Finite Group Databasebeta

← #4930673

Subgroups — #4930673

Order 512.

17624 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×2−1+4C211yesmaximal, characteristic, complemented
2C2×#2431C213yesmaximal, complemented
3C2×#2436C213yesmaximal, complemented
4C2×#2615C211yesmaximal, characteristic, complemented
5#14885C2112yesmaximal, complemented
6#14873C214yesmaximal, complemented
7#13937C213yesmaximal, complemented
8#13922C213yesmaximal, complemented
9#13851C211yesmaximal, characteristic, complemented
10C22×2−1+4—28no—
11C22×2−1+4—224no—
12C22×2−1+4—224no—
13C23×#41C2216yes—
14C23×#41C2211yescharacteristic
15C23×#41C2213yes—
16C24×Q8C2212yes—
17C24×Q8C2213yes—
18C22×#127C2211yescharacteristic
19C22×#124C2213yes—
20C22×#121C2213yes—
21#2615—28no—
22#2436—224no—
23#2431—224no—
24C2×#393C2212yes—
25C2×#393C22112yes—
26C2×#392C2216yes—
27C2×#392C22112yes—
28C2×#391C2213yes—
29C2×#391C2216yes—
30C2×#390C2213yes—
31C2×#386C22112yes—
32C2×#386C22112yes—
33C2×#380C2213yes—
34C2×C4×C22⋊C4C2213yes—
35C2×C4×C22⋊C4C2213yes—
36#1446C2212yes—
37#1446C22112yes—
38#1438C2216yes—
39#1438C22112yes—
40C22⋊#95C22112yes—
41C22⋊#95C22112yes—
42C2×2−1+4—2128no—
43C2×2−1+4—2384no—
44C2×2−1+4—2384no—
45C22×#41—224no—
46C22×#41—224no—
47C22×#41—224no—
48C22×#41—248no—
49C22×#41—224no—
50C22×#41—224no—
51C22×#41—248no—
52C22×#41—212no—
53C22×#41—212no—
54C22×#41—212no—
55C22×#41—212no—
56C22×#41D418yescomplemented
57C22×#41D4124yescomplemented
58C23×Q8—28no—
59C23×Q8—224no—
60C23×Q8—224no—
61C23×Q8—224no—
62C23×Q8—224no—
63C23×Q8—212no—
64C23×Q8—212no—
65C23×Q8D4124yescomplemented
66C23×Q8C2311yescharacteristic
67C23×Q8C2316yes—
68C23×Q8C2313yes—
69C23×D4C2311yescharacteristic
70C23×D4C2313yes—
71C23×D4C2316yes—
72C23×D4C2314yes—
73C23×D4C2311yescharacteristic
74C23×D4C2316yes—
75C24×C4C2316yes—
76C24×C4C2316yes—
77C24×C4C2313yes—
78C2×#127—212no—
79C2×#124—212no—
80C2×#124—224no—
81C2×#121—212no—
82C2×#121—212no—
83C2×#121—212no—
84C22×C4⋊C4C2313yes—
85C22×C22⋊C4C2316yes—
86C22×C22⋊C4C23112yes—
87C22×C22⋊C4C2316yes—
88C22×C22⋊C4C2316yes—
89C22×C22⋊C4C23112yes—
90C22×C22⋊C4C2316yes—
91C22×C42C2313yes—
92C22×C42C2311yescharacteristic
93#393—28no—
94#393—248no—
95#392—224no—
96#392—248no—
97#391—212no—
98#391—224no—
99#390—212no—
100#386—248no—
101#386—248no—
102#380—212no—
103C4×C22⋊C4—212no—
104C4×C22⋊C4—212no—
105C2×#95C2316yes—
106C2×#95C23112yes—
107C2×#95C2316yes—
108C2×#95C23124yes—
109C2×#95C2316yes—
110C25C2412yes—
111C25C2413yes—
112C25C2412yes—
1132−1+4—22048no—
114C2×#41—2192no—
115C2×#41—2384no—
116C2×#41—2384no—
117C2×#41—2192no—
118C2×#41—2192no—
119C2×#41—2192no—
120C2×#41—2192no—
121C2×#41—296no—
122C2×#41—232no—
123C2×#41—296no—
124C2×#41—296no—
125C2×#41—296no—
126C2×#41—296no—
127C22×Q8—2192no—
128C22×Q8—2192no—
129C22×Q8—296no—
130C22×Q8—296no—
131C22×Q8—2192no—
132C22×Q8—264no—
133C22×Q8—2192no—
134C22×Q8—296no—
