Finite Group Databasebeta

← #330440

Subgroups — #330440

Order 512.

573 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic, complemented
2#29981C211yesmaximal, characteristic
3#29868C211yesmaximal, characteristic
4#21266C211yesmaximal, characteristic
5#10601C212yesmaximal, complemented
6#10597C212yesmaximal, complemented
7#10481C212yesmaximal
8#10480C212yesmaximal
9#9903C211yesmaximal, characteristic, complemented
10#9702C211yesmaximal, characteristic, complemented
11#8807C211yesmaximal, characteristic, complemented
12C2×#476C2212yes—
13C2×#433—21no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2293C2211yescharacteristic
17#2100—21no—
18#2094—21no—
19#2084—21no—
20#2084—21no—
21#2071—21no—
22#2071—21no—
23#2058—21no—
24#2023—21no—
25#1990—21no—
26#1990—21no—
27#1990C2212yes—
28#1990C2212yes—
29#1987C2212yes—
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1948C2212yes—
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1923—21no—
36#1923—21no—
37#1834—21no—
38#1834—21no—
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1802C2212yes—
42#1767C2211yescharacteristic
43#1750—21no—
44C42×C8C2211yescharacteristic
45C2×#335C2212yes—
46C2×#334C2212yes—
47C2×#333C2212yes—
48C2×#333C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2312yes—
57C2×C4×D4C2311yescharacteristic
58C2×#117—21no—
59#476—42no—
60#476—42no—
61#433—44no—
62C2×#106—22no—
63C2×#106—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×#102—22no—
71C2×#102—22no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×#97—21no—
77C2×C4×C8D411yescharacteristic
78C2×C4×C8D411yescharacteristic
79C2×C4×C8D411yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#390D411yescharacteristic
84#390D411yescharacteristic
85#389D411yescharacteristic
86#389D411yescharacteristic
87#388—21no—
88#388—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96#384C2312yes—
97C4×C4⋊C4C2311yescharacteristic
98C4×C22⋊C4—21no—
99C43C2311yescharacteristic
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#335—42no—
116#334—42no—
117#333—42no—
118#333—42no—
119C22×Q8—22no—
120C22×Q8—41no—
121C22×D4C2×D411yescharacteristic
122C23×C4—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×D411yescharacteristic
130C22×C8C2×D411yescharacteristic
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137C4⋊Q8—82no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140C4⋊Q8—82no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—21no—
144C4×D4—21no—
145#117—42no—
146#117—42no—
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—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—21no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#106—44no—
178#106—82no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189#102—44no—
190#102—82no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195#97—42no—
196C4×C8—22no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—21no—
200C4×C8—21no—
201C4×C8—41no—
202C4×C8—41no—
203C4×C8SD1614yes—
204#95—21no—
205#95—21no—
206#95—21no—
207C24—21no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—82no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—22no—
218C2×D4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8C2×SD1612yes—
260C2×C8C2×SD1612yes—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
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—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—21no—
290C42—21no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42C2×SD1612yes—
298C42#13712yes—
299C42H(5,𝔽2)11yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C422−1+411yescharacteristic
302C422−1+411yescharacteristic
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C2×#12111yescharacteristic
311Q8—162no—
312Q8—162no—
313Q8—164no—
314D4—42no—
315D4—42no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#46112yes—
346C2×C4#46112yes—
347C2×C4#47112yes—
348C2×C4#47212yes—
349C2×C4#53511yescharacteristic
350C2×C4#53611yescharacteristic
351C2×C4#53711yescharacteristic
352C2×C4#53711yescharacteristic
353C2×C4#54211yescharacteristic
354C2×C4#54211yescharacteristic
355C2×C4C22×SD1611yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4#57511yescharacteristic
358C2×C4#57511yescharacteristic
359C2×C4#57712yes—
360C2×C4#57812yes—
361C8—22no—
362C8—22no—
363C8—41no—
364C8—41no—
365C8—82no—
366C8—82no—
367C8—81no—
368C8—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#262011yescharacteristic
377C22C2×#46111yescharacteristic
378C22#304011yescharacteristic
379C22#305711yescharacteristic
380C22#305911yescharacteristic
381C22C2×#47111yescharacteristic
382C22C2×#47211yescharacteristic
383C4—22no—
384C4—21no—
385C4—21no—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—162no—
393C4—162no—
394C4#322512yes—
395C4#324212yes—
396C4#324611yescharacteristic
397C4#324611yescharacteristic
398C4#325211yescharacteristic
399C4#325211yescharacteristic
400C2—161no—
401C2#1919111yescharacteristic
402C2#1919311yescharacteristic
403C2#1936211yescharacteristic
404C2#1936511yescharacteristic
405C2#2982811yescharacteristic
406C2#2984811yescharacteristic
407C2#2985811yescharacteristic
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
1. #33289C2semidirect
5. #10601C2semidirect
6. #10597C2semidirect
9. #9903C2semidirect
10. #9702C2semidirect
11. #8807C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#299812
#298682
#212662
#106012
#105972
#104812
#104802
#99032
#97022
#88072