Finite Group Databasebeta

← #480436

Subgroups — #480436

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60204C211yesmaximal, characteristic
2#33428C212yesmaximal, complemented
3#11539C212yesmaximal, complemented
4#11537C212yesmaximal, complemented
5#11524C212yesmaximal, complemented
6#11521C212yesmaximal, complemented
7#11485C212yesmaximal, complemented
8#11483C212yesmaximal, complemented
9#3444C411yescharacteristic
10#2966—22no—
11#2966—22no—
12#2965—22no—
13C8×#41—22no—
14#2960C411yescharacteristic
15#2960C2211yescharacteristic
16#2931C411yescharacteristic
17#2931C2211yescharacteristic
18#2929C411yescharacteristic
19#2929C411yescharacteristic
20#2929C2211yescharacteristic
21#2929C2211yescharacteristic
22#2922C411yescharacteristic
23#2922C2211yescharacteristic
24#2920C2211yescharacteristic
25C4×#131C411yescharacteristic
26C4×#131C2211yescharacteristic
27C22×C4×C8C411yescharacteristic
28#2166—22no—
29#2166C2214yes—
30#2165—22no—
31#2165C2214yes—
32C2×#364—22no—
33#2115C2214yes—
34#2110—22no—
35#1328C2214yes—
36#1327C2214yes—
37#1321C2214yes—
38#1320C2214yes—
39C2×#131C2×C411yescharacteristic
40C2×#131C2×C411yescharacteristic
41C2×#131C2×C411yescharacteristic
42C2×#131C2311yescharacteristic
43C23×C8C2×C411yescharacteristic
44C23×C8C2×C411yescharacteristic
45#533—21no—
46#533C2×C411yescharacteristic
47#532—21no—
48#532C2×C411yescharacteristic
49#529—21no—
50#529—21no—
51#529—22no—
52#529—22no—
53#525—22no—
54#525C2×C411yescharacteristic
55#525C2×C411yescharacteristic
56C4×#41—22no—
57C4×#41C2×C411yescharacteristic
58C4×#41C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C2×M32—24no—
61#446—22no—
62#446—22no—
63C8×Q8—22no—
64C8×Q8—22no—
65#436—22no—
66#436—22no—
67#435—22no—
68#435—22no—
69#435—22no—
70#435—22no—
71C8×D4—22no—
72C8×D4—22no—
73C8×D4—22no—
74C8×D4—22no—
75C8×D4—22no—
76C8×D4—22no—
77#433—22no—
78#433—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83#424C2×C411yescharacteristic
84#424C2×C411yescharacteristic
85#424C2×C411yescharacteristic
86#424C2311yescharacteristic
87#423C2×C411yescharacteristic
88#423C2311yescharacteristic
89#408C2×C411yescharacteristic
90#408C2×C411yescharacteristic
91#408C2×C411yescharacteristic
92#408C2311yescharacteristic
93#407C2×C411yescharacteristic
94#407C2311yescharacteristic
95#405—21no—
96#405—21no—
97#405C2×C411yescharacteristic
98#405C2×C411yescharacteristic
99#405C2×C411yescharacteristic
100#405D412yes—
101#405D412yes—
102#405C2311yescharacteristic
103C4×M16—21no—
104C4×M16C2×C411yescharacteristic
105C4×M16D412yes—
106C4×M16C2311yescharacteristic
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2×C411yescharacteristic
113C2×C4×C8C2×C411yescharacteristic
114C2×C4×C8C2×C411yescharacteristic
115C2×C4×C8C2×C411yescharacteristic
116C2×C4×C8D412yes—
117#364—24no—
118#364—24no—
119#349—24no—
120#349C2318yes—
121#347—24no—
122#347—24no—
123C2×#41—22no—
124C2×#41C22×C411yescharacteristic
125C2×#41C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4C22×C411yescharacteristic
128#131—42no—
129#131—41no—
130#131—41no—
131#131—41no—
132#131—41no—
133#131—42no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16C22⋊C412yes—
137C2×M16C22×C411yescharacteristic
138C2×M16C22×C411yescharacteristic
139C2×M16C22×C411yescharacteristic
140C2×M16C2×D412yes—
141C2×M16C2411yescharacteristic
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—42no—
151C22×C8C22⋊C412yes—
152C22×C8C22×C411yescharacteristic
153C22×C8C22×C411yescharacteristic
154C22×C8C22×C411yescharacteristic
155C22×C8C22×C411yescharacteristic
156C22×C8C2×D412yes—
157C4⋊Q8—21no—
158C4⋊Q8M1611yescharacteristic
159#127M1611yescharacteristic
160#126—22no—
161#126—22no—
