Finite Group Databasebeta

← #6498028

Subgroups — #6498028

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33295C211yesmaximal, characteristic, complemented
6#33293C211yesmaximal, characteristic, complemented
7#30232C214yesmaximal, complemented
8#29228C214yesmaximal, complemented
9#29224C212yesmaximal, complemented
10#29223C212yesmaximal, complemented
11#27970C214yesmaximal
12#27969C212yesmaximal
13#27968C212yesmaximal
14#21843C214yesmaximal
15#20465C211yesmaximal, characteristic
16C2×#544C2212yes—
17C2×#433C2212yes—
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2862C2211yescharacteristic
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2660C2212yes—
26#2660C2212yes—
27#2647C2212yes—
28#2629C2212yes—
29#2550—22no—
30#2550—22no—
31#2549—22no—
32#2549—22no—
33#2507—22no—
34#2497—22no—
35#2446—22no—
36#2429—22no—
37#2330C2212yes—
38#2317C2212yes—
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41#2285C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2101C2214yes—
46#2101C2214yes—
47#2101C2214yes—
48#2100C2214yes—
49#2100C2214yes—
50#2099C2214yes—
51#2099C2214yes—
52#1993C2212yes—
53#1993C2212yes—
54#1993C2214yes—
55#1993C2212yes—
56#1993C2212yes—
57#1992C2214yes—
58#1992C2214yes—
59#1991C2214yes—
60#1991C2214yes—
61#1990C2214yes—
62#1989C2214yes—
63#1989C2214yes—
64#1988C2214yes—
65#1987C2214yes—
66#1836C2214yes—
67#1836C2212yes—
68#1836C2212yes—
69#1836C2212yes—
70#1836C2212yes—
71#1834C2214yes—
72#1825C2214yes—
73#1825C2214yes—
74#1824C2214yes—
75#1823C2214yes—
76#1768C2212yes—
77#1767C2212yes—
78#1750C2212yes—
79#1749C2212yes—
80#1746C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#544—216no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×#125C2312yes—
91C2×#125C2312yes—
92C2×#125C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C4×Q8—22no—
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102#433—28no—
103C2×C4⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2312yes—
106C2×#97C2312yes—
107C2×#97C2312yes—
108C2×#97C2312yes—
109C2×C4×C8C2312yes—
110C2×C4×C8C2312yes—
111#400—24no—
112#398—24no—
113#398—24no—
114#398—22no—
115#398—22no—
116#398—22no—
117#398—22no—
118#396—24no—
119#395—22no—
120#395—22no—
121#395—22no—
122#395—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—24no—
140#389C2314yes—
141#389C2314yes—
142#388—24no—
143#387—24no—
144#387—24no—
145#385—22no—
146#385—22no—
147#384—22no—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—21no—
155#382—21no—
156#382C2312yes—
157#382C2312yes—
158#382C2312yes—
159#382C2311yescharacteristic
160#382C2311yescharacteristic
161#382C2311yescharacteristic
162#382C2311yescharacteristic
163#382C2312yes—
164#380—22no—
165#380—21no—
166#380—21no—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4D412yes—
176#376C2311yescharacteristic
177#376C2311yescharacteristic
178C2×#95—21no—
179C2×#95—21no—
180C43C2311yescharacteristic
181C43C2311yescharacteristic
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C23×C4—22no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—28no—
212C4⋊Q8—48no—
213C4⋊Q8—84no—
214#125—28no—
215#125—28no—
216#125—28no—
217#125—48no—
218C22⋊Q8—48no—
219C22⋊Q8—48no—
220C22⋊Q8—84no—
221C4×Q8—84no—
222#117—24no—
223#117—28no—
224#117#4118yes—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C4⋊C8—28no—
275C22⋊C8—28no—
276#97—24no—
277#97—24no—
278#97—24no—
279#97—24no—
280C4×C8—24no—
281C4×C8—24no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305C24C22×D411yescharacteristic
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D412yes—
339C22×C4C2×#4114yes—
340C22×C4C2511yescharacteristic
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4#12418yes—
376C22⋊C4—24no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4#12418yes—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—44no—
389C42—84no—
390C42C2×#4114yes—
391C42C2×#4114yes—
392C422−1+414yes—
393C422−1+414yes—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—21no—
398C23C2×#12412yes—
399C23C23×D411yescharacteristic
400Q8—88no—
401Q8—164no—
402Q8—164no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—164no—
442C2×C4C2×#12412yes—
443C2×C4C2×#12414yes—
444C2×C4C2×#12414yes—
445C2×C4#53612yes—
446C2×C4#53612yes—
447C2×C4#53614yes—
448C2×C4C22×#4112yes—
449C2×C4C2×2−1+412yes—
450C2×C4C2×2−1+412yes—
451C8—44no—
452C8—44no—
453C22—24no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—42no—
458C22C22×#12411yescharacteristic
459C22C2×#53611yescharacteristic
460C22C2×#53611yescharacteristic
461C22#358512yes—
462C22#358612yes—
463C4—22no—
464C4—22no—
465C4—24no—
466C4—24no—
467C4—24no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—84no—
478C4#350914yes—
479C2—42no—
480C2#3532011yescharacteristic
481C2#6062512yes—
482C2#6086811yescharacteristic
483C2#6086911yescharacteristic
484C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33297C2semidirect
4. #33296C2semidirect
5. #33295C2semidirect
6. #33293C2semidirect
7. #30232C2semidirect
8. #29228C2semidirect
9. #29224C2semidirect
10. #29223C2semidirect

Maximal subgroups

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

subgroupindex
#353302
#352842
#332972
#332962
#332952
#332932
#302322
#292282
#292242
#292232
#279702
#279692
#279682
#218432
#204652