Finite Group Databasebeta

← #480742

Subgroups — #480742

Order 512.

963 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35667C211yesmaximal, characteristic
2#17371C212yesmaximal, complemented
3#11568C214yesmaximal, complemented
4#11565C214yesmaximal, complemented
5#11510C212yesmaximal, complemented
6#11507C212yesmaximal
7C2×#536C411yescharacteristic
8C2×#536C411yescharacteristic
9C22×C4⋊Q8C411yescharacteristic
10C2×#519C2211yescharacteristic
11#2983—22no—
12#2546C412yes—
13#2546C2212yes—
14#2545C412yes—
15#2545C2212yes—
16#2429—22no—
17#2429—22no—
18#2413—22no—
19#2413—22no—
20#2310C2211yescharacteristic
21#2297C2211yescharacteristic
22#2296C411yescharacteristic
23#1467—22no—
24#1467C2214yes—
25#1464—22no—
26#1464C2214yes—
27#1461C2214yes—
28#1458—22no—
29#1319C2218yes—
30#1319C2214yes—
31#1318C2214yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34C23×Q8C2×C411yescharacteristic
35#536—44no—
36#536—44no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—44no—
39C2×C4⋊Q8C2×C412yes—
40C2×C4⋊Q8C2×C412yes—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C2×#124C2×C412yes—
44C2×#124C2×C412yes—
45C2×#123—21no—
46C2×#123—21no—
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51#519—44no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8C2312yes—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117D412yes—
59C2×#117D412yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C42C2×C411yescharacteristic
64#446—48no—
65#432—24no—
66#432—24no—
67#431—24no—
68#398—22no—
69#398—22no—
70#398—22no—
71#398—22no—
72#396—22no—
73#396—22no—
74#396—22no—
75#396—22no—
76#391—22no—
77#391C2×C412yes—
78#391C2×C411yescharacteristic
79#391C2312yes—
80#391C2311yescharacteristic
81#390C2×C411yescharacteristic
82#390C2311yescharacteristic
83#389—21no—
84#389—21no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#386—22no—
94#385—22no—
95#381—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C22⋊C4—22no—
99#324—24no—
100#324—24no—
101#324C2318yes—
102#323—24no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8—24no—
107C22×Q8—21no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8C22⋊C412yes—
114C22×Q8C22×C412yes—
115C22×Q8C22×C411yescharacteristic
116C22×Q8C22×C412yes—
117C22×D4—21no—
118C22×D4C22×C411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C4⋊Q8—44no—
126C4⋊Q8—48no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130#124—44no—
131#124—44no—
132#123—48no—
133#123—44no—
134#123—44no—
135C22⋊Q8—44no—
136C22⋊Q8—48no—
137C22⋊Q8—44no—
138C4×Q8—28no—
139C4×Q8—44no—
140C4×Q8—44no—
141C4×Q8—44no—
142#117—24no—
143#117—42no—
144#117—44no—
145#117—44no—
146#117—42no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2411yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C2×D412yes—
177C2×C42—21no—
178C2×C42—42no—
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C412yes—
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C4⋊C8—44no—
190C4⋊C8—44no—
191C22⋊C8—28no—
192#97—44no—
193#97—44no—
194C4×C8—44no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203C24—21no—
204C24C2×C22⋊C411yescharacteristic
205#41—88no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—416no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—88no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4#10014yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D412yes—
260C2×C8—48no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4#9914yes—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C42—24no—
290C42—24no—
291C42—24no—
292C42—24no—
293C42—42no—
294C42—42no—
295C42—41no—
296C42—41no—
297C42—42no—
298C42—42no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×#10012yes—
309C23C22×C22⋊C411yescharacteristic
310Q8—48no—
311Q8—48no—
312Q8—84no—
313Q8—82no—
314Q8—84no—
315D4—82no—
316D4—84no—
317C2×C4—28no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4C2×#9914yes—
355C2×C4C2×#9912yes—
356C2×C4C2×#10014yes—
357C2×C4#56812yes—
358C8—88no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—82no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#287911yescharacteristic
372C22C22×#9911yescharacteristic
373C22C22×#10011yescharacteristic
374C4—24no—
375C4—22no—
376C4—22no—
377C4—24no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—82no—
385C4—81no—
386C4—82no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—161no—
391C2#1739611yescharacteristic
392C2#3336011yescharacteristic
393C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #11568C2semidirect
4. #11565C2semidirect
5. #11510C2semidirect

Maximal subgroups

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

subgroupindex
#356672
#173712
#115682
#115652
#115102
#115072