Finite Group Databasebeta

← #287350

Subgroups — #287350

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29141C211yesmaximal, characteristic
2#29141C211yesmaximal, characteristic
3#28283C211yesmaximal, characteristic
4#23153C211yesmaximal, characteristic
5#10623C212yesmaximal
6#10486C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9602C211yesmaximal, characteristic
9#9228C212yesmaximal
10#8657C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#475C2212yes—
13C2×#446C2212yes—
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2066—22no—
18#2024—22no—
19#2024—22no—
20#1984C2212yes—
21#1981—21no—
22#1956—21no—
23#1956C2212yes—
24#1948—21no—
25#1939—21no—
26#1917—22no—
27#1842C2211yescharacteristic
28#1834—21no—
29#1834C2212yes—
30#1803—21no—
31#1803—21no—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42#1296C2212yes—
43#1296C2211yescharacteristic
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1277C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4×Q8—21no—
53C2×C4×Q8C2312yes—
54#475—24no—
55#475—42no—
56#446—24no—
57#446—42no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8D411yescharacteristic
63C2×C4⋊C8D411yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#391—21no—
80#391D412yes—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384D411yescharacteristic
89#384C2311yescharacteristic
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C43C2311yescharacteristic
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—22no—
102#336—22no—
103#336—21no—
104#336—21no—
105#336D412yes—
106#334—42no—
107#332—22no—
108#332—22no—
109C22×Q8—21no—
110C22×Q8—41no—
111C22×Q8C2×D412yes—
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—21no—
123C4⋊Q8—21no—
124C4⋊Q8—81no—
125C4⋊Q8—81no—
126C4⋊Q8—81no—
127C4⋊Q8—81no—
128C4⋊Q8—82no—
129C4⋊Q8#4112yes—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—42no—
135C4×Q8—42no—
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—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C2×D412yes—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42C2411yescharacteristic
163#106—44no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—22no—
170C4⋊C8—21no—
171C4⋊C8—21no—
172C4⋊C8—22no—
173C4⋊C8—21no—
174C4⋊C8—21no—
175C4⋊C8—44no—
176C4⋊C8—42no—
177C4⋊C8—81no—
178C4⋊C8#4112yes—
179#103—44no—
180#103—82no—
181#103—82no—
182#103—82no—
183#103—82no—
184#103—82no—
185#97—44no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—42no—
189C4×C8—42no—
190#95—22no—
191#95—22no—
192#95—41no—
193#95—41no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—82no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—42no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—41no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—22no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—82no—
277C42#13512yes—
278C42#13712yes—
279C42#13712yes—
280C42#13712yes—
281C42C2×#4111yescharacteristic
282C42C2×#4111yescharacteristic
283C42H(5,𝔽2)11yescharacteristic
284C422−1+411yescharacteristic
285C23C2×#12111yescharacteristic
286Q8—42no—
287Q8—42no—
288Q8—42no—
289Q8—42no—
290Q8—42no—
291Q8—42no—
292Q8—81no—
293Q8—81no—
294Q8—164no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4#46412yes—
320C2×C4#46412yes—
321C2×C4#47012yes—
322C2×C4#47012yes—
323C2×C4#47112yes—
324C2×C4#52911yescharacteristic
325C2×C4#54211yescharacteristic
326C2×C4#54611yescharacteristic
327C2×C4#54611yescharacteristic
328C2×C4#54711yescharacteristic
329C2×C4#54811yescharacteristic
330C2×C4C2×#13511yescharacteristic
331C2×C4C2×#13711yescharacteristic
332C2×C4C2×#13711yescharacteristic
333C2×C4C2×#13711yescharacteristic
334C2×C4#57611yescharacteristic
335C2×C4#57711yescharacteristic
336C2×C4#57711yescharacteristic
337C2×C4#57711yescharacteristic
338C8—42no—
339C8—42no—
340C8—42no—
341C8—81no—
342C8—82no—
343C8—81no—
344C8—81no—
345C22#242711yescharacteristic
346C22C2×#46411yescharacteristic
347C22#304311yescharacteristic
348C22C2×#47011yescharacteristic
349C22#306211yescharacteristic
350C22C2×#47111yescharacteristic
351C22#306911yescharacteristic
352C4—21no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—82no—
358C4—82no—
359C4—82no—
360C4—161no—
361C4—162no—
362C4#170414yes—
363C4#330311yescharacteristic
364C4#332911yescharacteristic
365C4#332911yescharacteristic
366C4#333311yescharacteristic
367C4#335611yescharacteristic
368C4#335711yescharacteristic
369C4#336011yescharacteristic
370C4#337211yescharacteristic
371C2C2×#170411yescharacteristic
372C2#1924511yescharacteristic
373C2#1943511yescharacteristic
374C2#1956411yescharacteristic
375C2#2531911yescharacteristic
376C2#2534911yescharacteristic
377C2#2536111yescharacteristic
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
#291412
#282832
#231532
#106232
#104862
#98962
#96022
#92282
#86572
C4×#3322