Finite Group Databasebeta

← #330573

Subgroups — #330573

Order 512.

583 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33199C211yesmaximal, characteristic
2#33197C211yesmaximal, characteristic
3#30230C211yesmaximal, characteristic, complemented
4#22008C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10593C212yesmaximal, complemented
7#10481C212yesmaximal
8#10481C212yesmaximal
9#9900C211yesmaximal, characteristic, complemented
10#9691C211yesmaximal, characteristic, complemented
11#8810C211yesmaximal, characteristic, complemented
12C2×#478—21no—
13C2×#476C2212yes—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C42×D4C2211yescharacteristic
17#2100—21no—
18#2100—21no—
19#2100—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1989—21no—
23#1987C2212yes—
24#1954C2212yes—
25#1949—21no—
26#1948—21no—
27#1948—21no—
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1939C2212yes—
31#1923—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1817—21no—
35#1804—21no—
36#1802—21no—
37#1802—21no—
38#1802C2211yescharacteristic
39#1802C2211yescharacteristic
40#1797—21no—
41C4×C4⋊C8C2211yescharacteristic
42#1744C2211yescharacteristic
43C2×#335C2212yes—
44C2×#335C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2312yes—
52C2×#125—21no—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C22×C42—21no—
57#478—44no—
58#476—42no—
59#476—42no—
60C2×#107—21no—
61C2×#106—21no—
62C2×#106—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×#102—21no—
73C2×#102—21no—
74C2×#102—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#382—21no—
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4C2311yescharacteristic
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—21no—
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—21no—
104#340—21no—
105#340—21no—
106#340—21no—
107#336—22no—
108#336—21no—
109#336D411yescharacteristic
110#336D411yescharacteristic
111#336D412yes—
112#335—42no—
113#335—42no—
114#333—42no—
115#333—42no—
116C22×Q8—22no—
117C22×D4C2×D411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8C2×D412yes—
126C22×C8C2×D412yes—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—82no—
132C4⋊Q8—82no—
133#125—42no—
134#125—42no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4Q1614yes—
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—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—82no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—42no—
173#106—44no—
174#106—42no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—42no—
182C4⋊C8—42no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8—82no—
186#102—44no—
187#102—42no—
188#102—42no—
189C22⋊C8—82no—
190C22⋊C8—82no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—81no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—41no—
198#95—41no—
199C24—21no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4C2×Q1612yes—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C2×#4111yescharacteristic
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4C2×Q1612yes—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C42—21no—
280C42—22no—
281C42—21no—
282C42—41no—
283C42—41no—
284C42—42no—
285C42—81no—
286C42—81no—
287C42C2×Q1612yes—
288C42#13512yes—
289C42AGL(1,ℤ/8)12yes—
290C42#13712yes—
291C42H(5,𝔽2)12yes—
292C422−1+412yes—
293C23—22no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×#12111yescharacteristic
301Q8—162no—
302Q8—162no—
303D4—22no—
304D4—22no—
305D4—41no—
306D4—41no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—84no—
327C2×C4—84no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#46712yes—
340C2×C4#46712yes—
341C2×C4#46812yes—
342C2×C4#46812yes—
343C2×C4#53511yescharacteristic
344C2×C4#53611yescharacteristic
345C2×C4#53712yes—
346C2×C4#54212yes—
347C2×C4C22×Q1611yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C4C2×#13711yescharacteristic
351C2×C4#57712yes—
352C2×C4#57711yescharacteristic
353C2×C4#57811yescharacteristic
354C8—42no—
355C8—42no—
356C8—82no—
357C8—81no—
358C8—81no—
359C8—82no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22#262011yescharacteristic
368C22#304211yescharacteristic
369C22#304611yescharacteristic
370C22C2×#46711yescharacteristic
371C22C2×#46811yescharacteristic
372C22#306811yescharacteristic
373C22#306811yescharacteristic
374C4—22no—
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—42no—
380C4—81no—
381C4—81no—
382C4—162no—
383C4—161no—
384C4—161no—
385C4#320711yescharacteristic
386C4#320911yescharacteristic
387C4#323011yescharacteristic
388C4#323111yescharacteristic
389C4#325812yes—
390C4#326012yes—
391C2—82no—
392C2#1939311yescharacteristic
393C2#1941711yescharacteristic
394C2#1942711yescharacteristic
395C2#1945011yescharacteristic
396C2#2983611yescharacteristic
397C2#2984411yescharacteristic
398C2#2986111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #30230C2semidirect
5. #10601C2semidirect
6. #10593C2semidirect
9. #9900C2semidirect
10. #9691C2semidirect
11. #8810C2semidirect

Maximal subgroups

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

subgroupindex
#331992
#331972
#302302
#220082
#106012
#105932
#104812
#99002
#96912
#88102