Finite Group Databasebeta

← #323574

Subgroups — #323574

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic, complemented
2#29190C211yesmaximal, characteristic, complemented
3#29190C211yesmaximal, characteristic, complemented
4#29139C211yesmaximal, characteristic, complemented
5#28508C211yesmaximal, characteristic, complemented
6#21690C211yesmaximal, characteristic
7#10678C211yesmaximal, characteristic, complemented
8#10677C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic, complemented
11#9832C211yesmaximal, characteristic
12#9832C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8819C211yesmaximal, characteristic, complemented
15#8263C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#474—22no—
18C2×#442—22no—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21C4×#127C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2093—22no—
24#2065—22no—
25#2065—22no—
26#2023—22no—
27#1993—22no—
28#1992—22no—
29#1972—22no—
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1957C2211yescharacteristic
40#1864—22no—
41#1864—22no—
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1802—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×#322C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×D8—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×#127C2311yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#493—22no—
77#493Q814yescomplemented
78#474—48no—
79#442—48no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×C4⋊C8D412yes—
86C2×C4⋊C8D412yes—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#97D412yes—
98C2×#97D412yes—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#390—21no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125C43C2311yescharacteristic
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—41no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C23×C4—41no—
146C23×C4—41no—
147C2×D8—28no—
148C2×D8—44no—
149C2×D8—44no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—82no—
159#127—21no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164#127—21no—
165#127C2×Q812yes—
166#127C2×Q812yes—
167#125—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—44no—
172C4×D4—82no—
173C4×D4—82no—
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—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210#107—44no—
211#107—44no—
212#106—44no—
213#106—82no—
214#106—82no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217#102—44no—
218#102—44no—
219#102—44no—
220#102—44no—
221#102—44no—
222#102—44no—
223#102—82no—
224#102—82no—
225#102—82no—
226#102—82no—
227#97—24no—
228#97—42no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8C2×Q812yes—
240C4×C8#4112yes—
241#95—41no—
242#95—41no—
243C24—41no—
244C24—41no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4#12711yescharacteristic
285C22×C4#12711yescharacteristic
286C22×C4#12711yescharacteristic
287C22×C4#12711yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291D8—416no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8C22⋊Q814yes—
309C2×C8AGL(1,ℤ/8)14yes—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42AGL(1,ℤ/8)14yes—
345C42C22×Q811yescharacteristic
346C42C2×#4111yescharacteristic
347C422−1+411yescharacteristic
348C422−1+411yescharacteristic
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23C2×#12711yescharacteristic
364Q8—164no—
365D4—44no—
366D4—44no—
367D4—44no—
368D4—44no—
369D4—44no—
370D4—44no—
371D4—82no—
372D4—82no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—164no—
406C2×C4—164no—
407C2×C4#49612yes—
408C2×C4#49612yes—
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#54711yescharacteristic
413C2×C4D4×Q811yescharacteristic
414C2×C4D4×Q811yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)12yes—
416C2×C4C2×AGL(1,ℤ/8)12yes—
417C2×C4#57512yes—
418C2×C4#57512yes—
419C2×C4#57612yes—
420C2×C4#57712yes—
421C8—24no—
422C8—24no—
423C8—42no—
424C8—42no—
425C8—42no—
426C8—42no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22#254211yescharacteristic
442C22C2×#49611yescharacteristic
443C22#315111yescharacteristic
444C22#315311yescharacteristic
445C22#315311yescharacteristic
446C22#315411yescharacteristic
447C22#315511yescharacteristic
448C4—22no—
449C4—22no—
450C4—41no—
451C4—41no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4#307712yes—
459C4#323512yes—
460C4#334512yes—
461C4#334712yes—
462C4#338412yes—
463C4#338912yes—
464C2—162no—
465C2—162no—
466C2#1967611yescharacteristic
467C2#1967811yescharacteristic
468C2#1972211yescharacteristic
469C2#1975011yescharacteristic
470C2#2852111yescharacteristic
471C2#2854411yescharacteristic
472C2#2854811yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #33297C2semidirect
2. #29190C2semidirect
3. #29190C2semidirect
4. #29139C2semidirect
5. #28508C2semidirect
7. #10678C2semidirect
8. #10677C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
13. #8821C2semidirect
14. #8819C2semidirect
15. #8263C2semidirect
77. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#291902
#291392
#285082
#216902
#106782
#106772
#106722
#106712
#98322
#88212
#88192
#82632