Finite Group Databasebeta

← #302864

Subgroups — #302864

Order 512.

739 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59871C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#29208C211yesmaximal, characteristic
5#10581C214yesmaximal
6#9864C211yesmaximal, characteristic
7#9864C211yesmaximal, characteristic
8#8922C214yesmaximal
9C4×#332C211yesmaximal, characteristic
10#2868C2211yescharacteristic
11#2806—22no—
12#2584—22no—
13#2582—22no—
14#2567—21no—
15#2552—21no—
16#2552C2212yes—
17#2507—22no—
18#2369—21no—
19#2365C2212yes—
20Q8×C4⋊C4—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2100—22no—
25#2057—22no—
26#2057—22no—
27#1990—21no—
28#1990—21no—
29#1990C2212yes—
30#1987—22no—
31#1947C2214yes—
32#1947C2214yes—
33#1845—21no—
34#1845—21no—
35#1834C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#332C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#107—24no—
55C2×#107—24no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61C2×C4×C8C2312yes—
62#398—24no—
63#398—24no—
64#398—24no—
65#395—24no—
66#391—24no—
67#391—22no—
68#391—21no—
69#391—21no—
70#391—21no—
71#389—22no—
72#389—22no—
73#389—24no—
74#389—24no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389D412yes—
82#389D412yes—
83#389D412yes—
84#387—24no—
85#384—22no—
86#384—22no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D411yescharacteristic
91#384D411yescharacteristic
92#384C2312yes—
93#384C2311yescharacteristic
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4C2312yes—
101C4×C4⋊C4C2312yes—
102C43C2311yescharacteristic
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#336—21no—
108#336—21no—
109#336—21no—
110#332Q818yescomplemented
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—44no—
123C4⋊Q8—82no—
124C4⋊Q8—82no—
125C4⋊Q8Q1614yes—
126C4⋊Q8C2×Q814yes—
127#125—84no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—82no—
131C4×Q8—82no—
132C4×Q8—84no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2411yescharacteristic
172#107—84no—
173#107—84no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—42no—
177C4⋊C8C2×Q814yes—
178C4⋊C8C2×Q814yes—
179C4×C8—24no—
180C4×C8—42no—
181C4×C8—42no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—81no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×Q8—81no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4#12011yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12011yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C4⋊C4—22no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—82no—
257C4⋊C4C4⋊Q814yes—
258C4⋊C4C2×Q1614yes—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—41no—
263C42—44no—
264C42—44no—
265C42—44no—
266C42—84no—
267C42C2×Q1612yes—
268C42#13512yes—
269C42#13712yes—
270C42#13712yes—
271C42C22×Q812yes—
272C422−1+412yes—
273C23C2×#12011yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—84no—
277Q8—84no—
278Q8—164no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4#45112yes—
310C2×C4#45112yes—
311C2×C4#45212yes—
312C2×C4#45212yes—
313C2×C4#45412yes—
314C2×C4C2×C4⋊Q811yescharacteristic
315C2×C4#53611yescharacteristic
316C2×C4D4×Q812yes—
317C2×C4D4×Q812yes—
318C2×C4C22×Q1611yescharacteristic
319C2×C4C2×#13511yescharacteristic
320C2×C4C2×#13711yescharacteristic
321C2×C4C2×#13711yescharacteristic
322C2×C4#57612yes—
323C2×C4#57612yes—
324C8—44no—
325C8—44no—
326C8—82no—
327C22#243411yescharacteristic
328C22C2×#45111yescharacteristic
329C22C2×#45211yescharacteristic
330C22#300811yescharacteristic
331C22#300811yescharacteristic
332C22C2×#45411yescharacteristic
333C22#302211yescharacteristic
334C4—22no—
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—44no—
340C4—44no—
341C4—82no—
342C4—82no—
343C4—82no—
344C4—82no—
345C4—162no—
346C4#163614yes—
347C4#338012yes—
348C4Q8×Q1612yes—
349C4#338512yes—
350C4#338512yes—
351C2C2×#163611yescharacteristic
352C2#1872711yescharacteristic
353C2#1890511yescharacteristic
354C2#1892711yescharacteristic
355C2#2582611yescharacteristic
356C2#2583911yescharacteristic
357C2#2584111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
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
110. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#598712
#292182
#292082
#105812
#98642
#89222
C4×#3322