Finite Group Databasebeta

← #4307536

Subgroups — #4307536

Order 512.

3826 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2351C212yesmaximal
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#15230C213yesmaximal
5#15228C216yesmaximal
6#15218C216yesmaximal
7#15202C213yesmaximal
8#15128C213yesmaximal
9C4×#395C213yesmaximal
10Q8×#95C211yesmaximal, characteristic
11Q8×#95C212yesmaximal
12C2×#557—28no—
13C2×#557—216no—
14C22×C4⋊Q8C2213yes—
15C22×#125C2216yes—
16C22×C4×Q8C2213yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C2×#395C2213yes—
23#2351—216no—
24Q8×C4⋊C4—28no—
25C2×#389C2216yes—
26C2×#389C2213yes—
27C2×#389C2213yes—
28C2×#389C2216yes—
29C2×#384C2213yes—
30C2×#382C2216yes—
31C2×C4×C4⋊C4C2213yes—
32C2×C4×C4⋊C4C2216yes—
33#1446C2216yes—
34#1446C2213yes—
35#1444C2216yes—
36#1444C2213yes—
37#1444C2216yes—
38#1444C22112yes—
39#1441C2216yes—
40#1441C2211yescharacteristic
41#1441C2216yes—
42#1441C22112yes—
43C4⋊#95C2213yes—
44C4⋊#95C2216yes—
45C4⋊#95C2216yes—
46C4⋊#95C2216yes—
47C4⋊#95C2216yes—
48C4×#95C2216yes—
49C4×#95C2213yes—
50C23×Q8C2313yes—
51C23×Q8C2311yescharacteristic
52#557—2128no—
53C2×C4⋊Q8—224no—
54C2×C4⋊Q8—212no—
55C2×#125—224no—
56C2×#125—224no—
57C2×#125—224no—
58C2×C4×Q8—24no—
59C2×C4×Q8—28no—
60C2×C4×Q8—28no—
61C2×C4×Q8—28no—
62C2×C4×Q8—28no—
63C2×C4×Q8—24no—
64C2×C4×Q8—28no—
65C2×C4×Q8—24no—
66C2×C4×Q8—28no—
67C2×C4×Q8—212no—
68C2×C4×Q8—224no—
69C2×C4×Q8D418yes—
70C2×C4×Q8D4116yes—
71C2×C4×Q8Q818yescomplemented
72C2×C4×Q8Q8116yescomplemented
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2316yes—
75C22×C4⋊C4C2313yes—
76C22×C4⋊C4C2316yes—
77C22×C4⋊C4C2316yes—
78C22×C4⋊C4C2313yes—
79C22×C4⋊C4C2313yes—
80C22×C4⋊C4C2316yes—
81C22×C4⋊C4C2313yes—
82C22×C4⋊C4C2316yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2316yes—
85C22×C4⋊C4C2316yes—
86C22×C42C2313yes—
87C22×C42C2316yes—
88C22×C42C2313yes—
89C22×C42C2316yes—
90C22×C42C2313yes—
91#395—212no—
92#389—212no—
93#389—224no—
94#389—212no—
95#389—224no—
96#384—212no—
97#382—224no—
98C4×C4⋊C4—212no—
99C4×C4⋊C4—224no—
100C2×#95C2316yes—
101C2×#95C2313yes—
102C2×#95C2312yes—
103C2×#95C2311yescharacteristic
104C2×#95C2312yes—
105C2×#95C2313yes—
106C2×#95C2316yes—
107C2×#95C23112yes—
108C2×#95C2313yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C23112yes—
114C22×Q8—212no—
115C22×Q8—26no—
116C22×Q8—26no—
117C22×Q8—26no—
118C22×Q8—46no—
119C22×Q8C2×D414yes—
120C22×Q8C2×D418yes—
121C22×Q8C2×Q814yes—
122C22×Q8C2×Q818yes—
123C22×Q8#4114yes—
124C23×C4C2413yes—
125C23×C4C2413yes—
126C23×C4C2412yes—
127C23×C4C2416yes—
128C23×C4C2416yes—
129C23×C4C2413yes—
130C23×C4C2412yes—
131C23×C4C2411yescharacteristic
132C23×C4C2411yescharacteristic
133C23×C4C2411yescharacteristic
134C23×C4C2413yes—
135C4⋊Q8—296no—
136#125—2192no—
137C4×Q8—296no—
138C4×Q8—264no—
139C4×Q8—232no—
140C4×Q8—232no—
141C4×Q8—264no—
142C4×Q8—416no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—212no—
148C2×C4⋊C4—212no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—212no—
151C2×C4⋊C4—26no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—224no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—212no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—26no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4—212no—
165C2×C4⋊C4—26no—
166C2×C4⋊C4—212no—
167C2×C4⋊C4—43no—
168C2×C4⋊C4—46no—
169C2×C4⋊C4—46no—
170C2×C4⋊C4—46no—
171C2×C4⋊C4—412no—
172C2×C4⋊C4C2×D4112yes—
173C2×C4⋊C4C2×D4124yes—
174C2×C4⋊C4C2×Q8124yes—
175C2×C4⋊C4C2×Q8112yes—
176C2×C4⋊C4#4114yes—
177C2×C42—212no—
178C2×C42—212no—
179C2×C42—212no—
180C2×C42—212no—
181C2×C42—26no—
182C2×C42—26no—
183C2×C42—212no—
184C2×C42—26no—
185C2×C42—212no—
186C2×C42C2×D4124yes—
187C2×C42C2×D4112yes—
188C2×C42C2×Q8124yes—
189C2×C42C2×Q8112yes—
190#95—24no—
191#95—212no—
192#95—26no—
193#95—26no—
