Finite Group Databasebeta

← #515601

Subgroups — #515601

Order 512.

4258 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4#14159C214yesmaximal
5#14139C214yesmaximal
6#14054C218yesmaximal, complemented
7#14053C218yesmaximal, complemented
8#13795C212yesmaximal
9#13756C212yesmaximal
10C2×#544—28no—
11C2×#544—28no—
12C2×#544C4116yescomplemented
13C22×C4⋊Q8C2214yes—
14C22×#125C2214yes—
15C22×C22⋊Q8C2214yes—
16C22×#117C2212yes—
17C22×#117C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#391C2218yes—
23C2×#389—28no—
24C2×#389C2218yes—
25#2274—28no—
26C2×C4×C4⋊C4C2218yes—
27C2×C4×C22⋊C4C2214yes—
28C22×#95C2212yes—
29#1446C2218yes—
30#1445C2218yes—
31#1441C22116yes—
32#1436C22116yes—
33C4⋊#95C2218yes—
34C4⋊#95C2214yes—
35C4⋊#95C2214yes—
36C4⋊#95C2214yes—
37C4⋊#95C2218yes—
38C4⋊#95C2214yes—
39C22⋊#95C2214yes—
40C22⋊#95C2218yes—
41C22⋊#95C2214yes—
42C4×#95C2218yes—
43C23×Q8C2314yes—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2312yes—
47C24×C4C2312yes—
48C24×C4C2311yescharacteristic
49#544—2128no—
50C2×C4⋊Q8—216no—
51C2×C4⋊Q8—216no—
52C2×C4⋊Q8C2×C4132yescomplemented
53C2×#125—216no—
54C2×#125—216no—
55C2×#125C2×C4132yescomplemented
56C2×C22⋊Q8—216no—
57C2×C22⋊Q8—216no—
58C2×C22⋊Q8C2×C4132yes—
59C2×#117—28no—
60C2×#117—28no—
61C2×#117—28no—
62C2×#117—28no—
63C2×#117C2×C4116yes—
64C2×#117Q8116yescomplemented
65C22×C4⋊C4—28no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—28no—
68C22×C4⋊C4—48no—
69C22×C4⋊C4C2×C418yes—
70C22×C4⋊C4D418yes—
71C22×C4⋊C4D418yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2318yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2318yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C42—24no—
83C22×C42—28no—
84C22×C42C2314yes—
85C22×C42C2314yes—
86C22×C42C2314yes—
87C22×C42C2318yes—
88#391—416no—
89#389—232no—
90C4×C4⋊C4—232no—
91C4×C22⋊C4—216no—
92C2×#95—28no—
93C2×#95C2318yes—
94C2×#95C2318yes—
95C2×#95C2318yes—
96C2×#95C2318yes—
97C2×#95C2318yes—
98C2×#95C2318yes—
99C2×#95C23116yes—
100C2×#95C23116yes—
101C25C2411yescharacteristic
102C22×Q8—28no—
103C22×Q8—28no—
104C22×Q8—48no—
105C22×Q8—48no—
106C22×Q8C22×C4116yes—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—28no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4C22×C418yes—
122C23×C4C22×C414yes—
123C23×C4C2×D418yes—
124C23×C4C2×D418yes—
125C23×C4C2×D414yes—
126C23×C4C2×D414yes—
127C23×C4C2×Q818yes—
128C23×C4#4118yes—
129C23×C4C2414yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2418yes—
133C23×C4C2412yes—
134C23×C4C2412yes—
135C23×C4C2414yes—
136C23×C4C2412yes—
137C4⋊Q8—2128no—
138#125—2128no—
139C22⋊Q8—2128no—
140#117—264no—
141#117—264no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—432no—
159C2×C4⋊C4—416no—
160C2×C4⋊C4C22⋊C4164yes—
161C2×C4⋊C4C22×C418yes—
162C2×C4⋊C4C22×C418yes—
163C2×C4⋊C4C22×C4116yes—
164C2×C4⋊C4C22×C4132yes—
165C2×C4⋊C4C22×C4116yes—
166C2×C4⋊C4C2×D4116yes—
167C2×C4⋊C4C2×D4116yes—
168C2×C4⋊C4C2×Q8116yes—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4C22×C4116yes—
174C2×C22⋊C4C2×Q8116yes—
175C2×C42—28no—
176C2×C42—28no—
177C2×C42—28no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—216no—
183C2×C42—416no—
184C2×C42—48no—
185C2×C42—48no—
186C2×C42C22×C4116yes—
187C2×C42C2×Q8116yes—
188#95—216no—
189#95—216no—
190#95—216no—
191#95—216no—
192#95—48no—
193#95—416no—
194#95—48no—
195#95—416no—
196C24—22no—
197C24—22no—
198C24—22no—
199C24—22no—
200C24C23×C412yes—
201C24C22×D412yes—
202C24C22×D412yes—
203C24C22×Q814yes—
204C24C2×#4114yes—
205C24C2511yescharacteristic
206C2×Q8—232no—
207C2×Q8—432no—
