Finite Group Databasebeta

← #194238

Subgroups — #194238

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34373C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic
4#21804C211yesmaximal, characteristic
5C2×#1497C211yesmaximal, characteristic
6#8975C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8046C211yesmaximal, characteristic
9#7894C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2214yes—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2378—22no—
22#2362—22no—
23#2355—22no—
24#2355C2212yes—
25Q8×C4⋊C4C2212yes—
26#2336—22no—
27#2331—21no—
28#2327—21no—
29#2314—22no—
30#2285C2211yescharacteristic
31#2283C2211yescharacteristic
32#1993—22no—
33#1987C2212yes—
34#1987C2212yes—
35#1981—22no—
36#1956C2214yes—
37#1823—22no—
38#1787—22no—
39#1497—24no—
40C2×Q8⋊C8C2214yes—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2212yes—
44C23×Q8—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533D414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D411yescharacteristic
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8D412yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C42C2311yescharacteristic
70#500—28no—
71#487—28no—
72#484—216no—
73#477—28no—
74#477C2×C4116yes—
75#432—22no—
76#432—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431C2×C414yes—
81C2×#107—24no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—24no—
86C2×#103—28no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×#97—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#391—24no—
94#391—22no—
95#389—22no—
96#389—24no—
97#388—24no—
98#388—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#387—24no—
104#384—22no—
105#384C2312yes—
106#383—21no—
107#382—22no—
108#382—21no—
109#382D412yes—
110#381—21no—
111#380—21no—
112#380D412yes—
113#380D412yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123#376—21no—
124#340—24no—
125#340—24no—
126Q8⋊C8—28no—
127C22×Q8—21no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—41no—
131C22×Q8—42no—
132C22×Q8—84no—
133C22×Q8C2×D412yes—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C4⋊Q8C22⋊C414yes—
153C4⋊Q8C22⋊C414yes—
154C4⋊Q8C22×C414yes—
155#125—24no—
156#125—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8C22⋊C418yes—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—84no—
166C4×Q8C22×C414yes—
167C4×Q8C2×D414yes—
168#117—24no—
169#117—24no—
170#117—82no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#107—216no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—42no—
214C4⋊C8C22×C414yes—
215#103—216no—
216#103—88no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—42no—
221C4×C8—24no—
222C4×C8—22no—
223C4×C8—22no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—21no—
234#95—21no—
235C24#12011yescharacteristic
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8—84no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8C2×C22⋊C414yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—84no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4#12011yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12112yes—
268C22×C4#12112yes—
269C22×C4#12312yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—44no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4C2×C22⋊C414yes—
301C22⋊C4—28no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—24no—
305C42—24no—
306C42—42no—
307C42—42no—
308C42—82no—
309C42—84no—
310C42—84no—
311C42C2×C22⋊C412yes—
312C42C4×D412yes—
313C42C4×D412yes—
314C42#12112yes—
315C42#12112yes—
316C42#12312yes—
317C42C23×C411yescharacteristic
318C42C22×D411yescharacteristic
319C42C2×#4111yescharacteristic
320C42C2×#4111yescharacteristic
321C23—21no—
322C23—21no—
323C23—22no—
324C23—22no—
325C23#38612yes—
326C23C2×#12011yescharacteristic
327Q8—28no—
328Q8—48no—
329Q8—44no—
330Q8—84no—
331Q8—84no—
332Q8—88no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—88no—
361C2×C4—84no—
362C2×C4#38612yes—
363C2×C4#38612yes—
364C2×C4#38612yes—
365C2×C4C22×C22⋊C411yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×#12111yescharacteristic
369C2×C4C2×#12111yescharacteristic
370C2×C4C2×#12311yescharacteristic
371C2×C4#57712yes—
372C2×C4#57812yes—
373C2×C4#57814yes—
374C8—44no—
375C8—82no—
376C8—82no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—42no—
381C22—42no—
382C22C2×#38611yescharacteristic
383C22#300911yescharacteristic
384C22#301112yes—
385C22#301111yescharacteristic
386C22#302112yes—
387C4—22no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—44no—
393C4—44no—
394C4—82no—
395C4—84no—
396C4—82no—
397C4—84no—
398C4#289812yes—
399C4#295012yes—
400C4#304412yes—
401C4#309712yes—
402C2—42no—
403C2#1895912yes—
404C2#1896112yes—
405C2#2091311yescharacteristic
406C2#2093511yescharacteristic
407C2#2094511yescharacteristic
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. #8975C2semidirect
7. #8974C2semidirect
13. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#343732
#218402
#218042
C2×#14972
#89752
#89742
#80462
#78942