194#95—224no—
195#95—26no—
196#95—212no—
197#95—22no—
198#95—24no—
199#95—46no—
200#95—412no—
201#95—46no—
202#95—46no—
203#95—46no—
204C24C2511yescharacteristic
205C2×Q8—224no—
206C2×Q8—424no—
207C2×Q8—424no—
208C2×Q8—412no—
209C2×Q8—412no—
210C2×Q8#120116yes—
211C2×Q8C22⋊Q8132yes—
212C2×Q8C22⋊Q8116yes—
213C2×Q8C4⋊Q8132yes—
214C2×Q8C4⋊Q8116yes—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—26no—
224C22×C4—23no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—22no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—26no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—26no—
262C22×C4C22×D4112yes—
263C22×C4C22×D414yes—
264C22×C4C22×D416yes—
265C22×C4C22×D412yes—
266C22×C4C22×D416yes—
267C22×C4C22×D4112yes—
268C22×C4C22×Q814yes—
269C22×C4C22×Q8112yes—
270C22×C4C22×Q816yes—
271C22×C4C22×Q812yes—
272C22×C4C22×Q816yes—
273C22×C4C22×Q8112yes—
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4116yes—
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C4⋊C4—28no—
281C4⋊C4—248no—
282C4⋊C4—248no—
283C4⋊C4—224no—
284C4⋊C4—248no—
285C4⋊C4—248no—
286C4⋊C4—224no—
287C4⋊C4—28no—
288C4⋊C4—412no—
289C4⋊C4—424no—
290C4⋊C4—424no—
291C4⋊C4—412no—
292C4⋊C4—812no—
293C42—248no—
294C42—224no—
295C42—224no—
296C42—248no—
297C42—412no—
298C23C23×D411yescharacteristic
299C23C23×D412yes—
300C23C23×Q811yescharacteristic
301C23C23×Q812yes—
302C23C22×#4111yescharacteristic
303C23C22×#4111yescharacteristic
304C23C2×2−1+411yescharacteristic
305C23C2×2−1+411yescharacteristic
306C23#58512yes—
307C23#58512yes—
308C23#58511yescharacteristic
309Q8—448no—
310Q8#395164yescomplemented
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—26no—
323C2×C4—212no—
324C2×C4—24no—
325C2×C4—212no—
326C2×C4—26no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—212no—
330C2×C4—212no—
331C2×C4—24no—
332C2×C4—26no—
333C2×C4—212no—
334C2×C4—24no—
335C2×C4—26no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—212no—
339C2×C4—212no—
340C2×C4—41no—
341C2×C4—46no—
342C2×C4—46no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—46no—
349C2×C4—43no—
350C2×C4—46no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—46no—
355C2×C4—42no—
356C2×C4—46no—
357C2×C4—42no—
358C2×C4—43no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—42no—
364C2×C4—43no—
365C2×C4—46no—
366C2×C4—43no—
367C2×C4—46no—
368C2×C4—43no—
369C2×C4C2×#120112yes—
370C2×C4C2×C22⋊Q8124yes—
371C2×C4C2×C22⋊Q8112yes—
372C2×C4C2×C4⋊Q8124yes—
373C2×C4C2×C4⋊Q8112yes—
374C2×C4#54814yes—
375C2×C4#55714yescomplemented
376C22C22×#12011yescharacteristic
377C22C22×C22⋊Q812yes—
378C22C22×C22⋊Q811yescharacteristic
379C22C22×C4⋊Q812yes—
380C22C22×C4⋊Q811yescharacteristic
381C22C2×#54811yescharacteristic
382C22D4×#4112yes—
383C22C2×#55711yescharacteristic
384C22Q8×#4112yes—
385C22#348011yescharacteristic
386C22#348711yescharacteristic
387C22#348712yes—
388C22#348912yes—
389C22#348911yescharacteristic
390C22#349712yes—
391C22#350011yescharacteristic
392C22#350212yes—
393C22#350211yescharacteristic
394C22#350212yes—
395C22#350412yes—
396C22#350412yes—
397C22#352311yescharacteristic
398C22#352412yes—
399C4—24no—
400C4—24no—
401C4—42no—
402C4—412no—
403C4—412no—
404C4—44no—
405C4—46no—
406C4—46no—
407C4—44no—
408C4—83no—
409C4C2×#395124yes—
410C2C22×#39511yescharacteristic
411C2#3575711yescharacteristic
412C2#3578612yes—
413C2#3595712yes—
414C2#3595711yescharacteristic
415C2#3604011yescharacteristic
416C2#3642811yescharacteristic
417C2#3643112yes—
418C2#3646612yes—
419C2#3647411yescharacteristic
420C2#3657711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
71. C2×C4×Q8Q8semidirect
72. C2×C4×Q8Q8semidirect
310. Q8#395semidirect
375. C2×C4#557semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#23512
C2×Q8×C4⋊C42
#152302
#152282
#152182
#152022
#151282
C4×#3952
Q8×#952