Finite Group Databasebeta

← #493983

Subgroups — #493983

Order 512.

1052 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2#20727C212yesmaximal
3#12860C212yesmaximal
4#12804C214yesmaximal, complemented
5#12700C212yesmaximal
6#12631C214yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×C4×Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—22no—
11#2327—22no—
12#2327—22no—
13#2327C412yes—
14#2327C412yes—
15#2327C2212yes—
16#2327C2212yes—
17C2×#391C411yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#387—22no—
20C2×#387—22no—
21C2×#382C411yescharacteristic
22C2×#376C411yescharacteristic
23C2×#356C2218yes—
24#1821—22no—
25#1801—22no—
26#1801C2214yes—
27#1784C2214yes—
28C4×#101—22no—
29C4×#101C2214yes—
30#1394C2218yes—
31C23×Q8C2×C411yescharacteristic
32C23×Q8C2311yescharacteristic
33C2×C4⋊Q8—22no—
34C2×C4⋊Q8—28no—
35C2×C4⋊Q8C2×C412yes—
36C2×C4⋊Q8C2312yes—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—42no—
40C2×C4×Q8—42no—
41C2×C4×Q8—42no—
42C2×C4×Q8—42no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C42—21no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61#432—42no—
62#432—42no—
63#431—42no—
64#431—42no—
65#423—24no—
66C2×#101—24no—
67C2×#101C2318yes—
68#407—24no—
69C4×M16—24no—
70#391—22no—
71#391—22no—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2312yes—
76#387—24no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#382—22no—
85#382C2×C412yes—
86#382C2×C412yes—
87#376—21no—
88#376—21no—
89#376C2×C412yes—
90#376C2×C412yes—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96#356—216no—
97#343—24no—
98#343—24no—
99C22×Q8—24no—
100C22×Q8—24no—
101C22×Q8—21no—
102C22×Q8—41no—
103C22×Q8—41no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8C22×C412yes—
109C22×Q8C22×C411yescharacteristic
110C22×Q8C2×D412yes—
111C22×Q8C2411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C2×D412yes—
121C2×M16—28no—
122C4⋊Q8—216no—
123C4⋊Q8—24no—
124C4⋊Q8—24no—
125C4⋊Q8—24no—
126C4⋊Q8#4118yes—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×Q8—42no—
135C4×Q8—44no—
136C4×Q8—44no—
137C4×Q8—44no—
138C4×Q8—44no—
139C2×C4⋊C4—216no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C412yes—
178C2×C42C22⋊C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C4⋊C8—44no—
186C4⋊C8—44no—
187#101—216no—
188#101—48no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191#97—44no—
192C4×C8—44no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203C24C2×C22⋊C411yescharacteristic
204C2×Q8—24no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—48no—
216C2×Q8—48no—
217C2×Q8—48no—
218C2×Q8—48no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8#11714yes—
223C2×Q8C2×#4114yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4#9912yes—
257C22×C4#9912yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268M16—48no—
269M16—48no—
270C2×C8—48no—
271C4⋊C4—216no—
272C4⋊C4—216no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C42—24no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—42no—
299C42—44no—
300C42—44no—
301C42—42no—
302C42#11714yes—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23C2×#9911yescharacteristic
308C23C2×#9911yescharacteristic
309C23C22×C22⋊C411yescharacteristic
310Q8—44no—
311Q8—48no—
312Q8—44no—
313Q8—44no—
314Q8—44no—
315Q8—48no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4C2×#9912yes—
363C2×C4C2×#9912yes—
364C2×C4#41012yes—
365C2×C4C2×#11712yes—
366C2×C4#52514yes—
367C8—88no—
368C22—24no—
369C22—24no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22#213312yes—
377C22#213512yes—
378C22#228011yescharacteristic
379C22C22×#9911yescharacteristic
380C22C2×#41011yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—22no—
385C4—22no—
386C4—44no—
387C4—41no—
388C4—41no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—82no—
394C4—82no—
395C2—21no—
396C2—21no—
397C2—22no—
398C2—22no—
399C2#2072211yescharacteristic
400C2C2×#213311yescharacteristic
401C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12804C2semidirect
6. #12631C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
#207272
#128602
#128042
#127002
#126312