Finite Group Databasebeta

← #6798307

Subgroups — #6798307

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31687C218yesmaximal, complemented
5#29876C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#27875C212yesmaximal, complemented
9#26445C214yesmaximal, complemented
10#26299C214yesmaximal
11#25812C212yesmaximal, complemented
12#21835C212yesmaximal, complemented
13C4×#498C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#533C2212yes—
16C2×#500C2212yes—
17C2×#498C2211yescharacteristic
18C2×#431C2212yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2857C2212yes—
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2552C2212yes—
26#2542—21no—
27#2538—21no—
28#2538—21no—
29#2497C2214yes—
30#2492—22no—
31#2441C2214yes—
32#2439—22no—
33#2429—22no—
34#2423—22no—
35#2369C2212yes—
36#2365—21no—
37#2355—21no—
38#2354—21no—
39#2354C2212yes—
40Q8×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2100C2214yes—
46#2086C2218yes—
47#2083C2214yes—
48#2077C2218yes—
49#2075C2218yes—
50#2059C2218yes—
51#1990C2212yes—
52#1990C2214yes—
53#1990C2212yes—
54#1990C2212yes—
55#1990C2212yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1989C2218yes—
59#1987C2214yes—
60#1954C2212yes—
61#1947C2214yes—
62#1941C2214yes—
63#1939C2212yes—
64#1923C2214yes—
65#1920C2218yes—
66#1854C2218yes—
67#1844C2211yescharacteristic
68#1834C2214yes—
69#1834C2212yes—
70#1825C2218yes—
71C4×#106C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75#533—28no—
76#533—44no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125C2314yes—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C2×#117—22no—
98C2×#117C2314yes—
99C22×C42C2312yes—
100#500—44no—
101#498—24no—
102#431—44no—
103C2×#107C2318yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C4×C8C2312yes—
111C2×C4×C8C2312yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2312yes—
115#400—24no—
116#399—24no—
117#398—24no—
118#396—24no—
119#396—24no—
120#393—24no—
121#391—21no—
122#391—21no—
123#391—22no—
124#391—22no—
125#391—24no—
126#390—22no—
127#390—22no—
128#389—22no—
129#389—22no—
130#389—21no—
131#389—21no—
132#389—22no—
133#389—22no—
134#389—24no—
135#389C2312yes—
136#389C2312yes—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#387—24no—
142#387—24no—
143#384—22no—
144#384—21no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380C2314yes—
160#380C2314yes—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C22⋊C4D412yes—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4C2312yes—
176C43C2311yescharacteristic
177#340C2318yes—
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2318yes—
182#340C2318yes—
183#340C2314yes—
184#336C2314yes—
185#336C2312yes—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—21no—
189C22×Q8—22no—
190C22×D4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2412yes—
194C22×C8C2414yes—
195C22×C8C2412yes—
196C22×C8C2412yes—
197C4⋊Q8—24no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—82no—
205C4⋊Q8—82no—
206C4⋊Q8—82no—
207C4⋊Q8#4114yes—
208#125—28no—
209#125—44no—
210C22⋊Q8—48no—
211C22⋊Q8—48no—
212C22⋊Q8—84no—
213C22⋊Q8—84no—
214C4×Q8—48no—
215C4×Q8—48no—
216C4×Q8—44no—
217C4×Q8—44no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221#117—28no—
222#117—44no—
223#117—84no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2412yes—
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—48no—
279#106—28no—
280#106—28no—
281C4⋊C8—24no—
282C4⋊C8—44no—
283C22⋊C8—28no—
284C22⋊C8—44no—
285C4×C8—24no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—24no—
289C4×C8—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95C2×D412yes—
301#95C2×D412yes—
302#95C2×D412yes—
303C24C22×D412yes—
304C2×Q8—24no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C2511yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C82−1+414yes—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—84no—
382C4⋊C4—84no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4C2×#4114yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—48no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—24no—
396C42—24no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—48no—
403C42—44no—
404C42—44no—
405C42—84no—
406C42#13514yes—
407C42#13714yes—
408C42C2×#4112yes—
409C42H(5,𝔽2)12yes—
410C422−1+414yes—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23C23×D411yescharacteristic
419Q8—48no—
420Q8—88no—
421Q8—84no—
422Q8—164no—
423D4—44no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—48no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4#53612yes—
462C2×C4#53612yes—
463C2×C4#54612yes—
464C2×C4#54612yes—
465C2×C4C2×#13512yes—
466C2×C4C2×#13512yes—
467C2×C4C2×#13512yes—
468C2×C4C2×#13712yes—
469C2×C4C2×#13712yes—
470C2×C4C2×#13712yes—
471C2×C4#57612yes—
472C2×C4#57712yes—
473C2×C4C22×#4111yescharacteristic
474C2×C4C2×H(5,𝔽2)11yescharacteristic
475C2×C4C2×2−1+412yes—
476C2×C4C2×2−1+412yes—
477C8—24no—
478C8—42no—
479C8—44no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22C2×#53611yescharacteristic
488C22C2×#54611yescharacteristic
489C22#349911yescharacteristic
490C22C22×#13511yescharacteristic
491C22C22×#13711yescharacteristic
492C22C2×#57611yescharacteristic
493C22C2×#57711yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4—82no—
505C4—84no—
506C4#323114yes—
507C4#330514yes—
508C4#333114yes—
509C4#352312yes—
510C4#355012yes—
511C2—82no—
512C2#3895411yescharacteristic
513C2C2×#323111yescharacteristic
514C2#6087711yescharacteristic
515C2C2×#330511yescharacteristic
516C2C2×#333111yescharacteristic
517C2#6247011yescharacteristic
518C2#6248011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #31687C2semidirect
5. #29876C2semidirect
6. #29218C2semidirect
7. #28689C2semidirect
8. #27875C2semidirect
9. #26445C2semidirect
11. #25812C2semidirect
12. #21835C2semidirect
13. C4×#498C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361342
#332892
#316872
#298762
#292182
#286892
#278752
#264452
#262992
#258122
#218352
C4×#4982
#212642