Finite Group Databasebeta

← #6806411

Subgroups — #6806411

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic
2#36560C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic
4#32082C218yesmaximal
5#30220C212yesmaximal
6#29191C212yesmaximal
7#28748C211yesmaximal, characteristic
8#28748C212yesmaximal
9#28512C211yesmaximal, characteristic
10#25647C214yesmaximal
11#25276C214yesmaximal
12#22218C212yesmaximal
13C4×#487C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×#557—21no—
16C2×#557C2212yes—
17C2×#487C2211yescharacteristic
18C2×#487C2212yes—
19C2×#487C2214yes—
20C2×#475C2214yes—
21#2868C2211yescharacteristic
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2859C2212yes—
25#2568—21no—
26#2565—22no—
27#2552—22no—
28#2552—21no—
29#2552—21no—
30#2552—22no—
31#2546—21no—
32#2546C2212yes—
33#2546C2214yes—
34#2546C2212yes—
35#2506—22no—
36#2506C2214yes—
37#2365—21no—
38#2365C2212yes—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C42×Q8C2211yescharacteristic
46#2096C2214yes—
47#2066C2218yes—
48#2043C2218yes—
49#2043C2218yes—
50#2043C2218yes—
51#2043C2214yes—
52#1990C2211yescharacteristic
53#1984C2212yes—
54#1981C2214yes—
55#1981C2218yes—
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1956C2212yes—
60#1956C2212yes—
61#1956C2214yes—
62#1939C2212yes—
63#1933C2214yes—
64#1927C2218yes—
65#1916C2214yes—
66#1865C2218yes—
67#1865C2218yes—
68#1844C2212yes—
69#1834C2211yescharacteristic
70C4×C4⋊C8C2212yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#103C2214yes—
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75#557—48no—
76#557—48no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2314yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#125—24no—
89C2×#125—22no—
90C2×#125—24no—
91C2×#125—22no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—21no—
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8D411yescharacteristic
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2311yescharacteristic
101#487—24no—
102#487—28no—
103#487—48no—
104#475—48no—
105C2×#106C2318yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2314yes—
110C2×#103C2318yes—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#103C2314yes—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2312yes—
120#400—24no—
121#398—24no—
122#398—24no—
123#398—24no—
124#398—24no—
125#398—24no—
126#391—22no—
127#391—22no—
128#391—21no—
129#391—22no—
130#391C2314yes—
131#391C2312yes—
132#389—22no—
133#389—24no—
134#389—22no—
135#389—24no—
136#389—21no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—22no—
144#389C2314yes—
145#389C2312yes—
146#389C2314yes—
147#389C2314yes—
148#387—24no—
149#387—24no—
150#384—21no—
151#384—22no—
152#384—22no—
153#384—21no—
154#384—21no—
155#384—21no—
156#384—22no—
157#384—22no—
158#384—21no—
159#384—21no—
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#384C2312yes—
163#384C2312yes—
164#382—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C4⋊C4C2312yes—
179C43C2312yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2314yes—
185#336C2314yes—
186#336C2314yes—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8C2×D411yescharacteristic
190C22×Q8C2412yes—
191C22×Q8C2414yes—
192C22×C8C2412yes—
193C22×C8C2414yes—
194C22×C8C2412yes—
195C4⋊Q8—24no—
196C4⋊Q8—22no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—22no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—48no—
203C4⋊Q8—44no—
204C4⋊Q8—48no—
205C4⋊Q8—48no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8#4114yes—
209#125—416no—
210#125—48no—
211#125—416no—
212#125—48no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—28no—
216C4×Q8—48no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—44no—
221C4×Q8SD1618yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2412yes—
273C2×C42C2412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277#106—48no—
278C4⋊C8—24no—
279C4⋊C8—44no—
280#103—216no—
281#103—28no—
282#103—44no—
283#103—48no—
284#103—48no—
285#103—48no—
286#103—48no—
287C4×C8—22no—
288C4×C8—24no—
289C4×C8—24no—
290C4×C8—22no—
291C4×C8—42no—
292#95—22no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—28no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—28no—
316C2×Q8—44no—
317C2×Q8—42no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—42no—
321C2×Q8C2×SD1614yes—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2511yescharacteristic
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—44no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—82no—
381C4⋊C4—84no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4C2×SD1614yes—
385C4⋊C4C2×SD1618yes—
386C4⋊C4C2×#4114yes—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—24no—
392C42—24no—
393C42—22no—
394C42—24no—
395C42—48no—
396C42—44no—
397C42—48no—
398C42—44no—
399C42C2×SD1618yes—
400C42C2×SD1614yes—
401C42#13714yes—
402C42C2×#4112yes—
403C42H(5,𝔽2)14yes—
404C422−1+412yes—
405C23C23×D411yescharacteristic
406Q8—28no—
407Q8—48no—
408Q8—44no—
409Q8—48no—
410Q8—44no—
411Q8—48no—
412Q8—88no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4#53512yes—
448C2×C4#53512yes—
449C2×C4#54812yes—
450C2×C4#54812yes—
451C2×C4C22×SD1614yes—
452C2×C4C22×SD1612yes—
453C2×C4C22×SD1614yes—
454C2×C4C22×SD1612yes—
455C2×C4C22×SD1612yes—
456C2×C4C2×#13712yes—
457C2×C4C2×#13712yes—
458C2×C4C2×#13712yes—
459C2×C4#57812yes—
460C2×C4#57812yes—
461C2×C4C22×#4111yescharacteristic
462C2×C4C2×H(5,𝔽2)12yes—
463C2×C4C2×2−1+411yescharacteristic
464C2×C4#58512yes—
465C8—42no—
466C8—44no—
467C8—42no—
468C22C2×#53511yescharacteristic
469C22C2×#54811yescharacteristic
470C22#350011yescharacteristic
471C22C23×SD1611yescharacteristic
472C22C22×#13711yescharacteristic
473C22C2×#57811yescharacteristic
474C22C2×#57811yescharacteristic
475C4—24no—
476C4—22no—
477C4—22no—
478C4—24no—
479C4—22no—
480C4—42no—
481C4—44no—
482C4—42no—
483C4—84no—
484C4—82no—
485C4—82no—
486C4—84no—
487C4—82no—
488C4#320614yes—
489C4#335914yes—
490C4#337414yes—
491C4#351712yes—
492C4#355412yes—
493C2#3917411yescharacteristic
494C2C2×#320611yescharacteristic
495C2#6082111yescharacteristic
496C2C2×#335911yescharacteristic
497C2C2×#337411yescharacteristic
498C2#6252611yescharacteristic
499C2#6253611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365932
#365602
#332252
#320822
#302202
#291912
#287482
#285122
#256472
#252762
#222182
C4×#4872
#213392