Finite Group Databasebeta

← #351105

Subgroups — #351105

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33202C211yesmaximal, characteristic
2#33196C211yesmaximal, characteristic
3#30205C211yesmaximal, characteristic
4#20569C211yesmaximal, characteristic
5#9896C211yesmaximal, characteristic
6#9697C211yesmaximal, characteristic
7#9567C212yesmaximal
8#9562C212yesmaximal
9#9220C212yesmaximal
10#9216C212yesmaximal
11#8787C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#475—21no—
14C2×#475C2211yescharacteristic
15C2×C8×Q8C2212yes—
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18C42×Q8C2211yescharacteristic
19#2072—22no—
20#2024—21no—
21#2024—21no—
22#1990C2212yes—
23#1988—21no—
24#1988C2211yescharacteristic
25#1987C2211yescharacteristic
26#1956C2212yes—
27#1949—21no—
28#1949—21no—
29#1948—21no—
30#1948C2212yes—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1805—21no—
34#1805—21no—
35#1803—21no—
36#1803C2212yes—
37C8×C4⋊C4—22no—
38C8×C4⋊C4C2212yes—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41#1744—21no—
42#1744C2211yescharacteristic
43C2×#333C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1277C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#477—24no—
58#475—24no—
59#475—44no—
60C8×Q8—24no—
61C8×Q8—24no—
62C2×#107—21no—
63C2×#106—22no—
64C2×#106—21no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×#103—22no—
75C2×#103—21no—
76C2×#103—21no—
77C2×#103—22no—
78C2×#103—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8C2312yes—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2312yes—
91#382—21no—
92#382—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C43—21no—
100C43C2311yescharacteristic
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—21no—
106#340—21no—
107#340—21no—
108#340—21no—
109#336—22no—
110#336—21no—
111#336D412yes—
112#336D411yescharacteristic
113#336D411yescharacteristic
114#333—24no—
115#332—24no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—41no—
119C22×Q8C2×D411yescharacteristic
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8C2×D412yes—
125C22×C8C2×D412yes—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—82no—
133C4⋊Q8—82no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8SD1614yes—
138C4×Q8Q1614yes—
139C4×Q8#4114yes—
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—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D411yescharacteristic
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42C2411yescharacteristic
169#107—42no—
170#106—42no—
171#106—44no—
172C4⋊C8—28no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8#4114yes—
183#103—42no—
184#103—42no—
185#103—44no—
186#103—82no—
187#103—82no—
188C4×C8—28no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—81no—
192C4×C8—82no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×Q8—81no—
211C2×Q8—81no—
212C2×Q8—82no—
213C2×Q8#12412yes—
214C2×Q8C2×SD1612yes—
215C2×Q8C2×Q1612yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4111yescharacteristic
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—24no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—42no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—82no—
268C4⋊C4#12412yes—
269C4⋊C4C2×SD1612yes—
270C4⋊C4C2×Q1612yes—
271C42—24no—
272C42—21no—
273C42—22no—
274C42—21no—
275C42—24no—
276C42—24no—
277C42—41no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—81no—
282C42—81no—
283C42C2×SD1612yes—
284C42C2×Q1612yes—
285C42#13512yes—
286C42#13512yes—
287C42C2×#4112yes—
288C42C2×#4112yes—
289C23C2×#12111yescharacteristic
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—82no—
294Q8—82no—
295Q8—162no—
296Q8—164no—
297Q8#48714yes—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4#46512yes—
330C2×C4#46512yes—
331C2×C4#46712yes—
332C2×C4#46712yes—
333C2×C4#52512yes—
334C2×C4C2×#12411yescharacteristic
335C2×C4#52912yes—
336C2×C4#53211yescharacteristic
337C2×C4C22×SD1611yescharacteristic
338C2×C4C22×Q1611yescharacteristic
339C2×C4C2×#13511yescharacteristic
340C2×C4C2×#13511yescharacteristic
341C2×C4#57712yes—
342C2×C4#57711yescharacteristic
343C2×C4#57811yescharacteristic
344C8—42no—
345C8—42no—
346C8—44no—
347C8—81no—
348C8—81no—
349C8—82no—
350C22#285011yescharacteristic
351C22#304311yescharacteristic
352C22#304611yescharacteristic
353C22C2×#46511yescharacteristic
354C22C2×#46711yescharacteristic
355C22#306811yescharacteristic
356C22#306911yescharacteristic
357C4—24no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—161no—
367C4—161no—
368C4—162no—
369C4#312112yes—
370C4#312512yes—
371C4C2×#48711yescharacteristic
372C4#313111yescharacteristic
373C4#316711yescharacteristic
374C4#317011yescharacteristic
375C2#1939611yescharacteristic
376C2#1939911yescharacteristic
377C2#1941511yescharacteristic
378C2#1941711yescharacteristic
379C2#3303711yescharacteristic
380C2#3304311yescharacteristic
381C2#3306311yescharacteristic
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
#332022
#331962
#302052
#205692
#98962
#96972
#95672
#95622
#92202
#92162
#87872