Finite Group Databasebeta

← #6806258

Subgroups — #6806258

Order 512.

1228 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36608C211yesmaximal, characteristic
2Q8×#125C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic
4#32239C214yesmaximal
5#32235C214yesmaximal
6#30214C212yesmaximal
7#29223C212yesmaximal
8#29150C211yesmaximal, characteristic
9#28695C211yesmaximal, characteristic
10#28660C212yesmaximal
11#26039C214yesmaximal
12#25004C214yesmaximal
13#22250C212yesmaximal
14#21657C211yesmaximal, characteristic
15#21346C211yesmaximal, characteristic
16C2×#501C2212yes—
17C2×#491C2211yescharacteristic
18#2862C2212yes—
19#2862C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2581—22no—
23#2581C2212yes—
24#2552C2212yes—
25#2552C2212yes—
26#2552C2212yes—
27#2514—22no—
28#2514—22no—
29#2514C2214yes—
30#2507—22no—
31#2507—22no—
32#2507—22no—
33#2356—22no—
34#2356C2212yes—
35Q8×C4⋊C4—22no—
36Q8×C4⋊C4C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×#125C2212yes—
41C42×Q8C2211yescharacteristic
42#2100C2214yes—
43#2099C2214yes—
44#2085C2214yes—
45#2072C2214yes—
46#2072C2214yes—
47#2068C2214yes—
48#2066C2214yes—
49#2026C2214yes—
50#2026C2214yes—
51#2026C2212yes—
52#2026C2212yes—
53#1993C2212yes—
54#1993C2212yes—
55#1993C2212yes—
56#1992C2211yescharacteristic
57#1991C2212yes—
58#1991C2212yes—
59#1989C2214yes—
60#1989C2214yes—
61#1985C2212yes—
62#1960C2211yescharacteristic
63#1960C2212yes—
64#1949C2214yes—
65#1940C2212yes—
66#1933C2214yes—
67#1928C2214yes—
68#1927C2214yes—
69#1921C2214yes—
70#1919C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1916C2214yes—
74#1836C2214yes—
75#1836C2211yescharacteristic
76#1836C2212yes—
77#1803C2214yes—
78#1767C2211yescharacteristic
79#1767C2212yes—
80#1763C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×#125C2311yescharacteristic
86C2×#125C2312yes—
87C2×#125C2312yes—
88C2×#125C2311yescharacteristic
89C2×#125C2311yescharacteristic
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8D412yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2311yescharacteristic
95#501—28no—
96#491—24no—
97C2×#107C2314yes—
98C2×#106C2314yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2314yes—
101C2×#103C2314yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×#97C2312yes—
106C2×#97C2311yescharacteristic
107C2×#97C2312yes—
108C2×#97C2311yescharacteristic
109C2×C4×C8C2312yes—
110#401—24no—
111#400—24no—
112#400—22no—
113#398—22no—
114#398—22no—
115#398—22no—
116#398—22no—
117#398—22no—
118#398—22no—
119#395—22no—
120#391—22no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391C2312yes—
126#391C2314yes—
127#391C2314yes—
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#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2314yes—
146#387—24no—
147#387—24no—
148#387—24no—
149#387—22no—
150#387—22no—
151#387—22no—
152#384—22no—
153#384—22no—
154#384—22no—
155#384C2311yescharacteristic
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#383—24no—
160#383—22no—
161#382—22no—
162#382—22no—
163#382C2311yescharacteristic
164#382C2312yes—
165#382C2312yes—
166#382C2312yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181#376—22no—
182C43C2311yescharacteristic
183C43C2312yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195#336C2312yes—
196#336C2314yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×Q8C2×D411yescharacteristic
200C22×Q8C2412yes—
201C22×C8C2412yes—
202C22×C8C2414yes—
203C22×C8C2412yes—
204C4⋊Q8—24no—
205C4⋊Q8—22no—
206C4⋊Q8—24no—
207C4⋊Q8—22no—
208#125—22no—
209#125—42no—
210#125—42no—
211#125—44no—
212#125—44no—
213#125#4114yes—
214C4×Q8—24no—
215C4×Q8—28no—
216C4×Q8—44no—
217C4×Q8—48no—
218C4×Q8—48no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
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—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2414yes—
250C2×C4⋊C4C2412yes—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2412yes—
263C2×C42C2412yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C2×C42C2411yescharacteristic
267#107—28no—
268#106—28no—
269C4⋊C8—24no—
270C4⋊C8—44no—
271#103—28no—
272#103—44no—
273#103—44no—
274#103—44no—
275#97—22no—
276#97—24no—
277#97—24no—
278#97—22no—
279C4×C8—42no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95C2×D411yescharacteristic
300#95C2×D412yes—
301#95C2×D411yescharacteristic
302#95C2×D412yes—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4C22×D412yes—
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C2511yescharacteristic
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—28no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—48no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4C2×#4114yes—
371C42—24no—
372C42—24no—
373C42—22no—
374C42—24no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—44no—
381C42—44no—
382C42—48no—
383C42—44no—
384C42—44no—
385C42C2×#4112yes—
386C422−1+414yes—
387C422−1+412yes—
388C23C23×D411yescharacteristic
389Q8—28no—
390Q8—44no—
391Q8—44no—
392Q8—88no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4#53612yes—
432C2×C4#53612yes—
433C2×C4#54814yes—
434C2×C4#57514yes—
435C2×C4C22×#4111yescharacteristic
436C2×C4C2×2−1+412yes—
437C2×C4C2×2−1+411yescharacteristic
438C2×C4#58512yes—
439C8—42no—
440C8—44no—
441C8—42no—
442C22C2×#53611yescharacteristic
443C22C2×#54811yescharacteristic
444C22#350011yescharacteristic
445C22C2×#57512yes—
446C22#358612yes—
447C4—24no—
448C4—22no—
449C4—24no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—42no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—84no—
461C4—82no—
462C4#351812yes—
463C4#355512yes—
464C2#3917211yescharacteristic
465C2#6086111yescharacteristic
466C2#6088411yescharacteristic
467C2#6132312yes—
468C2#6258712yes—
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
#366082
Q8×#1252
#332962
#322392
#322352
#302142
#292232
#291502
#286952
#286602
#260392
#250042
#222502
#216572
#213462