Finite Group Databasebeta

← #290364

Subgroups — #290364

Order 512.

615 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29312C211yesmaximal, characteristic, complemented
2#28514C211yesmaximal, characteristic
3#28279C211yesmaximal, characteristic
4#23151C211yesmaximal, characteristic, complemented
5#10653C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10591C211yesmaximal, characteristic
8#10591C211yesmaximal, characteristic
9#9904C211yesmaximal, characteristic, complemented
10#9617C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9601C211yesmaximal, characteristic, complemented
13#9571C211yesmaximal, characteristic
14#9563C211yesmaximal, characteristic
15#8234C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#478—21no—
18C2×#474—21no—
19C2×#446—21no—
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2092—21no—
24#2092—21no—
25#2083—21no—
26#2067—21no—
27#2067—21no—
28#2057—21no—
29#1985—21no—
30#1982C2211yescharacteristic
31#1982C2211yescharacteristic
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1942—21no—
41#1942—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1917—21no—
45#1916—21no—
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1842C2211yescharacteristic
49#1802—21no—
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8D411yescharacteristic
68C2×#127C2311yescharacteristic
69C2×#125—21no—
70C2×C4×Q8—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#478—44no—
75#478—44no—
76#474—44no—
77#446—44no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389D411yescharacteristic
124#389D411yescharacteristic
125#384D411yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C43C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—22no—
149#335—22no—
150#331—22no—
151#331Q814yescomplemented
152C22×Q8—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C23×C4—41no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—22no—
168C4⋊Q8—41no—
169#127—21no—
170#127—21no—
171#127—21no—
172#127C2×Q812yes—
173#125—42no—
174#125—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—82no—
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—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—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—42no—
215#107—42no—
216#107—42no—
217#107—42no—
218#107—42no—
219#107—42no—
220#107—81no—
221#107—81no—
222#106—42no—
223#106—42no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—81no—
238C4⋊C8D814yes—
239C4⋊C8C2×Q812yes—
240C4⋊C8C2×Q812yes—
241C4⋊C8#4112yes—
242#102—42no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—81no—
249#102—81no—
250#97—42no—
251#97—42no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257C4×C8—81no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264C24—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—81no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—81no—
283C2×D4—81no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
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×C8C2×D812yes—
330C2×C8C2×D812yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4AGL(1,ℤ/8)12yes—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42C2×D812yes—
377C42AGL(1,ℤ/8)12yes—
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C422−1+411yescharacteristic
381C422−1+411yescharacteristic
382C23—42no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23C2×#12111yescharacteristic
390Q8—84no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—42no—
395D4—82no—
396D4—82no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4—81no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
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—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4#45912yes—
436C2×C4#45912yes—
437C2×C4#53311yescharacteristic
438C2×C4#54111yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#54811yescharacteristic
441C2×C4D4×Q811yescharacteristic
442C2×C4D4×Q811yescharacteristic
443C2×C4C22×D811yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4#57511yescharacteristic
447C2×C4#57611yescharacteristic
448C2×C4#57811yescharacteristic
449C2×C4#57811yescharacteristic
450C2×C4#57811yescharacteristic
451C8—22no—
452C8—22no—
453C8—41no—
454C8—41no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22#242911yescharacteristic
469C22C2×#45911yescharacteristic
470C22#304411yescharacteristic
471C22#305511yescharacteristic
472C22#306111yescharacteristic
473C22#306711yescharacteristic
474C22#306811yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4#333411yescharacteristic
489C4#335311yescharacteristic
490C4#336611yescharacteristic
491C4#337711yescharacteristic
492C4Q8×D811yescharacteristic
493C4#338611yescharacteristic
494C4#338611yescharacteristic
495C4#338911yescharacteristic
496C2—162no—
497C2#1916211yescharacteristic
498C2#1919011yescharacteristic
499C2#1947911yescharacteristic
500C2#1953511yescharacteristic
501C2#2544911yescharacteristic
502C2#2548211yescharacteristic
503C2#2549311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #29312C2semidirect
4. #23151C2semidirect
5. #10653C2semidirect
6. #10650C2semidirect
9. #9904C2semidirect
10. #9617C2semidirect
11. #9614C2semidirect
12. #9601C2semidirect
151. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#293122
#285142
#282792
#231512
#106532
#106502
#105912
#99042
#96172
#96142
#96012
#95712
#95632
#82342