Finite Group Databasebeta

← #193061

Subgroups — #193061

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#21839C211yesmaximal, characteristic
4#20762C211yesmaximal, characteristic
5C2×#1531C211yesmaximal, characteristic
6#8915C214yesmaximal, complemented
7#8912C214yesmaximal, complemented
8#8034C211yesmaximal, characteristic
9#8031C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2382C2214yes—
15#2365—21no—
16#2365—21no—
17#2355—21no—
18#2355—21no—
19Q8×C4⋊C4—21no—
20Q8×C4⋊C4—21no—
21#2323—22no—
22#2323—22no—
23#2314—22no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1992C2212yes—
28#1992C2212yes—
29#1988—21no—
30#1987—21no—
31#1948—21no—
32#1947—21no—
33#1854—22no—
34#1847C2212yes—
35#1836—21no—
36#1836—21no—
37#1836C2212yes—
38#1836C2212yes—
39#1836C2212yes—
40#1825—22no—
41#1735—22no—
42#1531C418yescomplemented
43C2×#330C2214yes—
44#1295C2214yes—
45#1295C2214yes—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533C2×C414yes—
50#533Q814yes—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8D411yescharacteristic
53C2×#125C2312yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#117—21no—
61C2×#117D412yes—
62C2×#117D412yes—
63C2×#117D412yes—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#432—22no—
68#432C2×C414yes—
69C2×#107—22no—
70C2×#106—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×#97—21no—
76C2×#97—22no—
77C2×#97C2×C414yes—
78C2×#97C2312yes—
79C2×#97C2312yes—
80C2×#97C2312yes—
81#391—22no—
82#391—22no—
83#389—22no—
84#389—22no—
85#389—24no—
86#389—24no—
87#388—24no—
88#388—24no—
89#387—24no—
90#387—24no—
91#387—24no—
92#387—24no—
93#384—22no—
94#384D411yescharacteristic
95#384D411yescharacteristic
96#384D411yescharacteristic
97#384D411yescharacteristic
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#382—22no—
101#382—21no—
102#382C2312yes—
103#382C2312yes—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#330C2×C4116yes—
131C22×Q8—21no—
132C22×Q8—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C22×C8—24no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—42no—
141C22×C8—42no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8C4⋊C414yes—
145C4⋊Q8C4⋊C414yes—
146#125—22no—
147#125—22no—
148#125C22×C414yes—
149#125C2×Q814yes—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—84no—
159C4×Q8—84no—
160#117—24no—
161#117—44no—
162#117—44no—
163#117—82no—
164#117C4⋊C418yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—41no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—41no—
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q812yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42C2411yescharacteristic
209#107—82no—
210#106—82no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C22⋊C8—24no—
214C22⋊C8—42no—
215#97—24no—
216#97—22no—
217#97—22no—
218#97—22no—
219#97—22no—
220#97C22×C414yes—
221#97C22×C414yes—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—41no—
231#95—41no—
232C24#12011yescharacteristic
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×C4⋊C412yes—
254C22×C4C4×D412yes—
255C22×C4C4×D412yes—
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4C22⋊Q812yes—
260C22×C4C22⋊Q812yes—
261C22×C4#12312yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—48no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4C2×C4⋊C414yes—
305C4⋊C4C2×C4⋊C414yes—
306C22⋊C4—28no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C42—28no—
311C42—44no—
312C42—44no—
313C42—84no—
314C42—84no—
315C42—82no—
316C42C2×C4⋊C412yes—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C22⋊Q812yes—
320C42C22⋊Q812yes—
321C42#12312yes—
322C42C23×C411yescharacteristic
323C42C22×Q811yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23#38512yes—
333C23C2×#12011yescharacteristic
334Q8—48no—
335Q8—48no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#38512yes—
377C2×C4#38512yes—
378C2×C4#38512yes—
379C2×C4#45412yes—
380C2×C4#45512yes—
381C2×C4C22×C4⋊C411yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×C4×D411yescharacteristic
384C2×C4C2×C22⋊Q811yescharacteristic
385C2×C4C2×C22⋊Q811yescharacteristic
386C2×C4C2×#12311yescharacteristic
387C8—44no—
388C8—44no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22C2×#38511yescharacteristic
398C22C2×#45511yescharacteristic
399C22C2×#45411yescharacteristic
400C22#301911yescharacteristic
401C22#302011yescharacteristic
402C22#302011yescharacteristic
403C22#302011yescharacteristic
404C4—21no—
405C4—21no—
406C4—22no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—44no—
413C4—44no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—161no—
419C4—161no—
420C4#186014yes—
421C2—42no—
422C2#1905711yescharacteristic
423C2#1905811yescharacteristic
424C2#1906211yescharacteristic
425C2#1906311yescharacteristic
426C2#2085711yescharacteristic
427C2C2×#186011yescharacteristic
428C2#2086411yescharacteristic
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. #8915C2semidirect
7. #8912C2semidirect
42. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219882
#218392
#207622
C2×#15312
#89152
#89122
#80342
#80312