208C2×Q8—88no—
209C2×Q8—88no—
210C2×Q8—816no—
211C22×C4—28no—
212C22×C4—28no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—28no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—22no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—28no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—48no—
242C22×C4—416no—
243C22×C4—48no—
244C22×C4—416no—
245C22×C4—416no—
246C22×C4—416no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—816no—
255C22×C4C2×C22⋊C4132yes—
256C22×C4C2×C22⋊C4116yes—
257C22×C4C4×Q8132yes—
258C22×C4C22⋊Q8132yes—
259C22×C4#124116yes—
260C22×C4C4⋊Q8116yes—
261C22×C4C23×C418yes—
262C22×C4C23×C418yes—
263C22×C4C23×C418yes—
264C22×C4C23×C414yes—
265C22×C4C22×D418yes—
266C22×C4C22×D418yes—
267C22×C4C22×D414yes—
268C22×C4C22×D414yes—
269C22×C4C22×Q8116yes—
270C22×C4C22×Q814yes—
271C22×C4C22×Q814yes—
272C22×C4C2×#4114yes—
273C22×C4C2×#4114yes—
274C22×C4H(5,𝔽2)14yes—
275C22×C42−1+414yes—
276C4⋊C4—216no—
277C4⋊C4—216no—
278C4⋊C4—232no—
279C4⋊C4—264no—
280C4⋊C4—232no—
281C4⋊C4—432no—
282C4⋊C4—416no—
283C4⋊C4C2×C22⋊C4164yes—
284C22⋊C4—232no—
285C22⋊C4—232no—
286C42—232no—
287C42—232no—
288C42—416no—
289C42—816no—
290C23—24no—
291C23—24no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—28no—
297C23—28no—
298C23—24no—
299C23—24no—
300C23—28no—
301C23—28no—
302C23C22×C22⋊C414yes—
303C23C2×C4×Q818yes—
304C23C2×C22⋊Q818yes—
305C23C2×#12414yes—
306C23C2×C4⋊Q814yes—
307C23C24×C411yescharacteristic
308C23C23×D411yescharacteristic
309C23C23×D411yescharacteristic
310C23C23×Q812yes—
311C23C22×#4112yes—
312C23C2×H(5,𝔽2)12yes—
313C23C2×2−1+412yes—
314C23#58512yes—
315C23#58512yes—
316Q8—832no—
317C2×C4—216no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—216no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—216no—
324C2×C4—216no—
325C2×C4—216no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—216no—
331C2×C4—216no—
332C2×C4—432no—
333C2×C4—416no—
334C2×C4—416no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—48no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—416no—
355C2×C4—816no—
356C2×C4#391164yescomplemented
357C2×C4C22×C22⋊C4116yes—
358C2×C4C22×C22⋊C418yes—
359C2×C4C2×C4×Q818yes—
360C2×C4C2×C4×Q818yes—
361C2×C4C2×C22⋊Q818yes—
362C2×C4C2×C22⋊Q818yes—
363C2×C4C2×#12418yes—
364C2×C4C2×C4⋊Q818yes—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—216no—
371C22—216no—
372C22—24no—
373C22C2×#39118yes—
374C22C23×C22⋊C411yescharacteristic
375C22C22×C4×Q812yes—
376C22C4×H(5,𝔽2)12yes—
377C22C4×2−1+412yes—
378C22C22×C22⋊Q812yes—
379C22C22×#12411yescharacteristic
380C22C22×C4⋊Q811yescharacteristic
381C22#348411yescharacteristic
382C22#348511yescharacteristic
383C22#348611yescharacteristic
384C22#348711yescharacteristic
385C22#349414yes—
386C22#349514yes—
387C22#349612yes—
388C22#349712yes—
389C22#350314yes—
390C22#350414yes—
391C4—216no—
392C4—44no—
393C4—44no—
394C4—48no—
395C4—48no—
396C4—416no—
397C4—84no—
398C4C2×#391116yes—
399C2—28no—
400C2C22×#39111yescharacteristic
401C2#3425611yescharacteristic
402C2#3425711yescharacteristic
403C2#3427214yes—
404C2#3557514yes—
405C2#3557611yescharacteristic
406C2#3558411yescharacteristic
407C2#3564011yescharacteristic
408C2#3564411yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #14054C2semidirect
7. #14053C2semidirect
12. C2×#544C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
55. C2×#125C2×C4semidirect
64. C2×#117Q8semidirect
356. C2×C4#391semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C22×#3892
C2×#22742
#141592
#141392
#140542
#140532
#137952
#137562