Finite Group Databasebeta

← #286689

Subgroups — #286689

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29290C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic
4#23247C211yesmaximal, characteristic
5#10593C212yesmaximal
6#10543C212yesmaximal
7#9730C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9568C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#500C2212yes—
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2100—22no—
17#2087—22no—
18#2083—22no—
19#1990—21no—
20#1990C2212yes—
21#1987—21no—
22#1987C2212yes—
23#1987C2212yes—
24#1987C2212yes—
25#1984—22no—
26#1947—21no—
27#1947—21no—
28#1947—21no—
29#1947C2212yes—
30#1939—21no—
31#1939C2212yes—
32#1842C2211yescharacteristic
33#1834—21no—
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2211yescharacteristic
43C2×#333C2212yes—
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#500—24no—
55#500—42no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—22no—
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#335—42no—
105#333—22no—
106#333—22no—
107#333—24no—
108#333—42no—
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—21no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—21no—
124C4⋊Q8—22no—
125C4⋊Q8—21no—
126C4⋊Q8—22no—
127C4⋊Q8—82no—
128C4⋊Q8—82no—
129C4⋊Q8Q1614yes—
130C4⋊Q8#4112yes—
131C4×Q8—81no—
132C4×Q8—81no—
133C4×Q8—82no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C42—22no—
153C2×C42—41no—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42C2411yescharacteristic
161#107—44no—
162#107—44no—
163#107—82no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—21no—
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⋊C8Q1614yes—
178C4⋊C8#4112yes—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—42no—
182C4×C8—42no—
183C4×C8—82no—
184#95—22no—
185#95—22no—
186#95—41no—
187#95—41no—
188C2×Q8—24no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—81no—
194C2×Q8—81no—
195C2×Q8—81no—
196C2×Q8—81no—
197C2×Q8—82no—
198C2×Q8—82no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4111yescharacteristic
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8C2×Q1614yes—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4C2×Q1614yes—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—22no—
263C42—41no—
264C42—42no—
265C42—42no—
266C42—82no—
267C42C2×Q1612yes—
268C42C2×Q1612yes—
269C42#13512yes—
270C42#13712yes—
271C42C2×#4111yescharacteristic
272C42C2×#4111yescharacteristic
273C42H(5,𝔽2)11yescharacteristic
274C422−1+411yescharacteristic
275C23C2×#12111yescharacteristic
276Q8—84no—
277Q8—82no—
278Q8—82no—
279Q8—162no—
280Q8—162no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4#46212yes—
306C2×C4#46212yes—
307C2×C4#46712yes—
308C2×C4#46712yes—
309C2×C4#47212yes—
310C2×C4#52911yescharacteristic
311C2×C4#54211yescharacteristic
312C2×C4#54611yescharacteristic
313C2×C4#54811yescharacteristic
314C2×C4#54811yescharacteristic
315C2×C4#55011yescharacteristic
316C2×C4C22×Q1611yescharacteristic
317C2×C4C22×Q1611yescharacteristic
318C2×C4C2×#13511yescharacteristic
319C2×C4C2×#13711yescharacteristic
320C2×C4#57711yescharacteristic
321C2×C4#57711yescharacteristic
322C2×C4#57711yescharacteristic
323C2×C4#57811yescharacteristic
324C8—24no—
325C8—42no—
326C8—42no—
327C8—81no—
328C8—81no—
329C8—81no—
330C8—82no—
331C22#242611yescharacteristic
332C22C2×#46211yescharacteristic
333C22#304211yescharacteristic
334C22C2×#46711yescharacteristic
335C22#306311yescharacteristic
336C22C2×#47211yescharacteristic
337C22#306811yescharacteristic
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—44no—
344C4—82no—
345C4—82no—
346C4—161no—
347C4—162no—
348C4#167614yes—
349C4#329911yescharacteristic
350C4#332911yescharacteristic
351C4#336311yescharacteristic
352C4#336311yescharacteristic
353C4#336411yescharacteristic
354C4#337211yescharacteristic
355C4#339411yescharacteristic
356C4#339511yescharacteristic
357C2C2×#167611yescharacteristic
358C2#1924111yescharacteristic
359C2#1939311yescharacteristic
360C2#1956111yescharacteristic
361C2#2527811yescharacteristic
362C2#2529411yescharacteristic
363C2#2531211yescharacteristic
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
#292902
#292182
#286892
#232472
#105932
#105432
#97302
#96912
#96012
#95682
C4×#3332