Finite Group Databasebeta

← #287771

Subgroups — #287771

Order 512.

657 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28748C211yesmaximal, characteristic
2#28512C211yesmaximal, characteristic
3#28284C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5C2×#1684C212yesmaximal
6#10487C212yesmaximal
7#10277C212yesmaximal
8#9694C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9220C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#487C2212yes—
13C2×#477C2212yes—
14C2×#475C2212yes—
15C2×#475C2212yes—
16C2×C8×Q8C2212yes—
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2096—22no—
21#2068—22no—
22#2043—22no—
23#1984C2212yes—
24#1981—21no—
25#1981—21no—
26#1981—21no—
27#1956—21no—
28#1956C2212yes—
29#1947—21no—
30#1939—21no—
31#1916—22no—
32#1844—21no—
33#1842C2211yescharacteristic
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1684—28no—
40C2×#335C2212yes—
41C2×#333C2211yescharacteristic
42C2×#333C2211yescharacteristic
43C2×#332C2212yes—
44C2×#332C2212yes—
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1277C2212yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8C2312yes—
55#487—24no—
56#487—42no—
57#477—24no—
58#477—42no—
59#475—24no—
60#475—24no—
61#475—42no—
62#475—42no—
63C8×Q8—24no—
64C8×Q8—42no—
65C2×#107—22no—
66C2×#106—22no—
67C2×#106—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#391—21no—
91#391D412yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384C2311yescharacteristic
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C43C2311yescharacteristic
107#340—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336D412yes—
113#335—24no—
114#333—22no—
115#333D414yes—
116#332—24no—
117#332—42no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8C2×D412yes—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—21no—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—42no—
137C4⋊Q8—42no—
138C4⋊Q8—81no—
139C4⋊Q8—81no—
140C4⋊Q8Q1614yes—
141C4⋊Q8C2×D412yes—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—82no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D412yes—
165C2×C42—22no—
166C2×C42—41no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42C2411yescharacteristic
174#107—44no—
175#106—44no—
176#106—44no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—44no—
187C4⋊C8—42no—
188C4⋊C8—81no—
189C4⋊C8C2×D412yes—
190C4⋊C8C2×D412yes—
191C4⋊C8#4112yes—
192#103—44no—
193#103—44no—
194#103—44no—
195#103—44no—
196#103—44no—
197#103—82no—
198#103—82no—
199#103—82no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—44no—
203C4×C8—42no—
204C4×C8—42no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—81no—
223C2×Q8—81no—
224C2×Q8C2×Q1614yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—41no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—22no—
289C42—41no—
290C42—42no—
291C42—42no—
292C42—82no—
293C42C2×Q1612yes—
294C42#13512yes—
295C42#13512yes—
296C42#13712yes—
297C42C22×D411yescharacteristic
298C42C2×#4111yescharacteristic
299C42H(5,𝔽2)11yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C23C2×#12111yescharacteristic
302Q8—24no—
303Q8—24no—
304Q8—42no—
305Q8—42no—
306Q8—82no—
307Q8—82no—
308Q8—82no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4#46412yes—
337C2×C4#46412yes—
338C2×C4#46712yes—
339C2×C4#46712yes—
340C2×C4#47112yes—
341C2×C4#53211yescharacteristic
342C2×C4#54011yescharacteristic
343C2×C4D4211yescharacteristic
344C2×C4D4211yescharacteristic
345C2×C4#54611yescharacteristic
346C2×C4#55011yescharacteristic
347C2×C4C22×Q1611yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×#13511yescharacteristic
350C2×C4C2×#13711yescharacteristic
351C2×C4#57711yescharacteristic
352C2×C4#57711yescharacteristic
353C2×C4#57711yescharacteristic
354C2×C4#57811yescharacteristic
355C8—42no—
356C8—42no—
357C8—42no—
358C8—42no—
359C8—42no—
360C8—81no—
361C8—82no—
362C22#242811yescharacteristic
363C22C2×#46411yescharacteristic
364C22#304311yescharacteristic
365C22C2×#46711yescharacteristic
366C22#306311yescharacteristic
367C22C2×#47111yescharacteristic
368C22#306911yescharacteristic
369C4—22no—
370C4—21no—
371C4—21no—
372C4—22no—
373C4—44no—
374C4—41no—
375C4—82no—
376C4—82no—
377C4—162no—
378C4—161no—
379C4#168414yescomplemented
380C4#327811yescharacteristic
381C4#328411yescharacteristic
382C4#328411yescharacteristic
383C4D4×Q1611yescharacteristic
384C4#329911yescharacteristic
385C4#330211yescharacteristic
386C4#339411yescharacteristic
387C4#340011yescharacteristic
388C2C2×#168411yescharacteristic
389C2#1924311yescharacteristic
390C2#1939911yescharacteristic
391C2#1956511yescharacteristic
392C2#2536911yescharacteristic
393C2#2538611yescharacteristic
394C2#2540511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
379. C4#1684semidirect

Maximal subgroups

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

subgroupindex
#287482
#285122
#282842
#231492
C2×#16842
#104872
#102772
#96942
#96012
#92202
C4×#3332