88no—
319Q8—82no—
320Q8—81no—
321Q8—81no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—416no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#38612yes—
348C2×C4#38612yes—
349C2×C4#38612yes—
350C2×C4C22×C22⋊C411yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×#12111yescharacteristic
354C2×C4C2×#12111yescharacteristic
355C2×C4C2×#12311yescharacteristic
356C2×C4#57712yes—
357C2×C4#57812yes—
358C2×C4#57814yes—
359C8—44no—
360C8—82no—
361C8—82no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—44no—
366C22C2×#38611yescharacteristic
367C22#301112yes—
368C22#301312yes—
369C22#301912yes—
370C4—22no—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—48no—
376C4—82no—
377C4—82no—
378C4—84no—
379C4—84no—
380C4#294712yes—
381C4#295012yes—
382C4#304311yescharacteristic
383C4#304411yescharacteristic
384C4#304412yes—
385C2—42no—
386C2#1893314yes—
387C2#2093011yescharacteristic
388C2#2093511yescharacteristic
389C2#2095311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9066C2semidirect
7. #9060C2semidirect
13. #3209C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#343652
#218922
#218372
C2×#15482
#90662
#90602
#78992
#78942