Finite Group Databasebeta

← #323839

Subgroups — #323839

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30305C211yesmaximal, characteristic, complemented
2#29192C211yesmaximal, characteristic, complemented
3#29139C211yesmaximal, characteristic, complemented
4#29139C211yesmaximal, characteristic, complemented
5#28514C211yesmaximal, characteristic, complemented
6#22269C211yesmaximal, characteristic
7#10677C211yesmaximal, characteristic, complemented
8#10677C211yesmaximal, characteristic, complemented
9#10672C211yesmaximal, characteristic, complemented
10#10671C211yesmaximal, characteristic, complemented
11#9837C211yesmaximal, characteristic
12#9832C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8821C211yesmaximal, characteristic, complemented
15#8265C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#478—22no—
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#127C2211yescharacteristic
22#2092—22no—
23#2084—22no—
24#2067—22no—
25#2058—22no—
26#2023—22no—
27#2023—22no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2211yescharacteristic
35#1957C2211yescharacteristic
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1879—22no—
39#1879—22no—
40#1862—22no—
41#1845C2211yescharacteristic
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1834C2211yescharacteristic
51#1834C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1802—22no—
55#1802—22no—
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
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#478—48no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—22no—
84C2×C4⋊C8D412yes—
85C2×C4⋊C8D412yes—
86C2×#102—22no—
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×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122C43C2311yescharacteristic
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#336D412yes—
132#336D412yes—
133#336D412yes—
134#336D412yes—
135#322—22no—
136#322—22no—
137C22×Q8—41no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C23×C4—41no—
145C23×C4—41no—
146C2×D8—28no—
147C2×D8—82no—
148C2×D8—82no—
149C22×C8C2×D412yes—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C4⋊Q8—82no—
156C4⋊Q8—82no—
157C4⋊Q8—82no—
158#127—21no—
159#127—21no—
160#127—21no—
161#127—21no—
162#127—21no—
163#127—21no—
164#127C2×Q812yes—
165#127C2×Q812yes—
166#125—44no—
167C4×D4—44no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—82no—
171C4×D4—82no—
172C4×D4—82no—
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—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209#107—44no—
210#107—82no—
211#106—44no—
212#106—82no—
213#106—82no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216#102—44no—
217#102—44no—
218#102—44no—
219#102—82no—
220#102—82no—
221#102—82no—
222#102—82no—
223#102—82no—
224#102—82no—
225#102—82no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8C2×Q812yes—
237C4×C8#4112yes—
238#95—41no—
239#95—41no—
240C24—41no—
241C24—41no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
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—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4#12711yescharacteristic
282C22×C4#12711yescharacteristic
283C22×C4#12711yescharacteristic
284C22×C4#12711yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288D8—88no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
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×C8C22⋊Q814yes—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
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—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42AGL(1,ℤ/8)14yes—
342C42C22×Q811yescharacteristic
343C42C2×#4111yescharacteristic
344C422−1+411yescharacteristic
345C422−1+411yescharacteristic
346C23—42no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23C2×#12711yescharacteristic
361Q8—164no—
362D4—44no—
363D4—44no—
364D4—44no—
365D4—44no—
366D4—44no—
367D4—44no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373D4—82no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
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—164no—
403C2×C4—164no—
404C2×C4#49612yes—
405C2×C4#49612yes—
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4#53611yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4D4×Q811yescharacteristic
411C2×C4D4×Q811yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)12yes—
413C2×C4#57512yes—
414C2×C4#57612yes—
415C2×C4#57612yes—
416C2×C4#57712yes—
417C2×C4#57712yes—
418C8—24no—
419C8—42no—
420C8—42no—
421C8—42no—
422C8—42no—
423C8—42no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22#254211yescharacteristic
439C22C2×#49611yescharacteristic
440C22#315311yescharacteristic
441C22#315411yescharacteristic
442C22#315411yescharacteristic
443C22#315511yescharacteristic
444C22#315511yescharacteristic
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#308112yes—
456C4#324112yes—
457C4#334712yes—
458C4#335412yes—
459C4#338412yes—
460C4#338912yes—
461C2—162no—
462C2—162no—
463C2#1967311yescharacteristic
464C2#1970111yescharacteristic
465C2#1971811yescharacteristic
466C2#1973511yescharacteristic
467C2#2852511yescharacteristic
468C2#2854111yescharacteristic
469C2#2854411yescharacteristic
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. #30305C2semidirect
2. #29192C2semidirect
3. #29139C2semidirect
4. #29139C2semidirect
5. #28514C2semidirect
7. #10677C2semidirect
8. #10677C2semidirect
9. #10672C2semidirect
10. #10671C2semidirect
13. #8821C2semidirect
14. #8821C2semidirect
15. #8265C2semidirect
77. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#303052
#291922
#291392
#285142
#222692
#106772
#106722
#106712
#98372
#98322
#88212
#82652