162#125—22no—
163#125—21no—
164#124—22no—
165#124M1611yescharacteristic
166#124M1611yescharacteristic
167#123—22no—
168#123—22no—
169#123—21no—
170#123M1612yes—
171C22⋊Q8—22no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C22⋊Q8—22no—
175#121—22no—
176#121—22no—
177#121—21no—
178#121—21no—
179#120—21no—
180#120M1612yes—
181C4×Q8—22no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—21no—
199#117C22⋊C412yes—
200#117C22×C411yescharacteristic
201#117C22×C411yescharacteristic
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22⋊C412yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211M32—28no—
212M32—44no—
213C2×C16—44no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#97—21no—
227#97—21no—
228#97—21no—
229#97C22⋊C412yes—
230#97C2×D412yes—
231#97C2×D412yes—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8C22⋊C411yescharacteristic
242C4×C8C22⋊C411yescharacteristic
243C4×C8C22⋊C411yescharacteristic
244C4×C8C22⋊C412yes—
245C4×C8C22⋊C411yescharacteristic
246C4×C8C2×D412yes—
247C4×C8C2×D412yes—
248C24C2×M1611yescharacteristic
249#41—41no—
250#41—41no—
251#41—48no—
252#41—41no—
253#41—41no—
254C2×Q8—22no—
255C2×Q8C2×M1611yescharacteristic
256C2×Q8C2×M1611yescharacteristic
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4C2×M1611yescharacteristic
262C2×D4C2×M1611yescharacteristic
263C2×D4C2×M1612yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—44no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×M1611yescharacteristic
280C22×C4C23×C411yescharacteristic
281M16—42no—
282M16—41no—
283M16—42no—
284M16—41no—
285M16—42no—
286M16—42no—
287M16—41no—
288M16—41no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8C2×C22⋊C411yescharacteristic
310C2×C8C2×C22⋊C411yescharacteristic
311C2×C8C2×C22⋊C412yes—
312C2×C8C2×C22⋊C411yescharacteristic
313C2×C8C2×C22⋊C411yescharacteristic
314C2×C8C22×D412yes—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—21no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4C2×M1612yes—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—22no—
344C42C2×C22⋊C411yescharacteristic
345C42C2×C22⋊C411yescharacteristic
346C42C2×C22⋊C412yes—
347C42C2×M1611yescharacteristic
348C42C2×M1611yescharacteristic
349C16—48no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—41no—
354C23#40712yes—
355C23C22×M1611yescharacteristic
356Q8—41no—
357Q8—41no—
358Q8—44no—
359D4—41no—
360D4—42no—
361D4—41no—
362D4—44no—
363D4—44no—
364D4—44no—
365D4—42no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4#40712yes—
393C2×C4C22×C22⋊C411yescharacteristic
394C2×C4C22×M1611yescharacteristic
395C8—22no—
396C8—21no—
397C8—21no—
398C8—22no—
399C8—41no—
400C8—41no—
401C8—41no—
402C8—41no—
403C8#41011yescharacteristic
404C8#41011yescharacteristic
405C8#41312yes—
406C22—22no—
407C22—21no—
408C22—22no—
409C22—41no—
410C22—42no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22C2×#40711yescharacteristic
418C4—22no—
419C4—22no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4C2×#41011yescharacteristic
429C4C2×#41311yescharacteristic
430C2—21no—
431C2—42no—
432C2—42no—
433C2—41no—
434C2—81no—
435C2#1736311yescharacteristic
subgroupG/H
centerC8#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC2#17363
prime radical#3444C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C8 ◁ C4×C8 ◁ C4×#131 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33428C2semidirect
3. #11539C2semidirect
4. #11537C2semidirect
5. #11524C2semidirect
6. #11521C2semidirect
7. #11485C2semidirect
8. #11483C2semidirect

Maximal subgroups

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

subgroupindex
#602042
#334282
#115392
#115372
#115242
#115212
#114852
#114832