Finite Group Databasebeta

← #218265

Subgroups — #218265

Order 512.

593 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22307C211yesmaximal, characteristic, complemented
2#22008C211yesmaximal, characteristic
3C4×#474C211yesmaximal, characteristic
4#9864C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9595C211yesmaximal, characteristic, complemented
7#9594C211yesmaximal, characteristic, complemented
8#9556C211yesmaximal, characteristic
9#9554C211yesmaximal, characteristic
10#9542C211yesmaximal, characteristic
11#9541C211yesmaximal, characteristic
12#9536C211yesmaximal, characteristic, complemented
13#9533C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic
15#8264C211yesmaximal, characteristic, complemented
16C2×#474C411yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C42×D4C2211yescharacteristic
19#1988C411yescharacteristic
20#1984—21no—
21#1984—21no—
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1947C411yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1923—21no—
30#1923—21no—
31#1918C411yescharacteristic
32#1918C411yescharacteristic
33#1916—21no—
34#1916—21no—
35#1862C411yescharacteristic
36#1862C411yescharacteristic
37#1831C2211yescharacteristic
38#1817—21no—
39#1802C411yescharacteristic
40#1797—21no—
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73#474C818yescomplemented
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#107C2×C411yescharacteristic
77C2×#107C2×C411yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2×C411yescharacteristic
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#389—21no—
108#389C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111C4×C4⋊C4—21no—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C4⋊C4C2311yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C43C2311yescharacteristic
121#340C2×C411yescharacteristic
122#340C2×C411yescharacteristic
123#340C2×C411yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#336—21no—
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—41no—
146#334—41no—
147D4⋊C8—41no—
148D4⋊C8—41no—
149#322—41no—
150C82—41no—
151C22×Q8—21no—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—22no—
171C4⋊Q8C2×C814yes—
172C4×Q8—81no—
173C4×Q8—81no—
174C4×D4—22no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4C2×C814yes—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—81no—
203C2×C42—81no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42C2411yescharacteristic
211#107—41no—
212#107—41no—
213#107C2×C814yes—
214#107C2×C814yes—
215C4⋊C8—22no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—42no—
225C4⋊C8—81no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8C2×C814yes—
230#102—41no—
231#102—41no—
232#102C2×C814yes—
233#102C2×C814yes—
234C22⋊C8—81no—
235C22⋊C8—81no—
236C22⋊C8—81no—
237C22⋊C8—81no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—81no—
251C4×C8—81no—
252C4×C8—81no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—24no—
262C2×Q8—81no—
263C2×Q8—81no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4C22×C812yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8C22×C812yes—
337C2×C8C22×C812yes—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—24no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4C22×C812yes—
355C4⋊C4C22×C812yes—
356C4⋊C4C22×C812yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—81no—
376C42—81no—
377C42C22×C812yes—
378C42#13112yes—
379C42H(5,𝔽2)11yescharacteristic
380C422−1+411yescharacteristic
381C422−1+411yescharacteristic
382C422−1+411yescharacteristic
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23C2×C4×D411yescharacteristic
391Q8—84no—
392D4—22no—
393D4—22no—
394D4—41no—
395D4—41no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4C8×D412yes—
436C2×C4C8×D412yes—
437C2×C4#51911yescharacteristic
438C2×C4#52011yescharacteristic
439C2×C4#53611yescharacteristic
440C2×C4#53711yescharacteristic
441C2×C4#54111yescharacteristic
442C2×C4#54211yescharacteristic
443C2×C4C23×C811yescharacteristic
444C2×C4C2×#13111yescharacteristic
445C2×C4#56811yescharacteristic
446C2×C4#56811yescharacteristic
447C2×C4#57611yescharacteristic
448C2×C4#57611yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—22no—
453C8—42no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C8—81no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#232311yescharacteristic
473C22C2×C8×D411yescharacteristic
474C22#293011yescharacteristic
475C22#294711yescharacteristic
476C22#294811yescharacteristic
477C22#295611yescharacteristic
478C22#295711yescharacteristic
479C4—21no—
480C4—21no—
481C4—22no—
482C4—22no—
483C4—41no—
484C4—41no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4#298211yescharacteristic
493C4#298511yescharacteristic
494C4#298811yescharacteristic
495C4#299311yescharacteristic
496C4#324111yescharacteristic
497C4#324311yescharacteristic
498C4#325711yescharacteristic
499C4#326211yescharacteristic
500C2—82no—
501C2#1785311yescharacteristic
502C2#1785411yescharacteristic
503C2#1794911yescharacteristic
504C2#1795011yescharacteristic
505C2#2209811yescharacteristic
506C2#2212711yescharacteristic
507C2#2213211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22307C2semidirect
4. #9864C2semidirect
5. #9603C2semidirect
6. #9595C2semidirect
7. #9594C2semidirect
12. #9536C2semidirect
13. #9533C2semidirect
15. #8264C2semidirect
73. #474C8semidirect

Maximal subgroups

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

subgroupindex
#223072
#220082
C4×#4742
#98642
#96032
#95952
#95942
#95562
#95542
#95422
#95412
#95362
#95332
#85652
#82642