Finite Group Databasebeta

← #204737

Subgroups — #204737

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35335C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic
3#20480C211yesmaximal, characteristic
4C2×#1537C211yesmaximal, characteristic
5C2×#1528C211yesmaximal, characteristic
6C2×#1528C211yesmaximal, characteristic
7C2×#1526C211yesmaximal, characteristic
8#9690C214yesmaximal, complemented
9#9686C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2859C2214yes—
17#2660C412yes—
18#2660C412yes—
19#2584—22no—
20#2584—21no—
21#2584—21no—
22#2583—22no—
23#2583—22no—
24#2317C412yes—
25#2317C412yes—
26C4×#117C2211yescharacteristic
27#1939—22no—
28#1851—22no—
29#1770—21no—
30C8×C4⋊C4—21no—
31#1767C2214yes—
32#1753—22no—
33#1745C2212yes—
34C4×#97C2212yes—
35#1537—24no—
36#1528—24no—
37#1528—24no—
38#1526—24no—
39C2×#330C2212yes—
40C2×#330C2212yes—
41C2×#330C2214yes—
42C2×#330C2214yes—
43C2×#330C2212yes—
44C2×#330C2212yes—
45C22×M16—21no—
46C22×M16—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C412yes—
60C2×#125C2312yes—
61C2×#125C2312yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2×C412yes—
67C2×#117C2×C412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#432—22no—
76#432D414yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C4×M16—22no—
83C4×M16—22no—
84C4×M16—22no—
85C4×M16D414yes—
86C2×#97—22no—
87C2×#97C2312yes—
88C2×#97C2312yes—
89C2×#97C2311yescharacteristic
90C2×#97C2311yescharacteristic
91C2×#97C2312yes—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—22no—
95#398—22no—
96#398—24no—
97#398—22no—
98#395—22no—
99#395—24no—
100#395—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—24no—
106#389—24no—
107#389—24no—
108#389—24no—
109#385—24no—
110#385—24no—
111#384C2×C414yes—
112#384C2×C414yes—
113#382C2×C414yes—
114#382C2×C414yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120#376C2×C412yes—
121C43C2312yes—
122#336—24no—
123#336—22no—
124#336—22no—
125#330—28no—
126#330—24no—
127#330—24no—
128#330—28no—
129#330—24no—
130#330—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16—44no—
142C2×M16—42no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152#125—24no—
153#125—24no—
154#125—22no—
155#125—22no—
156#125—22no—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161#125—22no—
162#125—42no—
163#125—42no—
164#125—42no—
165#125—42no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174#117—28no—
175#117—24no—
176#117—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C414yes—
207C2×C42—22no—
208C2×C42C22×C414yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D414yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—24no—
221C4⋊C8—48no—
222C4⋊C8—42no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225#97—22no—
226#97—22no—
227#97—22no—
228#97—22no—
229#97—21no—
230#97—21no—
231#97—22no—
232#97—44no—
233#97C2×D412yes—
234#97C2×D412yes—
235#97C2×D414yes—
236C4×C8—24no—
237C4×C8—44no—
238C4×C8—42no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95C22⋊C412yes—
244#95C22⋊C414yes—
245#95C22⋊C412yes—
246C24C2×C22⋊C411yescharacteristic
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—82no—
250C2×Q8—81no—
251C2×Q8—81no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C414yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12412yes—
275C22×C4#12712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—48no—
280M16—48no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4#10118yes—
323C22⋊C4—28no—
324C22⋊C4—44no—
325C42—28no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42C4×D412yes—
331C42C4×D412yes—
332C42#12112yes—
333C42#12112yes—
334C42#12412yes—
335C42#12712yes—
336C42C22×D411yescharacteristic
337C42C22×D411yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C23—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23#39012yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—84no—
347Q8—84no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#39012yes—
380C2×C4#39012yes—
381C2×C4#39012yes—
382C2×C4C2×#10118yes—
383C2×C4C2×#10114yes—
384C2×C4C2×C4×D411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×#12111yescharacteristic
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12411yescharacteristic
389C2×C4C2×#12711yescharacteristic
390C8—24no—
391C8—44no—
392C8—44no—
393C8—42no—
394C8—82no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22C2×#39011yescharacteristic
401C22C22×#10112yes—
402C22#290112yes—
403C22#290111yescharacteristic
404C22#290111yescharacteristic
405C4—21no—
406C4—21no—
407C4—22no—
408C4—28no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—82no—
419C4#196614yes—
420C2—42no—
421C2#1768912yes—
422C2#1769212yes—
423C2#2121011yescharacteristic
424C2C2×#196611yescharacteristic
425C2#2125311yescharacteristic
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
8. #9690C2semidirect
9. #9686C2semidirect

Maximal subgroups

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

subgroupindex
#353352
#213562
#204802
C2×#15372
C2×#15282
C2×#15262
#96902
#96862