Finite Group Databasebeta

← #6436342

Subgroups — #6436342

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#30232C218yesmaximal, complemented
4#30230C212yesmaximal, complemented
5#30229C212yesmaximal, complemented
6#30229C212yesmaximal, complemented
7#30228C212yesmaximal, complemented
8#23749C214yesmaximal
9#23748C214yesmaximal
10#21324C211yesmaximal, characteristic
11#21322C211yesmaximal, characteristic
12#21318C211yesmaximal, characteristic
13C2×#1990C211yesmaximal, characteristic
14#20366C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2868C2212yes—
20#2862C2212yes—
21#2859C2212yes—
22#2859C2212yes—
23#2660C2214yes—
24#2647C2214yes—
25#2571—21no—
26#2571—21no—
27#2567—21no—
28#2567—21no—
29#2552—21no—
30#2552—21no—
31#2546—21no—
32#2546—21no—
33#2502—22no—
34#2442—22no—
35#2442—22no—
36#2434—22no—
37#2317C2212yes—
38#2317C2212yes—
39#2317C2212yes—
40#2316C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2101C2218yes—
45#2101C2218yes—
46#2101C2218yes—
47#2101C2218yes—
48#2100C2218yes—
49#2100C2218yes—
50#2099C2218yes—
51#2099C2218yes—
52#1993C2214yes—
53#1992C2214yes—
54#1991C2214yes—
55#1990C2211yescharacteristic
56#1990C2212yes—
57#1990C2212yes—
58#1990C2212yes—
59#1990C2211yescharacteristic
60#1990C2211yescharacteristic
61#1990C2212yes—
62#1990C2212yes—
63#1990C2212yes—
64#1990C2211yescharacteristic
65#1990C2214yes—
66#1746C2214yes—
67#1744C2212yes—
68#1744C2212yes—
69#1736C2214yes—
70#1736C2214yes—
71#1735C2214yes—
72C2×#340C2214yes—
73#1729C2212yes—
74#1729C2212yes—
75C23×Q8—21no—
76C22×M16C2312yes—
77C23×C8C2312yes—
78#533—24no—
79#533—24no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×#125—22no—
91C2×#125—22no—
92C2×#125C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8—21no—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C42C2311yescharacteristic
107C4×M16—24no—
108C2×#97C2312yes—
109C2×C4×C8—24no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114#400—24no—
115#400—24no—
116#398—24no—
117#398—24no—
118#398—24no—
119#398—24no—
120#395—24no—
121#395—24no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2314yes—
137#389C2314yes—
138#389C2314yes—
139#389C2314yes—
140#385—24no—
141#385—24no—
142#385—24no—
143#384—22no—
144#384—22no—
145#384—22no—
146#384—21no—
147#384—21no—
148#384C2314yes—
149#384C2314yes—
150#384C2312yes—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2312yes—
154#382—22no—
155#382—22no—
156#382—22no—
157#382C2314yes—
158#382C2314yes—
159#382C2312yes—
160#380—21no—
161#380—21no—
162#380—24no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C22⋊C4D412yes—
170C4×C22⋊C4D412yes—
171#376C2312yes—
172C43C2312yes—
173#340C2314yes—
174#340C2318yes—
175#340C2318yes—
176#340C2318yes—
177#340C2314yes—
178#340C2318yes—
179#340C2318yes—
180#340C2314yes—
181#340C2314yes—
182#340C2318yes—
183#336C2314yes—
184#336C2314yes—
185#336C2314yes—
186#336C2314yes—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—44no—
192C22×Q8—82no—
193C22×Q8—82no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C22×C8—24no—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8C2412yes—
206C22×C8C2412yes—
207C22×C8C2414yes—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—44no—
217C4⋊Q8—44no—
218C4⋊Q8—84no—
219C4⋊Q8—84no—
220#125—24no—
221#125—24no—
222#125—84no—
223#125—84no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C4×Q8—44no—
227C4×Q8—44no—
228#117—24no—
229#117—24no—
230#117—24no—
231#117—24no—
232#117—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
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⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—82no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4114yes—
286C2×C42#4114yes—
287C2×C42C2412yes—
288C2×C42C2412yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#97—24no—
292C4×C8—24no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310C24C22×D411yescharacteristic
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—42no—
316C2×Q8—44no—
317C2×Q8—42no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×Q8—816no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4#12414yes—
341C22×C4#12414yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C42−1+414yes—
354C22×C42−1+414yes—
355C22×C4C2511yescharacteristic
356M16—48no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—48no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—44no—
415C42—44no—
416C42#12414yes—
417C42#12414yes—
418C42C2×#4112yes—
419C42C2×#4112yes—
420C42C2×#4112yes—
421C42C2×#4112yes—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23C2×#12412yes—
427C23#53612yes—
428C23#53612yes—
429C23C23×D411yescharacteristic
430Q8—88no—
431Q8—84no—
432Q8—88no—
433Q8—84no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—88no—
471C2×C4—164no—
472C2×C4C2×#12412yes—
473C2×C4C2×#12412yes—
474C2×C4C2×#12412yes—
475C2×C4C2×#12412yes—
476C2×C4C2×#12412yes—
477C2×C4#53612yes—
478C2×C4#53612yes—
479C2×C4#54118yes—
480C2×C4#54118yes—
481C2×C4#57612yes—
482C2×C4#57712yes—
483C2×C4#57712yes—
484C2×C4#57812yes—
485C2×C4C22×#4111yescharacteristic
486C2×C4C22×#4111yescharacteristic
487C2×C4C2×2−1+412yes—
488C2×C4C2×2−1+412yes—
489C8—42no—
490C8—42no—
491C8—44no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22#264214yes—
499C22C22×#12411yescharacteristic
500C22C2×#53611yescharacteristic
501C22C2×#53611yescharacteristic
502C22C2×#57611yescharacteristic
503C22C2×#57711yescharacteristic
504C22C2×#57711yescharacteristic
505C22C2×#57811yescharacteristic
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—84no—
513C4—82no—
514C4—82no—
515C4—84no—
516C4—82no—
517C4—82no—
518C4#264214yes—
519C4#313614yes—
520C4#313714yes—
521C4C2×#54112yes—
522C4C2×#54112yes—
523C2—24no—
524C2C2×#264211yescharacteristic
525C2C2×#313611yescharacteristic
526C2C2×#313711yescharacteristic
527C2#6087711yescharacteristic
528C2#6087811yescharacteristic
529C2#6088011yescharacteristic
530C2#6088111yescharacteristic
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. #30232C2semidirect
4. #30230C2semidirect
5. #30229C2semidirect
6. #30229C2semidirect
7. #30228C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#352812
#302322
#302302
#302292
#302282
#237492
#237482
#213242
#213222
#213182
C2×#19902
#203662