Finite Group Databasebeta

← #4435090

Subgroups — #4435090

Order 512.

2660 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2611C212yesmaximal
3C2×#2621C211yesmaximal, characteristic
4C2×#2660C212yesmaximal, complemented
5#17062C214yesmaximal
6#16964C214yesmaximal, complemented
7#15856C214yesmaximal, complemented
8#15430C214yesmaximal
9#15145C214yesmaximal, complemented
10#15020C214yesmaximal
11#13805C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2212yes—
14C22×C22⋊Q8C2212yes—
15C22×#117C2211yescharacteristic
16#2822—28no—
17#2660—216no—
18#2621—28no—
19#2611—216no—
20C2×#400C2212yes—
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#399C2214yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#397C2212yes—
31C2×#397C2214yes—
32C2×#395C2218yes—
33C2×#391C2212yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#385C2214yes—
38C2×#384C2214yes—
39C2×#384C2214yes—
40C2×#382C2214yes—
41C2×#382C2211yescharacteristic
42C2×#376C2212yes—
43#1445C2214yes—
44#1445C2218yes—
45#1445C2218yes—
46#1444C2218yes—
47#1444C2218yes—
48#1439C2218yes—
49#1437C2214yes—
50#1436C2214yes—
51C4⋊#95C2218yes—
52#1434C2214yes—
53C4×#95C2214yes—
54C23×Q8C2312yes—
55C24×C4C2311yescharacteristic
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—28no—
58C2×#125—28no—
59C2×#125—28no—
60C2×#125—28no—
61C2×C22⋊Q8—28no—
62C2×C22⋊Q8—44no—
63C2×C22⋊Q8—44no—
64C2×#117—28no—
65C2×#117—24no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2318yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C42C2311yescharacteristic
84C22×C42C2314yes—
85C22×C42C2314yes—
86C22×C42C2311yescharacteristic
87#400—216no—
88#400—28no—
89#399—28no—
90#399—48no—
91#398—216no—
92#398—216no—
93#398—28no—
94#398—216no—
95#398—216no—
96#398—216no—
97#397—28no—
98#397—216no—
99#395—232no—
100#391—28no—
101#389—216no—
102#388—216no—
103#387—216no—
104#385—216no—
105#384—216no—
106#384—48no—
107#382—216no—
108#382D418yes—
109#376D4116yes—
110C2×#95C2318yes—
111C2×#95C2318yes—
112C2×#95C2318yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C25C2411yescharacteristic
126C22×Q8—24no—
127C22×Q8—28no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—44no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—42no—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2414yes—
139C23×C4C2412yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2411yescharacteristic
143C23×C4C2412yes—
144C23×C4C2411yescharacteristic
145C23×C4C2414yes—
146C4⋊Q8—416no—
147#125—432no—
148C22⋊Q8—432no—
149#117—416no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—216no—
175C2×C4⋊C4—216no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C2×D418yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—216no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C42—216no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—216no—
203C2×C42—44no—
204C2×C42C2×D4116yes—
205C2×C42C2×D418yes—
206C2×C42C2×D414yes—
207#95—216no—
208#95—216no—
209#95—28no—
210#95—28no—
211#95—28no—
212#95—28no—
213#95—216no—
214#95—216no—
215#95—28no—
216#95—216no—
217#95—44no—
218#95C2×D4116yes—
219#95C2×D418yes—
220#95C2×D418yes—
221#95C2×D4116yes—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—28no—
226C24—22no—
227C24H(5,𝔽2)12yes—
228C24C2511yescharacteristic
229C2×Q8—48no—
230C2×Q8—48no—
231C2×Q8—48no—
232C2×Q8—88no—
233C2×Q8—88no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—28no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4C22×D418yes—
280C22×C4C22×D418yes—
281C22×C4C22×D414yes—
282C22×C4C22×D414yes—
283C22×C4C22×D414yes—
284C22×C4C22×D412yes—
285C22×C4C22×D414yes—
286C22×C4C22×D418yes—
287C22×C4H(5,𝔽2)12yes—
288C22×C42−1+4116yes—
289C4⋊C4—48no—
290C4⋊C4—416no—
291C4⋊C4—416no—
292C4⋊C4—48no—
293C4⋊C4—416no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—416no—
297C4⋊C4—416no—
298C4⋊C4—416no—
299C4⋊C4—816no—
300C4⋊C4—88no—
301C4⋊C4—816no—
302C22⋊C4—416no—
303C22⋊C4—416no—
304C22⋊C4—48no—
305C22⋊C4—416no—
306C42—232no—
307C42—416no—
308C42—44no—
309C42#127116yes—
310C23—22no—
311C23—24no—
312C23—28no—
313C23—41no—
314C23—44no—
315C23—41no—
316C23—42no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—48no—
321C23C23×D412yes—
322C23C23×D411yescharacteristic
323C23C2×H(5,𝔽2)11yescharacteristic
324C23C2×H(5,𝔽2)11yescharacteristic
325C23C2×2−1+418yes—
326C23#58512yes—
327Q8—816no—
328C2×C4—28no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—216no—
338C2×C4—216no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—216no—
345C2×C4—28no—
346C2×C4—416no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4C2×#12714yes—
377C2×C4C2×#12718yes—
378C2×C4#536116yes—
379C22—44no—
380C22—48no—
381C22—42no—
382C22—82no—
383C22—84no—
384C22—82no—
385C22C22×#12711yescharacteristic
386C22C2×#53614yes—
387C22#348411yescharacteristic
388C22#348711yescharacteristic
389C22#348812yes—
390C22#348812yes—
391C22#349914yes—
392C22#349914yes—
393C22#353814yes—
394C22#354414yes—
395C22#356318yes—
396C4—216no—
397C4—24no—
398C4—28no—
399C4—48no—
400C4—48no—
401C4—42no—
402C4—84no—
403C4—84no—
404C4—82no—
405C4—84no—
406C2—82no—
407C2#3531712yes—
408C2#3556511yescharacteristic
409C2#4193914yes—
410C2#4296414yes—
411C2#5154212yes—
412C2#5154712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
4. C2×#2660C2semidirect
6. #16964C2semidirect
7. #15856C2semidirect
9. #15145C2semidirect
11. #13805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#26112
C2×#26212
C2×#26602
#170622
#169642
#158562
#154302
#151452
#150202
#138052