Finite Group Databasebeta

← #6498026

Subgroups — #6498026

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35330C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#33298C211yesmaximal, characteristic, complemented
4#33296C211yesmaximal, characteristic, complemented
5#33294C211yesmaximal, characteristic, complemented
6#33293C211yesmaximal, characteristic, complemented
7#30232C214yesmaximal, complemented
8#29229C214yesmaximal, complemented
9#29224C212yesmaximal, complemented
10#29223C212yesmaximal, complemented
11#27971C214yesmaximal
12#27969C212yesmaximal
13#27968C212yesmaximal
14#21843C214yesmaximal
15#20465C211yesmaximal, characteristic
16C2×#544C2212yes—
17C2×#433C2212yes—
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2862C2211yescharacteristic
21#2862C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2660C2212yes—
26#2660C2212yes—
27#2647C2212yes—
28#2628C2212yes—
29#2550—22no—
30#2550—22no—
31#2549—22no—
32#2549—22no—
33#2511—22no—
34#2492—22no—
35#2451—22no—
36#2423—22no—
37#2331C2212yes—
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#1993C2214yes—
53#1993C2212yes—
54#1993C2212yes—
55#1993C2214yes—
56#1993C2212yes—
57#1993C2212yes—
58#1992C2214yes—
59#1991C2214yes—
60#1990C2214yes—
61#1990C2214yes—
62#1989C2214yes—
63#1989C2214yes—
64#1988C2214yes—
65#1987C2214yes—
66#1836C2212yes—
67#1836C2212yes—
68#1836C2212yes—
69#1836C2212yes—
70#1836C2214yes—
71#1835C2214yes—
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×C4⋊Q8C2312yes—
90C2×#125—22no—
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—22no—
112#400—24no—
113#399—22no—
114#398—24no—
115#398—22no—
116#398—22no—
117#398—22no—
118#398—22no—
119#395—22no—
120#395—22no—
121#395—22no—
122#395—22no—
123#395—22no—
124#393—22no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391—22no—
129#391—22no—
130#391—22no—
131#391—22no—
132#391—22no—
133#391—24no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389C2314yes—
143#389C2314yes—
144#388—24no—
145#387—24no—
146#387—24no—
147#385—22no—
148#385—22no—
149#384—22no—
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#384C2312yes—
155#382—22no—
156#382C2312yes—
157#382C2312yes—
158#382C2312yes—
159#382C2312yes—
160#382C2311yescharacteristic
161#382C2311yescharacteristic
162#382C2311yescharacteristic
163#382C2311yescharacteristic
164#382C2312yes—
165#380—22no—
166#380—22no—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2314yes—
173C4×C22⋊C4D412yes—
174C4×C22⋊C4D412yes—
175#376C2311yescharacteristic
176#376C2311yescharacteristic
177C2×#95—22no—
178C43C2311yescharacteristic
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
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#336C2314yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—28no—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—48no—
212C4⋊Q8—84no—
213#125—28no—
214#125—28no—
215#125—48no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218C22⋊Q8—84no—
219C4×Q8—84no—
220#117—24no—
221#117—28no—
222#117#4118yes—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
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—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2414yes—
252C2×C4⋊C4C2414yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D412yes—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—28no—
273C22⋊C8—28no—
274#97—24no—
275#97—24no—
276#97—24no—
277#97—24no—
278C4×C8—24no—
279C4×C8—24no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95C2×D412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301C24C22×D411yescharacteristic
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—84no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—82no—
312C2×Q8—82no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C2×#4114yes—
337C22×C4C2511yescharacteristic
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—44no—
353C4⋊C4—44no—
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—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4#12418yes—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—44no—
376C22⋊C4—84no—
377C22⋊C4#12418yes—
378C42—24no—
379C42—22no—
380C42—22no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—84no—
387C42C2×#4114yes—
388C42C2×#4114yes—
389C422−1+414yes—
390C422−1+414yes—
391C23—22no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23C2×#12412yes—
396C23C23×D411yescharacteristic
397Q8—88no—
398Q8—164no—
399Q8—164no—
400Q8—164no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—84no—
438C2×C4—164no—
439C2×C4C2×#12412yes—
440C2×C4C2×#12414yes—
441C2×C4C2×#12414yes—
442C2×C4#53614yes—
443C2×C4#53612yes—
444C2×C4#53612yes—
445C2×C4C22×#4112yes—
446C2×C4C2×2−1+412yes—
447C2×C4C2×2−1+412yes—
448C8—44no—
449C8—44no—
450C22—24no—
451C22—22no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22C22×#12411yescharacteristic
456C22C2×#53611yescharacteristic
457C22C2×#53611yescharacteristic
458C22#358512yes—
459C22#358612yes—
460C4—24no—
461C4—24no—
462C4—24no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—84no—
475C4#350914yes—
476C2—42no—
477C2#3532011yescharacteristic
478C2#6062512yes—
479C2#6087012yes—
480C2#6088311yescharacteristic
481C2#6088411yescharacteristic
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. #33298C2semidirect
4. #33296C2semidirect
5. #33294C2semidirect
6. #33293C2semidirect
7. #30232C2semidirect
8. #29229C2semidirect
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
#352832
#332982
#332962
#332942
#332932
#302322
#292292
#292242
#292232
#279712
#279692
#279682
#218432
#204652