Finite Group Databasebeta

← #413350

Subgroups — #413350

Order 512.

1088 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
1C2×#2084C212yesmaximal
2C2×#2085C212yesmaximal
3C2×#1847C211yesmaximal, characteristic
4C2×#1844C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#14276C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11323C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×#106C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#2085—28no—
29#2084—28no—
30#1847—24no—
31#1844—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—24no—
39#1441—24no—
40C4⋊#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—42no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#106—42no—
68C2×#106—42no—
69C2×#97—22no—
70C2×#97—22no—
71C2×#97Q814yes—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8Q814yes—
75#389—24no—
76#389—24no—
77#384—24no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—42no—
82#384D414yes—
83#382—24no—
84#382—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#382—42no—
89#376—22no—
90#376—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—24no—
97C2×#95D414yes—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336D418yes—
103#336Q818yes—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8C2×D414yes—
123C22×C8C2×D414yes—
124C22×C8C2×Q814yes—
125C22×C8C2×Q814yes—
126C22×C8C2×Q814yes—
127C22×C8C2×Q814yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—84no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4C2×D414yes—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×D414yes—
173C2×C42C2×D412yes—
174C2×C42C2×Q814yes—
175C2×C42C2×Q812yes—
176C2×C42C2×Q812yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180#107—48no—
181#106—48no—
182#106—48no—
183#106—48no—
184#97—28no—
185C4×C8—28no—
186#95—28no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—84no—
193C24C22×D411yescharacteristic
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4#12114yes—
231C22×C4#12314yes—
232C22×C4#12714yes—
233C22×C4C4⋊Q814yes—
234C22×C4C4⋊Q814yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D412yes—
237C22×C4C22×Q811yescharacteristic
238C22×C4C22×Q811yescharacteristic
239C22×C4C22×Q812yes—
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4111yescharacteristic
242C22×C4C2×#4111yescharacteristic
243C22×C4H(5,𝔽2)12yes—
244C22×C42−1+412yes—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8C4⋊Q818yes—
258C2×C8C4⋊Q818yes—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—88no—
276C4⋊C4—88no—
277C4⋊C4—88no—
278C4⋊C4—88no—
279C4⋊C4—88no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42C22⋊Q814yes—
285C42C22⋊Q814yes—
286C42#12514yes—
287C42#12514yes—
288C23C2×#12111yescharacteristic
289C23C2×#12311yescharacteristic
290C23C2×#12711yescharacteristic
291C23C2×C4⋊Q811yescharacteristic
292C23C2×C4⋊Q811yescharacteristic
293C23#53511yescharacteristic
294C23#53611yescharacteristic
295C23#57714yes—
296C23#57714yes—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—28no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#39818yes—
338C2×C4C2×C22⋊Q811yescharacteristic
339C2×C4C2×C22⋊Q811yescharacteristic
340C2×C4C2×#12511yescharacteristic
341C2×C4C2×#12511yescharacteristic
342C2×C4C2×C4⋊Q812yes—
343C2×C4C2×C4⋊Q812yes—
344C2×C4#54412yes—
345C2×C4#54412yes—
346C2×C4D4212yes—
347C2×C4#54712yes—
348C2×C4#54712yes—
349C2×C4D4×Q812yes—
350C2×C4#55012yes—
351C2×C4#55412yes—
352C2×C4#55412yes—
353C2×C4#55712yes—
354C8—28no—
355C8—28no—
356C22C2×#39811yescharacteristic
357C22#250111yescharacteristic
358C22#250211yescharacteristic
359C22#254111yescharacteristic
360C22#254611yescharacteristic
361C22#266011yescharacteristic
362C22#286811yescharacteristic
363C22#304212yes—
364C22#304312yes—
365C22#309914yes—
366C22#315412yes—
367C22#315412yes—
368C22#316314yes—
369C22#316314yes—
370C22#322112yes—
371C22#322212yes—
372C22#324414yes—
373C4—28no—
374C4—44no—
375C4—44no—
376C4—84no—
377C4—162no—
378C4—161no—
379C4—161no—
380C4—161no—
381C4—161no—
382C4#244012yes—
383C4#244012yes—
384C4#255212yes—
385C4#255212yes—
386C2#1514511yescharacteristic
387C2#2366112yes—
388C2#2792012yes—
389C2#2795712yes—
390C2#2845511yescharacteristic
391C2#2845711yescharacteristic
392C2#2869612yes—
393C2#3031412yes—
394C2#3331812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#20852
C2×#18472
C2×#18442
#143542
#142762
#113252
#113232
#108432
#108422
#106942