Finite Group Databasebeta

← #315694

Subgroups — #315694

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#27975C211yesmaximal, characteristic
3#27962C211yesmaximal, characteristic
4#27878C211yesmaximal, characteristic
5#27877C211yesmaximal, characteristic
6#20816C211yesmaximal, characteristic
7#10682C211yesmaximal, characteristic, complemented
8#10682C211yesmaximal, characteristic, complemented
9#10679C211yesmaximal, characteristic, complemented
10#10679C211yesmaximal, characteristic, complemented
11#9826C211yesmaximal, characteristic, complemented
12#9826C211yesmaximal, characteristic, complemented
13#8821C211yesmaximal, characteristic, complemented
14#8821C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22#2101—22no—
23#2076—22no—
24#2075—22no—
25#2074—22no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2211yescharacteristic
35#1956C2211yescharacteristic
36#1933—22no—
37#1921—22no—
38#1918—22no—
39#1892—22no—
40#1873—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1844C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1825—22no—
55#1807—22no—
56C4×#106—22no—
57C4×C22⋊C8C2211yescharacteristic
58C4×C22⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C23×C8D412yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C42C2311yescharacteristic
71#487—22no—
72#487—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×C22⋊C8D412yes—
82C2×C22⋊C8D412yes—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8D412yes—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#391—21no—
93#391—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#382—21no—
105#382—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#336D412yes—
135#336D412yes—
136#322—22no—
137#322—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C22×C8—24no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C22⋊Q8—44no—
161C22⋊Q8—44no—
162#117—44no—
163#117—82no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2411yescharacteristic
202#107—44no—
203#107—44no—
204#106—44no—
205#106—44no—
206#103—44no—
207#103—44no—
208#103—82no—
209#103—82no—
210C22⋊C8—24no—
211C22⋊C8—42no—
212C4×C8—24no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8#4112yes—
224C4×C8#4112yes—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233C24#12711yescharacteristic
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4#12711yescharacteristic
262C22×C4#12711yescharacteristic
263C22×C4#12711yescharacteristic
264C22×C4#13514yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—48no—
286C2×C8#12314yes—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—82no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—82no—
326C42—82no—
327C42C2×#4111yescharacteristic
328C42C2×#4111yescharacteristic
329C422−1+411yescharacteristic
330C422−1+411yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23#49512yes—
338C23C2×#12711yescharacteristic
339Q8—82no—
340Q8—82no—
341Q8—82no—
342Q8—82no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4#49512yes—
382C2×C4C2×#12311yescharacteristic
383C2×C4#53611yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4#54711yescharacteristic
386C2×C4#54711yescharacteristic
387C2×C4#54711yescharacteristic
388C2×C4C2×#13512yes—
389C2×C4#57512yes—
390C2×C4#57512yes—
391C2×C4#57512yes—
392C2×C4#57712yes—
393C2×C4#57812yes—
394C8—24no—
395C8—42no—
396C8—42no—
397C8—42no—
398C8—42no—
399C8—42no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22#250111yescharacteristic
408C22C2×#49511yescharacteristic
409C22#315011yescharacteristic
410C22#315011yescharacteristic
411C22#315311yescharacteristic
412C22#315411yescharacteristic
413C22#315611yescharacteristic
414C4—22no—
415C4—41no—
416C4—41no—
417C4—161no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4#309312yes—
426C4#324012yes—
427C4#333712yes—
428C4#333712yes—
429C4#334912yes—
430C4#334912yes—
431C2—42no—
432C2#1961311yescharacteristic
433C2#1962811yescharacteristic
434C2#1969211yescharacteristic
435C2#1970711yescharacteristic
436C2#2791511yescharacteristic
437C2#2792611yescharacteristic
438C2#2793611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10682C2semidirect
8. #10682C2semidirect
9. #10679C2semidirect
10. #10679C2semidirect
11. #9826C2semidirect
12. #9826C2semidirect
13. #8821C2semidirect
14. #8821C2semidirect

Maximal subgroups

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

subgroupindex
#330432
#279752
#279622
#278782
#278772
#208162
#106822
#106792
#98262
#88212
#81452