Finite Group Databasebeta

← #6490718

Subgroups — #6490718

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#33155C214yesmaximal
4#30104C214yesmaximal, complemented
5#29282C214yesmaximal, complemented
6#29226C211yesmaximal, characteristic, complemented
7#29225C211yesmaximal, characteristic, complemented
8#28695C211yesmaximal, characteristic, complemented
9#28695C211yesmaximal, characteristic, complemented
10#27968C212yesmaximal
11#27623C214yesmaximal
12#25819C212yesmaximal
13#22251C212yesmaximal, complemented
14#22250C212yesmaximal, complemented
15#20783C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C2×#501C2212yes—
18C2×#501C2212yes—
19#2868C2211yescharacteristic
20#2862C2211yescharacteristic
21#2857C2212yes—
22#2647C2212yes—
23#2628C2212yes—
24#2511—22no—
25#2492—22no—
26#2451—22no—
27#2423—22no—
28#2370—22no—
29#2370—22no—
30#2345—22no—
31#2345—22no—
32#2331C2212yes—
33#2330C2212yes—
34#2330C2212yes—
35#2316C2212yes—
36#2314C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×#125C2211yescharacteristic
39C4×#125C2211yescharacteristic
40C4×#125C2211yescharacteristic
41C4×#125C2211yescharacteristic
42#2285C2211yescharacteristic
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2100C2214yes—
46#2100C2214yes—
47#2099C2214yes—
48#2099C2214yes—
49#2085C2214yes—
50#2085C2214yes—
51#2059C2214yes—
52#2059C2214yes—
53#1993C2212yes—
54#1993C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1988C2214yes—
58#1987C2214yes—
59#1949C2214yes—
60#1949C2214yes—
61#1921C2214yes—
62#1921C2214yes—
63#1920C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1918C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1840C2212yes—
72#1836C2212yes—
73#1836C2212yes—
74#1832C2212yes—
75#1825C2214yes—
76#1825C2214yes—
77#1818C2212yes—
78#1798C2212yes—
79#1767C2211yescharacteristic
80#1767C2211yescharacteristic
81#1749C2212yes—
82#1745C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—22no—
89C2×#125C2312yes—
90C2×#125C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×C22⋊Q8—22no—
93C2×C4×Q8—22no—
94C2×#117C2312yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2312yes—
99C22×C42C2311yescharacteristic
100#501—28no—
101#501—28no—
102C2×#107C2314yes—
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×#106C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×C22⋊C8C2314yes—
109C2×#97C2312yes—
110C2×#97C2312yes—
111#400—22no—
112#400—24no—
113#399—22no—
114#398—24no—
115#395—22no—
116#393—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#391—22no—
121#391—24no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389C2312yes—
131#389C2312yes—
132#389C2312yes—
133#389C2312yes—
134#389C2314yes—
135#389C2314yes—
136#388—24no—
137#387—24no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—24no—
147#384—22no—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#382C2311yescharacteristic
152#382C2311yescharacteristic
153#382C2312yes—
154#382C2312yes—
155#380—22no—
156#380—22no—
157#380—22no—
158#380—22no—
159#380C2314yes—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181C43C2311yescharacteristic
182C43C2311yescharacteristic
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#336C2314yes—
196#336C2314yes—
197#336C2312yes—
198#336C2312yes—
199#336C2314yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C23×C4—22no—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—24no—
208C4⋊Q8—24no—
209C4⋊Q8—28no—
210C4⋊Q8—84no—
211#125—216no—
212#125—24no—
213#125—24no—
214#125—22no—
215#125—22no—
216#125#4114yes—
217#125#4114yes—
218C22⋊Q8—84no—
219C4×Q8—84no—
220#117—24no—
221#117—24no—
222#117—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
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⋊C4C2414yes—
250C2×C4⋊C4C2414yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D412yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277#107—28no—
278#107—28no—
279#106—28no—
280#106—28no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C22⋊C8—28no—
284#97—24no—
285#97—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—24no—
302#95—22no—
303#95C2×D412yes—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D412yes—
307C24C22×D411yescharacteristic
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—84no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C42−1+414yes—
338C22×C4C2511yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—28no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4C2×#4114yes—
374C4⋊C4C2×#4114yes—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—44no—
379C22⋊C4—84no—
380C42—24no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—44no—
391C42—82no—
392C42—82no—
393C42—82no—
394C42—82no—
395C42—84no—
396C42C2×#4112yes—
397C42C2×#4112yes—
398C422−1+414yes—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23#53612yes—
404C23C23×D411yescharacteristic
405Q8—88no—
406Q8—164no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4#52514yes—
449C2×C4#53612yes—
450C2×C4#53612yes—
451C2×C4#53612yes—
452C2×C4C22×#4111yescharacteristic
453C2×C4C22×#4111yescharacteristic
454C2×C4C2×2−1+412yes—
455C2×C4C2×2−1+412yes—
456C8—44no—
457C8—44no—
458C22—22no—
459C22—24no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22C2×#52511yescharacteristic
464C22C2×#53611yescharacteristic
465C22C2×#53611yescharacteristic
466C22#358512yes—
467C22#358612yes—
468C4—24no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—44no—
474C4—44no—
475C4—41no—
476C4—41no—
477C4—42no—
478C4—42no—
479C4—84no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4#352312yes—
484C4#352312yes—
485C2—42no—
486C2#3524811yescharacteristic
487C2#6056712yes—
488C2#6088311yescharacteristic
489C2#6088411yescharacteristic
490C2#6088712yes—
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
4. #30104C2semidirect
5. #29282C2semidirect
6. #29226C2semidirect
7. #29225C2semidirect
8. #28695C2semidirect
9. #28695C2semidirect
13. #22251C2semidirect
14. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#352832
#342172
#331552
#301042
#292822
#292262
#292252
#286952
#279682
#276232
#258192
#222512
#222502
#207832