Finite Group Databasebeta

← #641244

Subgroups — #641244

Order 512.

2550 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4C2×#2295C211yesmaximal, characteristic
5#16806C218yesmaximal
6#15234C214yesmaximal
7#15184C214yesmaximal
8#15180C214yesmaximal
9#14276C214yesmaximal
10#14161C211yesmaximal, characteristic
11#14155C211yesmaximal, characteristic
12C4×#384C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17#2862—28no—
18#2567—28no—
19#2546—28no—
20C2×#398C2214yes—
21C2×#395C2214yes—
22#2295—28no—
23C2×#391C2214yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2214yes—
32C2×#387C2214yes—
33C2×#384C2212yes—
34C2×#384C2212yes—
35C2×#382C2212yes—
36C2×#382C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2214yes—
42C2×#376C2212yes—
43C2×C43C2211yescharacteristic
44#1446C2214yes—
45#1445C2214yes—
46#1442C22116yes—
47#1441C22116yes—
48#1441C22116yes—
49#1441C22116yes—
50C4⋊#95C2214yes—
51C4⋊#95C2214yes—
52C4⋊#95C2214yes—
53C4⋊#95C2212yes—
54C4⋊#95C2214yes—
55C23×Q8C2312yes—
56C23×Q8C2311yescharacteristic
57C23×Q8C2312yes—
58C2×C4⋊Q8—28no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—24no—
65C2×C4⋊Q8—48no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8D418yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2318yes—
79C22×C4⋊C4C2318yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2318yes—
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C22×C42C2312yes—
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#398—216no—
92#395—216no—
93#391—28no—
94#391—28no—
95#391—216no—
96#391—216no—
97#391—28no—
98#391—48no—
99#389—216no—
100#389—216no—
101#389—28no—
102#387—216no—
103#384—28no—
104#384Q8116yescomplemented
105#382—28no—
106#382—28no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—48no—
112#376—28no—
113C2×#95C2312yes—
114C2×#95C2318yes—
115C2×#95C2312yes—
116C2×#95C2318yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C23116yes—
124C2×#95C2318yes—
125C2×#95C23116yes—
126C43D418yes—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—28no—
137C22×Q8—24no—
138C22×Q8—28no—
139C22×Q8—48no—
140C22×Q8C2×D414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2414yes—
144C23×C4C2412yes—
145C23×C4C2412yes—
146C23×C4C2412yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2411yescharacteristic
150C23×C4C2412yes—
151C23×C4C2411yescharacteristic
152C23×C4C2411yescharacteristic
153C4⋊Q8—432no—
154C4⋊Q8—416no—
155C4⋊Q8—416no—
156C4×Q8—232no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—216no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—216no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—416no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—416no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D418yes—
190C2×C4⋊C4C2×Q818yes—
191C2×C4⋊C4C2×Q818yes—
192C2×C4⋊C4C2×Q8132yes—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—28no—
202C2×C42—28no—
203C2×C42—28no—
204C2×C42—44no—
205C2×C42C2×D418yes—
206C2×C42C2×D414yes—
207C2×C42C2×D418yes—
208C2×C42C2×D418yes—
209C2×C42C2×D414yes—
210C2×C42C2×D414yes—
211C2×C42C2×D414yes—
212C2×C42C2×Q818yes—
213C2×C42#4114yes—
214#95—28no—
215#95—28no—
216#95—24no—
217#95—28no—
218#95—24no—
219#95—24no—
220#95—216no—
221#95—216no—
222#95—216no—
223#95—216no—
224#95—416no—
225#95—48no—
226#95—416no—
227C24C2511yescharacteristic
228C2×Q8—28no—
229C2×Q8—28no—
230C2×Q8—28no—
231C2×Q8—416no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—416no—
238C2×Q8—48no—
239C2×Q8—416no—
240C2×Q8—88no—
241C2×Q8—88no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—216no—
279C22×C4C22×D414yes—
280C22×C4C22×D414yes—
281C22×C4C22×D414yes—
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C22×D414yes—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C22×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×Q818yes—
290C22×C4C22×Q818yes—
291C22×C4C22×Q818yes—
292C22×C4C22×Q814yes—
293C22×C4C2×#4112yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)14yes—
297C22×C42−1+414yes—
298C22×C42−1+414yes—
299C4⋊C4—28no—
300C4⋊C4—216no—
301C4⋊C4—216no—
302C4⋊C4—216no—
303C4⋊C4—432no—
304C4⋊C4—416no—
305C4⋊C4—432no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—816no—
312C42—216no—
313C42—28no—
314C42—216no—
315C42—48no—
316C42—48no—
317C42—48no—
318C42—48no—
319C42#121116yes—
320C42C4⋊Q8116yescomplemented
321C23C23×D411yescharacteristic
322C23C23×D411yescharacteristic
323C23C23×Q812yes—
324C23C22×#4111yescharacteristic
325C23C2×H(5,𝔽2)12yes—
326C23C2×2−1+412yes—
327C23C2×2−1+412yes—
328C23#58514yes—
329Q8—416no—
330Q8—816no—
331Q8—816no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—216no—
350C2×C4—216no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4C2×#12114yes—
378C2×C4C2×#12114yes—
379C2×C4C2×#12114yes—
380C2×C4C2×C4⋊Q818yes—
381C2×C4C2×C4⋊Q814yes—
382C2×C4#53514yes—
383C2×C4#53614yes—
384C2×C4#53614yes—
385C2×C4#53718yes—
386C2×C4#54218yes—
387C2×C4D4×Q818yes—
388C2×C4#55718yes—
389C22C22×#12111yescharacteristic
390C22C22×C4⋊Q811yescharacteristic
391C22C2×#53511yescharacteristic
392C22C2×#53611yescharacteristic
393C22C2×#53611yescharacteristic
394C22C2×#53712yes—
395C22#344912yes—
396C22C2×#54212yes—
397C22C2×D4×Q812yes—
398C22C2×#55712yes—
399C22#350012yes—
400C22#350114yes—
401C22#350214yes—
402C22#351012yes—
403C22#354714yes—
404C22#355514yes—
405C4—28no—
406C4—24no—
407C4—44no—
408C4—44no—
409C4—48no—
410C4—48no—
411C4—44no—
412C4—84no—
413C4—84no—
414C4—84no—
415C4#254618yes—
416C4#262018yes—
417C2C2×#262011yescharacteristic
418C2C2×#254611yescharacteristic
419C2#3517312yes—
420C2#3532711yescharacteristic
421C2#3763312yes—
422C2#3961812yes—
423C2#3961912yes—
424C2#3964314yes—
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
104. #384Q8semidirect
320. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#25672
C2×#25462
C2×#22952
#168062
#152342
#151842
#151802
#142762
#141612
#141552
C4×#3842