Finite Group Databasebeta

← #497660

Subgroups — #497660

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39168C211yesmaximal, characteristic
2Q8×#101C212yesmaximal, complemented
3#13228C214yesmaximal, complemented
4#13226C212yesmaximal
5#12803C212yesmaximal, complemented
6#12802C211yesmaximal, characteristic, complemented
7#12632C212yesmaximal, complemented
8C4×#356C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic
10C2×Q82C2211yescharacteristic
11Q8×M16—22no—
12#2858C411yescharacteristic
13#2674—24no—
14#2640C412yes—
15#2567C411yescharacteristic
16#2548C2211yescharacteristic
17#2545C2212yes—
18#2542C412yes—
19#2452—22no—
20#2448—22no—
21#2326C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#356C2212yes—
25C2×#356C2212yes—
26C2×#355C2214yes—
27C2×#355C2214yes—
28#1997—24no—
29#1997—22no—
30#1801C2212yes—
31#1801C2214yes—
32C4×#101C2212yes—
33C4×#101C2214yes—
34#1394C2212yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37#559—42no—
38#559—42no—
39Q82—42no—
40Q82—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8D414yes—
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#124—21no—
48C2×#124—22no—
49C2×#124C2×C412yes—
50C2×#124C2312yes—
51C2×#123—22no—
52C2×C22⋊Q8—24no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8C2311yescharacteristic
57C2×C4×Q8C2312yes—
58C2×C4×D4—21no—
59C2×C4×D4C2×C412yes—
60#446—42no—
61#446—44no—
62C8×Q8—42no—
63#423—22no—
64#423—24no—
65C2×#101C2314yes—
66C2×#101C2312yes—
67C2×#101C2312yes—
68C4×M16—22no—
69C4×M16—24no—
70#399—24no—
71#397—24no—
72#396—24no—
73#395—22no—
74#394—22no—
75#393—24no—
76#391—21no—
77#391C2×C411yescharacteristic
78#391C2×C412yes—
79#391C2311yescharacteristic
80#390—22no—
81#390—22no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#389—22no—
86#389—21no—
87#389—22no—
88#388—24no—
89#388—24no—
90#387—22no—
91#385—24no—
92#384C2×C411yescharacteristic
93#384C2×C412yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4C2×C412yes—
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#356—24no—
100#356—42no—
101#355—28no—
102#355—44no—
103#343—24no—
104#343—22no—
105#343—24no—
106#343—22no—
107C22×Q8C22⋊C414yes—
108C22×Q8C22⋊C412yes—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22×C411yescharacteristic
111C22×Q8C2411yescharacteristic
112C22×D4—21no—
113C22×D4—21no—
114C22×D4C22×C412yes—
115C23×C4—22no—
116C2×M16—22no—
117C2×M16—24no—
118C2×M16—22no—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—44no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—42no—
126C4⋊Q8—44no—
127C4⋊Q8—42no—
128C4⋊Q8—44no—
129#127—41no—
130#127—41no—
131#124—24no—
132#124—22no—
133#124—22no—
134#124—44no—
135#124—42no—
136#124—42no—
137#124—84no—
138#123—48no—
139C22⋊Q8—88no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—24no—
145C4×Q8—41no—
146C4×Q8—44no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×D4—44no—
151C4×D4—42no—
152C4×D4—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C22×C412yes—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×D414yes—
185C4⋊C8—44no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8—44no—
189#101—22no—
190#101—24no—
191#101—22no—
192#101—22no—
193#101—24no—
194#101#4114yes—
195#97—42no—
196#97—44no—
197C4×C8—44no—
198C4×C8—42no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—22no—
204#95—24no—
205#95—22no—
206C24—22no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—84no—
217C2×Q8—82no—
218C2×Q8#9912yes—
219C2×Q8#9912yes—
220C2×Q8C2×#4112yes—
221C2×Q8H(5,𝔽2)11yescharacteristic
222C2×Q82−1+411yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—82no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—44no—
238C22×C4—84no—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C23×C411yescharacteristic
246C22×C4C22×D412yes—
247M16—24no—
248M16—42no—
249M16—44no—
250M16—42no—
251M16—44no—
252M16—42no—
253C2×C8—42no—
254C2×C8—44no—
255C2×C8—42no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—24no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—84no—
275C22⋊C4—44no—
276C22⋊C4—48no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—84no—
282C22⋊C4—84no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—24no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—21no—
292C42—22no—
293C42—21no—
294C42—41no—
295C42—42no—
296C42—42no—
297C42—41no—
298C42—41no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23C22×C22⋊C411yescharacteristic
305Q8—24no—
306Q8—42no—
307Q8—42no—
308Q8—41no—
309Q8—41no—
310Q8—81no—
311Q8—81no—
312Q8—164no—
313Q8—162no—
314D4—82no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—162no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—164no—
348C2×C4C2×#9911yescharacteristic
349C2×C4C2×#9914yes—
350C2×C4C2×#9911yescharacteristic
351C2×C4C4×#4111yescharacteristic
352C2×C4#52011yescharacteristic
353C2×C4#54812yes—
354C2×C4#55012yes—
355C8—44no—
356C8—84no—
357C8—82no—
358C22—21no—
359C22—21no—
360C22—84no—
361C22—84no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22#234111yescharacteristic
366C22C22×#9911yescharacteristic
367C22#288411yescharacteristic
368C4—22no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—24no—
374C4—42no—
375C4—42no—
376C4—41no—
377C4—44no—
378C4—42no—
379C4—82no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4#213312yes—
384C2—21no—
385C2—21no—
386C2—162no—
387C2#2253411yescharacteristic
388C2C2×#213311yescharacteristic
389C2#3337311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×Q82 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#101C2semidirect
3. #13228C2semidirect
5. #12803C2semidirect
6. #12802C2semidirect
7. #12632C2semidirect
8. C4×#356C2semidirect

Maximal subgroups

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

subgroupindex
#391682
Q8×#1012
#132282
#132262
#128032
#128022
#126322
C4×#3562