Finite Group Databasebeta

← #195682

Subgroups — #195682

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21839C211yesmaximal, characteristic
4#21392C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#9065C214yesmaximal, complemented
7#9059C214yesmaximal, complemented
8#7902C211yesmaximal, characteristic
9#7900C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229C418yescomplemented
14C2×#498C2212yes—
15C2×#487C2212yes—
16C2×#483—22no—
17C2×#475—22no—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2369—22no—
22#2365C2214yes—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1992—22no—
32#1988C2212yes—
33#1987C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1873—21no—
37#1873—21no—
38#1825—22no—
39#1763—22no—
40#1556—24no—
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—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⋊Q8D414yes—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8—24no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C42C2311yescharacteristic
72#498—24no—
73#498C2×C418yes—
74#487—24no—
75#487C2×C418yes—
76#483—216no—
77#475—216no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432—22no—
82#431—22no—
83#431C2×C414yes—
84C2×#106—24no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8C2312yes—
97#391—28no—
98#391—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#387—28no—
103#384D412yes—
104#382—21no—
105#380—24no—
106#380—21no—
107#380D412yes—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119C43—21no—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#332—28no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—84no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8C22⋊C418yes—
153C4⋊Q8C22×C412yes—
154C4⋊Q8C22×C412yes—
155C4⋊Q8C22×C414yes—
156C4⋊Q8C2×D414yes—
157#125—24no—
158#125—42no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—88no—
168#117—24no—
169#117—24no—
170#117—42no—
171#117—44no—
172#117—42no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C22⋊C418yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D412yes—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—28no—
203C2×C42—41no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212#106—216no—
213C4⋊C8—24no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218#103—216no—
219#103—44no—
220#103—44no—
221C22⋊C8—24no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224#97—42no—
225#97—42no—
226C4×C8—22no—
227C4×C8C22×C414yes—
228#95—28no—
229#95—28no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234C24#12011yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—28no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—48no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—84no—
247C2×Q8—88no—
248C2×Q8—88no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
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—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—88no—
302C4⋊C4C2×C22⋊C418yes—
303C4⋊C4#13714yes—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—24no—
309C42—44no—
310C42—42no—
311C42—44no—
312C42—44no—
313C42—44no—
314C42—88no—
315C42C2×C22⋊C412yes—
316C42C4×D412yes—
317C42C4×D412yes—
318C42#12112yes—
319C42#12112yes—
320C42#12312yes—
321C42C23×C411yescharacteristic
322C42C22×D411yescharacteristic
323C42C2×#4111yescharacteristic
324C42C2×#4111yescharacteristic
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23#38612yes—
330C23C2×#12011yescharacteristic
331Q8—44no—
332Q8—48no—
333Q8—48no—
334Q8—44no—
335Q8—88no—
336Q8—82no—
337Q8—82no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—28no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—88no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4#38612yes—
367C2×C4#38612yes—
368C2×C4#38612yes—
369C2×C4C22×C22⋊C411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×#12111yescharacteristic
373C2×C4C2×#12111yescharacteristic
374C2×C4C2×#12311yescharacteristic
375C2×C4C2×#13712yes—
376C2×C4#57512yes—
377C2×C4#57714yes—
378C8—44no—
379C8—82no—
380C8—82no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—42no—
385C22—42no—
386C22C2×#38611yescharacteristic
387C22#300411yescharacteristic
388C22#300511yescharacteristic
389C22#300912yes—
390C22#302012yes—
391C4—22no—
392C4—22no—
393C4—21no—
394C4—21no—
395C4—22no—
396C4—44no—
397C4—48no—
398C4—82no—
399C4—84no—
400C4—82no—
401C4—82no—
402C4C4×#13712yes—
403C4#294912yes—
404C4#303712yes—
405C4#304212yes—
406C2—42no—
407C2#1882912yes—
408C2#1883312yes—
409C2#2092611yescharacteristic
410C2#2093311yescharacteristic
411C2#2095411yescharacteristic
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. #9065C2semidirect
7. #9059C2semidirect
13. #3229C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#342162
#218392
#213922
C2×#15562
#90652
#90592
#79022
#79002