Finite Group Databasebeta

← #286805

Subgroups — #286805

Order 512.

545 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29287C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#28717C211yesmaximal, characteristic
4#23250C211yesmaximal, characteristic
5#10597C212yesmaximal
6#10541C212yesmaximal
7#9745C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9575C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#487C2212yes—
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2100—22no—
17#2088—22no—
18#2072—22no—
19#1990—21no—
20#1990C2212yes—
21#1988C2212yes—
22#1987—21no—
23#1987C2212yes—
24#1987C2212yes—
25#1985—22no—
26#1948—21no—
27#1948C2212yes—
28#1939—21no—
29#1939C2212yes—
30#1842C2211yescharacteristic
31#1834—21no—
32#1803—21no—
33#1803—21no—
34#1770—21no—
35#1770C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
42C2×#332C2211yescharacteristic
43C2×#332C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8C2312yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54#487—24no—
55#487—42no—
56C2×#106—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8D411yescharacteristic
61C2×C4⋊C8D411yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72C2×C4×C8C2312yes—
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389D412yes—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384D411yescharacteristic
84#384C2311yescharacteristic
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2312yes—
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#336—22no—
100#336—22no—
101#336—21no—
102#336—21no—
103#336D412yes—
104#334—42no—
105#333—24no—
106#333—42no—
107#332—22no—
108#332—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—22no—
126C4⋊Q8—21no—
127C4⋊Q8—22no—
128C4⋊Q8—82no—
129C4⋊Q8—82no—
130C4⋊Q8#4112yes—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—81no—
134C4×Q8—81no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C42—22no—
154C2×C42—41no—
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42C2411yescharacteristic
162#106—82no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—22no—
171C4⋊C8—21no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—42no—
175C4⋊C8—81no—
176C4⋊C8—82no—
177C4⋊C8#4112yes—
178#103—44no—
179#103—44no—
180#97—82no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—42no—
184C4×C8—42no—
185#95—22no—
186#95—22no—
187#95—41no—
188#95—41no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—81no—
199C2×Q8—81no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—81no—
254C4⋊C4—82no—
255C4⋊C4—81no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—41no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42#13512yes—
271C42#13712yes—
272C42#13712yes—
273C42#13712yes—
274C42C2×#4111yescharacteristic
275C42C2×#4111yescharacteristic
276C42H(5,𝔽2)11yescharacteristic
277C422−1+411yescharacteristic
278C23C2×#12111yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—84no—
282Q8—81no—
283Q8—81no—
284Q8—162no—
285Q8—162no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4#46412yes—
311C2×C4#46412yes—
312C2×C4#47012yes—
313C2×C4#47012yes—
314C2×C4#47112yes—
315C2×C4#52911yescharacteristic
316C2×C4#54211yescharacteristic
317C2×C4#54611yescharacteristic
318C2×C4#54811yescharacteristic
319C2×C4#54811yescharacteristic
320C2×C4#55011yescharacteristic
321C2×C4C2×#13511yescharacteristic
322C2×C4C2×#13711yescharacteristic
323C2×C4C2×#13711yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C2×C4#57711yescharacteristic
326C2×C4#57711yescharacteristic
327C2×C4#57711yescharacteristic
328C2×C4#57811yescharacteristic
329C8—42no—
330C8—42no—
331C8—42no—
332C8—81no—
333C8—81no—
334C8—82no—
335C8—81no—
336C22#242611yescharacteristic
337C22C2×#46411yescharacteristic
338C22#304211yescharacteristic
339C22C2×#47011yescharacteristic
340C22#306311yescharacteristic
341C22C2×#47111yescharacteristic
342C22#306811yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—82no—
349C4—82no—
350C4—82no—
351C4—161no—
352C4—162no—
353C4#170414yes—
354C4#330311yescharacteristic
355C4#333111yescharacteristic
356C4#336011yescharacteristic
357C4#336911yescharacteristic
358C4#336911yescharacteristic
359C4#337411yescharacteristic
360C4#339811yescharacteristic
361C4#340011yescharacteristic
362C2C2×#170411yescharacteristic
363C2#1924411yescharacteristic
364C2#1942911yescharacteristic
365C2#1955911yescharacteristic
366C2#2527311yescharacteristic
367C2#2530211yescharacteristic
368C2#2531511yescharacteristic
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

Maximal subgroups

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

subgroupindex
#292872
#292182
#287172
#232502
#105972
#105412
#97452
#96912
#96022
#95752
C4×#3322