Finite Group Databasebeta

← #493987

Subgroups — #493987

Order 512.

1053 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#20728C212yesmaximal
3#12863C212yesmaximal
4#12804C214yesmaximal, complemented
5#12704C212yesmaximal
6C4×#356C214yesmaximal, complemented
7C2×#536C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic
10#2920—22no—
11#2327C2212yes—
12#2326C412yes—
13#2323—22no—
14#2322—22no—
15#2317C411yescharacteristic
16#2311C2211yescharacteristic
17#2310C411yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#124C412yes—
20C4×C22⋊Q8—22no—
21C4×#123—22no—
22C4×#117C411yescharacteristic
23C2×#356C2218yes—
24#1822—22no—
25#1801C2214yes—
26#1800—22no—
27#1785C2214yes—
28C4×#101C2214yes—
29C4×#100—22no—
30#1394C2218yes—
31C22×#41C2311yescharacteristic
32C23×Q8C2×C411yescharacteristic
33#536—28no—
34C2×C4⋊Q8C2312yes—
35C2×#124C2×C412yes—
36C2×#123—22no—
37C2×C22⋊Q8—22no—
38C4×#41—42no—
39C4×#41—42no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—42no—
43C2×C4×Q8—42no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2311yescharacteristic
47C2×C4×Q8C2311yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117D412yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C42—21no—
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61#432—42no—
62#432—42no—
63#431—42no—
64#431—42no—
65#423—24no—
66C2×#101C2318yes—
67C2×#100—24no—
68#407—24no—
69C4×M16—24no—
70#391—22no—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2312yes—
74#390C2×C412yes—
75#389—24no—
76#389—22no—
77#388—24no—
78#388—24no—
79#387—24no—
80#387—24no—
81#386—22no—
82#385—24no—
83#385—22no—
84#384C2×C412yes—
85#382C2×C412yes—
86#381—22no—
87#380—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—24no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4—24no—
94#376C2×C412yes—
95C43C2×C412yes—
96#356—216no—
97#343—24no—
98#343—24no—
99C2×#41—24no—
100C2×#41—42no—
101C2×#41—42no—
102C22×Q8—24no—
103C22×Q8—21no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8C22×C412yes—
107C22×Q8C2411yescharacteristic
108C22×D4C22×C411yescharacteristic
109C22×D4C2×D412yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C22⋊C412yes—
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D412yes—
118C2×M16—28no—
119C4⋊Q8—24no—
120C4⋊Q8#4118yes—
121#124—24no—
122#124—24no—
123#123—216no—
124C22⋊Q8—216no—
125C4×Q8—41no—
126C4×Q8—42no—
127C4×Q8—41no—
128C4×Q8—44no—
129C4×Q8—44no—
130C4×Q8—42no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—42no—
134C4×D4—42no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141#117—24no—
142#117—42no—
143#117—42no—
144#117—42no—
145#117—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C22⋊C412yes—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C22×C412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C4⋊C8—44no—
190#101—216no—
191#100—48no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194#97—44no—
195C4×C8—44no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202C24C2×C22⋊C412yes—
203C24C2×C22⋊C411yescharacteristic
204#41—416no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—416no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8#11714yes—
215C2×Q8C2×#4114yes—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—41no—
220C2×D4—42no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—81no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D412yes—
258M16—48no—
259M16—48no—
260C2×C8—48no—
261C4⋊C4—24no—
262C4⋊C4—216no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—84no—
273C22⋊C4—24no—
274C22⋊C4—216no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—84no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—21no—
289C42—22no—
290C42—21no—
291C42—42no—
292C42—44no—
293C42—44no—
294C42—42no—
295C42—42no—
296C42—82no—
297C42#9912yes—
298C42#9912yes—
299C42#11714yes—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23#41012yes—
310C23C22×C22⋊C411yescharacteristic
311Q8—44no—
312Q8—44no—
313Q8—48no—
314Q8—44no—
315D4—48no—
316D4—44no—
317C2×C4—28no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4C2×#9911yescharacteristic
357C2×C4C2×#9912yes—
358C2×C4C2×#9912yes—
359C2×C4C2×#9911yescharacteristic
360C2×C4C2×#11712yes—
361C2×C4#52514yes—
362C8—88no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—82no—
375C22#228011yescharacteristic
376C22C22×#9911yescharacteristic
377C22C2×#41011yescharacteristic
378C4—21no—
379C4—22no—
380C4—24no—
381C4—22no—
382C4—22no—
383C4—44no—
384C4—41no—
385C4—42no—
386C4—42no—
387C4—41no—
388C4—44no—
389C4—82no—
390C4—82no—
391C4#213512yes—
392C2—21no—
393C2—21no—
394C2—42no—
395C2—82no—
396C2#2072211yescharacteristic
397C2C2×#213511yescharacteristic
398C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12804C2semidirect
6. C4×#356C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#207282
#128632
#128042
#127042
C4×#3562