135C22×Q8—224no—
136C22×Q8—26no—
137C22×Q8—224no—
138C22×Q8—212no—
139C22×Q8—26no—
140C22×Q8—212no—
141C22×Q8—26no—
142C22×Q8—212no—
143C22×Q8—26no—
144C22×Q8C2×D414yes—
145C22×Q8C2×D4124yes—
146C22×Q8C2×D4112yes—
147C22×Q8C2×D4124yes—
148C22×D4—224no—
149C22×D4—212no—
150C22×D4—212no—
151C22×D4—212no—
152C22×D4—212no—
153C22×D4—212no—
154C22×D4—224no—
155C22×D4—212no—
156C22×D4—212no—
157C22×D4—224no—
158C22×D4—26no—
159C22×D4—26no—
160C22×D4—26no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—26no—
164C22×D4—26no—
165C22×D4—28no—
166C22×D4—412no—
167C22×D4—424no—
168C22×D4C2×D4112yes—
169C22×D4C2×D4124yes—
170C22×D4C2×D4112yes—
171C23×C4—224no—
172C23×C4—212no—
173C23×C4—212no—
174C23×C4—224no—
175C23×C4—224no—
176C23×C4—212no—
177C23×C4—224no—
178C23×C4—212no—
179C23×C4—212no—
180C23×C4—212no—
181C23×C4C2×D4124yes—
182C23×C4C2×D4124yes—
183C23×C4C2×D4112yes—
184C23×C4C2×D4112yes—
185C23×C4C2×D4112yes—
186C23×C4C2416yes—
187C23×C4C2416yes—
188C23×C4C2412yes—
189C23×C4C2416yes—
190C23×C4C2413yes—
191C23×C4C2411yescharacteristic
192#127—416no—
193#124—448no—
194#121—448no—
195C2×C4⋊C4—26no—
196C2×C4⋊C4—26no—
197C2×C4⋊C4—26no—
198C2×C22⋊C4—212no—
199C2×C22⋊C4—212no—
200C2×C22⋊C4—212no—
201C2×C22⋊C4—212no—
202C2×C22⋊C4—212no—
203C2×C22⋊C4—224no—
204C2×C22⋊C4—248no—
205C2×C22⋊C4—212no—
206C2×C22⋊C4—212no—
207C2×C22⋊C4—224no—
208C2×C22⋊C4—46no—
209C2×C22⋊C4—412no—
210C2×C22⋊C4—424no—
211C2×C22⋊C4—412no—
212C2×C42—26no—
213C2×C42—212no—
214C2×C42—26no—
215#95—212no—
216#95—212no—
217#95—212no—
218#95—224no—
219#95—248no—
220C24—26no—
221C24—26no—
222C24—26no—
223C24—22no—
224C24—26no—
225C24—24no—
226C24—26no—
227C24—212no—
228C24—212no—
229C24—26no—
230C24—26no—
231C24—22no—
232C24—24no—
233C24—212no—
234C24C22×D416yes—
235C24C22×D4112yes—
236C24C22×D416yes—
237C24C22×D416yes—
238C24C2511yescharacteristic
239#41—21536no—
240#41—2768no—
241#41—2256no—
242C2×Q8—296no—
243C2×Q8—248no—
244C2×Q8—224no—
245C2×Q8—248no—
246C2×Q8—28no—
247C2×Q8—2512no—
248C2×Q8—2768no—
249C2×Q8—296no—
250C2×Q8—224no—
251C2×Q8—248no—
252C2×Q8—248no—
253C2×Q8—224no—
254C2×Q8—224no—
255C2×Q8—248no—
256C2×Q8#120148yes—
257C2×D4—296no—
258C2×D4—296no—
259C2×D4—248no—
260C2×D4—224no—
261C2×D4—224no—
262C2×D4—248no—
263C2×D4—248no—
264C2×D4—248no—
265C2×D4—248no—
266C2×D4—224no—
267C2×D4—224no—
268C2×D4—224no—
269C2×D4—448no—
270C2×D4—448no—
271C2×D4—424no—
272C2×D4—416no—
273C2×D4—412no—
274C2×D4—412no—
275C2×D4—44no—
276C2×D4—824no—
277C2×D4—824no—
278C2×D4—824no—
279C2×D4#120116yes—
280C22×C4—296no—
281C22×C4—296no—
282C22×C4—248no—
283C22×C4—296no—
284C22×C4—248no—
285C22×C4—248no—
286C22×C4—248no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—212no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—248no—
293C22×C4—296no—
294C22×C4—248no—
295C22×C4—224no—
296C22×C4—248no—
297C22×C4—224no—
298C22×C4—248no—
299C22×C4—26no—
300C22×C4—26no—
