Finite Group Databasebeta

← #287374

Subgroups — #287374

Order 512.

559 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29218C211yesmaximal, characteristic
2#29141C211yesmaximal, characteristic
3#28284C211yesmaximal, characteristic
4#23153C211yesmaximal, characteristic
5#10623C212yesmaximal
6#10482C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9602C211yesmaximal, characteristic
9#9224C212yesmaximal
10#8657C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#477C2212yes—
13C2×#446C2212yes—
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2100—22no—
18#2068—22no—
19#2024—22no—
20#1990C2212yes—
21#1987—21no—
22#1987—21no—
23#1984C2212yes—
24#1981—21no—
25#1956—21no—
26#1947—21no—
27#1939—21no—
28#1917—22no—
29#1842C2211yescharacteristic
30#1834—21no—
31#1834C2212yes—
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
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#477—24no—
55#477—42no—
56#446—24no—
57#446—42no—
58C2×#107—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×#97—22no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8C2312yes—
77#391—21no—
78#391D412yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384D411yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
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#333—22no—
108#333—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—42no—
125C4⋊Q8—42no—
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—82no—
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—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
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#107—44no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8—22no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—22no—
172C4⋊C8—21no—
173C4⋊C8—21no—
174C4⋊C8—44no—
175C4⋊C8—42no—
176C4⋊C8—81no—
177C4⋊C8#4112yes—
178#103—44no—
179#103—82no—
180#103—82no—
181#103—82no—
182#97—44no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—42no—
186C4×C8—42no—
187#95—22no—
188#95—22no—
189#95—41no—
190#95—41no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—81no—
202C2×Q8—81no—
203C2×Q8—82no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C2×#4111yescharacteristic
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—41no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—41no—
269C42—42no—
270C42—42no—
271C42—82no—
272C42#13512yes—
273C42#13712yes—
274C42#13712yes—
275C42#13712yes—
276C42C2×#4111yescharacteristic
277C42C2×#4111yescharacteristic
278C42H(5,𝔽2)11yescharacteristic
279C422−1+411yescharacteristic
280C23C2×#12111yescharacteristic
281Q8—42no—
282Q8—42no—
283Q8—42no—
284Q8—42no—
285Q8—82no—
286Q8—82no—
287Q8—164no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4#46412yes—
313C2×C4#46412yes—
314C2×C4#47012yes—
315C2×C4#47012yes—
316C2×C4#47112yes—
317C2×C4#52911yescharacteristic
318C2×C4#54211yescharacteristic
319C2×C4#54611yescharacteristic
320C2×C4#54611yescharacteristic
321C2×C4#54711yescharacteristic
322C2×C4#54811yescharacteristic
323C2×C4C2×#13511yescharacteristic
324C2×C4C2×#13711yescharacteristic
325C2×C4C2×#13711yescharacteristic
326C2×C4C2×#13711yescharacteristic
327C2×C4#57611yescharacteristic
328C2×C4#57711yescharacteristic
329C2×C4#57811yescharacteristic
330C2×C4#57811yescharacteristic
331C8—42no—
332C8—42no—
333C8—42no—
334C8—81no—
335C8—82no—
336C8—81no—
337C8—81no—
338C22#242711yescharacteristic
339C22C2×#46411yescharacteristic
340C22#304411yescharacteristic
341C22C2×#47011yescharacteristic
342C22#306211yescharacteristic
343C22C2×#47111yescharacteristic
344C22#306811yescharacteristic
345C4—21no—
346C4—21no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—82no—
351C4—82no—
352C4—82no—
353C4—162no—
354C4—161no—
355C4#170414yes—
356C4#330211yescharacteristic
357C4#332811yescharacteristic
358C4#332911yescharacteristic
359C4#333311yescharacteristic
360C4#335611yescharacteristic
361C4#335711yescharacteristic
362C4#336411yescharacteristic
363C4#337411yescharacteristic
364C2C2×#170411yescharacteristic
365C2#1924211yescharacteristic
366C2#1944111yescharacteristic
367C2#1957011yescharacteristic
368C2#2532311yescharacteristic
369C2#2534911yescharacteristic
370C2#2535911yescharacteristic
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
#292182
#291412
#282842
#231532
#106232
#104822
#98982
#96022
#92242
#86572
C4×#3332