Finite Group Databasebeta

← #352606

Subgroups — #352606

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30307C211yesmaximal, characteristic, complemented
2#29998C211yesmaximal, characteristic
3#29986C211yesmaximal, characteristic
4#20562C211yesmaximal, characteristic
5#9900C211yesmaximal, characteristic, complemented
6#9899C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9618C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9613C211yesmaximal, characteristic, complemented
11#8812C211yesmaximal, characteristic, complemented
12#8654C211yesmaximal, characteristic
13#8652C211yesmaximal, characteristic
14#8648C211yesmaximal, characteristic
15#8642C211yesmaximal, characteristic
16C2×#436C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2859C2211yescharacteristic
19#2859C2211yescharacteristic
20#2293C2211yescharacteristic
21#2101—21no—
22#2092—21no—
23#2088—21no—
24#2088—21no—
25#2085—21no—
26#2085—21no—
27#2065—21no—
28#2065—21no—
29#2059—21no—
30#2059—21no—
31#2059—21no—
32#2025—21no—
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1842C2211yescharacteristic
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1831C2211yescharacteristic
43#1802C2211yescharacteristic
44#1802C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50#1753—21no—
51C8×C22⋊C4—21no—
52#1750—21no—
53#1744C2211yescharacteristic
54#1729—21no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#125—21no—
70C2×#125—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#436—22no—
74#436Q814yes—
75C8×D4—22no—
76C8×D4Q814yes—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#390D411yescharacteristic
105#390D411yescharacteristic
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389D411yescharacteristic
115#389D411yescharacteristic
116#388—21no—
117#388—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C43C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—22no—
149#335—22no—
150#334—22no—
151#334—22no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C2×M16—24no—
155C2×M16—81no—
156C2×M16—81no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—24no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—81no—
165C22×C8—81no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170#125—81no—
171#125—81no—
172#125—81no—
173#125—81no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4C2×Q812yes—
177C4×D4C2×Q812yes—
178#117—81no—
179#117—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—81no—
215#107—81no—
216#107—81no—
217#106—81no—
218#106—81no—
219#106—81no—
220#106—81no—
221#106—81no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8C2×Q812yes—
235C4⋊C8C2×Q812yes—
236#102—81no—
237#102—81no—
238#102—81no—
239#102—81no—
240C22⋊C8—24no—
241C22⋊C8—24no—
242#97—81no—
243C4×C8—81no—
244C4×C8—81no—
245C4×C8C2×Q812yes—
246C4×C8C2×Q812yes—
247C4×C8#4112yes—
248C4×C8#4112yes—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4C4⋊Q812yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283M16—84no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—84no—
311C2×C8C22⋊Q812yes—
312C2×C8C22⋊Q812yes—
313C2×C8C22⋊Q812yes—
314C2×C8C22⋊Q812yes—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4C4⋊Q812yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42C22×Q811yescharacteristic
366C42C22×Q811yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C23—22no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23C2×#12111yescharacteristic
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4C2×C22⋊Q811yescharacteristic
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4C2×C4⋊Q811yescharacteristic
423C2×C4#53211yescharacteristic
424C2×C4#53311yescharacteristic
425C2×C4#53311yescharacteristic
426C2×C4#57611yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57811yescharacteristic
434C8—22no—
435C8—22no—
436C8—22no—
437C8—22no—
438C8—42no—
439C8—42no—
440C8—81no—
441C8—81no—
442C8—81no—
443C8—81no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#285711yescharacteristic
452C22#304211yescharacteristic
453C22#304511yescharacteristic
454C22#306211yescharacteristic
455C22#306311yescharacteristic
456C22#306711yescharacteristic
457C22#306711yescharacteristic
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—81no—
466C4—81no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#308111yescharacteristic
472C4#308111yescharacteristic
473C4#308411yescharacteristic
474C4#308411yescharacteristic
475C4#316311yescharacteristic
476C4#316411yescharacteristic
477C4#317811yescharacteristic
478C4#317911yescharacteristic
479C2—82no—
480C2#1956211yescharacteristic
481C2#1956311yescharacteristic
482C2#1957611yescharacteristic
483C2#1957711yescharacteristic
484C2#3320511yescharacteristic
485C2#3321911yescharacteristic
486C2#3322111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #30307C2semidirect
5. #9900C2semidirect
6. #9899C2semidirect
7. #9622C2semidirect
8. #9618C2semidirect
9. #9615C2semidirect
10. #9613C2semidirect
11. #8812C2semidirect

Maximal subgroups

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

subgroupindex
#303072
#299982
#299862
#205622
#99002
#98992
#96222
#96182
#96152
#96132
#88122
#86542
#86522
#86482
#86422