301C22×C4—26no—
302C22×C4—212no—
303C22×C4—26no—
304C22×C4—26no—
305C22×C4—26no—
306C22×C4—26no—
307C22×C4—26no—
308C22×C4—212no—
309C22×C4—224no—
310C22×C4#120148yes—
311C22×C4C22×D4124yes—
312C22×C4C22×D416yes—
313C22×C4C22×D4112yes—
314C22×C4C22×D416yes—
315C22×C4C22×D416yes—
316C22×C4C22×D416yes—
317C22×C4C22×D416yes—
318C22×C4C22×D412yes—
319C22×C42−1+414yescomplemented
320C4⋊C4—412no—
321C22⋊C4—424no—
322C22⋊C4—424no—
323C22⋊C4—424no—
324C22⋊C4—448no—
325C22⋊C4—824no—
326C22⋊C4—812no—
327C42—412no—
328C42—44no—
329C23—212no—
330C23—212no—
331C23—212no—
332C23—26no—
333C23—212no—
334C23—26no—
335C23—212no—
336C23—212no—
337C23—224no—
338C23—212no—
339C23—26no—
340C23—26no—
341C23—224no—
342C23—212no—
343C23—212no—
344C23—412no—
345C23—412no—
346C23—412no—
347C23—412no—
348C23—412no—
349C23—42no—
350C23—46no—
351C23—412no—
352C23—46no—
353C23—46no—
354C23—42no—
355C23—412no—
356C23—412no—
357C23—412no—
358C23—42no—
359C23—46no—
360C23—412no—
361C23C2×#12018yes—
362C23C2×#120112yes—
363C23C23×D413yes—
364C23C23×D413yes—
365C23C23×D411yescharacteristic
366C23C2×2−1+412yes—
367C23#58516yes—
368Q8—2192no—
369Q8—232no—
370Q8—296no—
371D4—2192no—
372D4—296no—
373D4—232no—
374D4—832no—
375D4—848no—
376D4—88no—
377C2×C4—2192no—
378C2×C4—2192no—
379C2×C4—224no—
380C2×C4—212no—
381C2×C4—212no—
382C2×C4—224no—
383C2×C4—212no—
384C2×C4—26no—
385C2×C4—212no—
386C2×C4—26no—
387C2×C4—296no—
388C2×C4—212no—
389C2×C4—42no—
390C2×C4—46no—
391C2×C4—46no—
392C2×C4—46no—
393C2×C4—412no—
394C2×C4—412no—
395C2×C4—46no—
396C2×C4—412no—
397C2×C4—46no—
398C2×C4—412no—
399C2×C4—412no—
400C2×C4C2×#120124yes—
401C2×C4C2×#120112yes—
402C2×C4C2×#120112yes—
403C2×C4C2×#12014yes—
404C2×C4C2×#120112yes—
405C22—28no—
406C22—212no—
407C22—212no—
408C22—212no—
409C22—216no—
410C22—412no—
411C22—412no—
412C22—412no—
413C22—412no—
414C22—424no—
415C22—412no—
416C22—424no—
417C22—412no—
418C22—82no—
419C22—86no—
420C22—86no—
421C22—812no—
422C22—84no—
423C22C22×#12013yes—
424C22C22×#12013yes—
425C22C22×#12011yescharacteristic
426C22#348716yes—
427C22#348916yes—
428C22#349716yes—
429C22#349911yescharacteristic
430C22#350013yes—
431C22#350016yes—
432C4—224no—
433C4—212no—
434C4—44no—
435C4—86no—
436C4—86no—
437C4#284618yes—
438C2—412no—
439C2—48no—
440C2—162no—
441C2#3566013yes—
442C2#3567613yes—
443C2#3575716yes—
444C2#3607812yes—
445C2C2×#284611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×2−1+4C2semidirect
2. C2×#2431C2semidirect
3. C2×#2436C2semidirect
4. C2×#2615C2semidirect
5. #14885C2semidirect
6. #14873C2semidirect
7. #13937C2semidirect
8. #13922C2semidirect
9. #13851C2semidirect
56. C22×#41D4semidirect
57. C22×#41D4semidirect
65. C23×Q8D4semidirect
319. C22×C42−1+4semidirect

Maximal subgroups

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

subgroupindex
C23×2−1+42
C2×#24312
C2×#24362
C2×#26152
#148852
#148732
#139372
#139222
#138512