Finite Group Databasebeta

← #213580

Subgroups — #213580

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2#20717C212yesmaximal
3#9086C214yesmaximal, complemented
4#9082C214yesmaximal, complemented
5#8700C212yesmaximal
6#8308C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C22×#117C2211yescharacteristic
9C2×#432C2212yes—
10C2×#431C2212yes—
11#2499—22no—
12#2499—22no—
13#2499—22no—
14#2499—22no—
15C2×#397C414yes—
16C2×#391C411yescharacteristic
17C2×#391C411yescharacteristic
18C2×#387C2214yes—
19C2×#384C411yescharacteristic
20C2×#382C411yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1817C2214yes—
23#1798—22no—
24#1798—22no—
25#1779C2214yes—
26#1750—22no—
27#1750—22no—
28C2×#324C2212yes—
29C2×#324C2214yes—
30C2×#324C2212yes—
31#1280C2212yes—
32#1280C2214yes—
33#1280C2212yes—
34C23×Q8C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—41no—
39C2×C22⋊Q8—41no—
40C2×C22⋊Q8—42no—
41C2×#117—41no—
42C2×#117—42no—
43C2×#117—41no—
44C2×#117D412yes—
45C2×#117D412yes—
46C2×#117D412yes—
47C2×#117D412yes—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C412yes—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2312yes—
59#432—42no—
60#432—42no—
61#431—42no—
62#431—42no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8C2314yes—
68C2×C22⋊C8C2314yes—
69C2×#97—22no—
70C2×C4×C8—22no—
71#397—24no—
72#397—24no—
73#397—24no—
74#397—24no—
75#391—21no—
76#391—21no—
77#391—22no—
78#391—21no—
79#391—21no—
80#391—22no—
81#387—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#384—22no—
86#384—21no—
87#384—21no—
88#382—22no—
89#382—21no—
90#382—21no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95C2×C414yes—
94C2×#95C2×C414yes—
95C2×#95C2×C414yes—
96C2×#95C2312yes—
97#336—24no—
98#336—24no—
99#324—42no—
100#324—44no—
101#324—42no—
102C25C22×C411yescharacteristic
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8—42no—
108C22×Q8—42no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4C22×C412yes—
115C23×C4C22×C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C412yes—
120C23×C4C22×C412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C22×C8—24no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130#117—42no—
131#117—42no—
132#117—42no—
133#117—42no—
134#117—42no—
135#117—42no—
136#117—42no—
137#117—42no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C412yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—48no—
193C22⋊C8—48no—
194#97—44no—
195C4×C8—44no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—41no—
208C24—41no—
209C24—42no—
210C24C2×C22⋊C412yes—
211C24C23×C411yescharacteristic
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—84no—
219C2×Q8—84no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4#13114yes—
270C22×C4C22×D412yes—
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)12yes—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—48no—
297C22⋊C4—48no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—24no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23C22×C22⋊C411yescharacteristic
330C23#51811yescharacteristic
331C23#51811yescharacteristic
332C23C2×#13112yes—
333C23#56812yes—
334Q8—84no—
335Q8—84no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4#41012yes—
385C2×C4#41012yes—
386C2×C4#53412yes—
387C2×C4#53412yes—
388C8—84no—
389C8—84no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—82no—
405C22—84no—
406C22#230811yescharacteristic
407C22#288012yes—
408C22C2×#41011yescharacteristic
409C22C2×#41011yescharacteristic
410C22#288912yes—
411C22#297911yescharacteristic
412C22#298011yescharacteristic
413C22#298112yes—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—44no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—82no—
425C4—82no—
426C2—21no—
427C2—21no—
428C2—22no—
429C2—82no—
430C2#1741312yes—
431C2#1742012yes—
432C2#2174511yescharacteristic
433C2#2174711yescharacteristic
434C2#2175611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9086C2semidirect
4. #9082C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#207172
#90862
#90822
#87002
#83082