Finite Group Databasebeta

← #229289

Subgroups — #229289

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23215C211yesmaximal, characteristic
2#23095C211yesmaximal, characteristic
3#23063C211yesmaximal, characteristic, complemented
4#22739C211yesmaximal, characteristic, complemented
5C2×#1678C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic
7#9613C211yesmaximal, characteristic, complemented
8#9568C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic
10#8978C211yesmaximal, characteristic
11#8974C211yesmaximal, characteristic, complemented
12#8927C211yesmaximal, characteristic, complemented
13#8920C211yesmaximal, characteristic
14C8×#106C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20#2369C2211yescharacteristic
21#2366C2211yescharacteristic
22#2355C2211yescharacteristic
23#2355C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1982C2211yescharacteristic
27#1968—21no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1949—21no—
32#1949—21no—
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1934—21no—
38#1933—21no—
39#1928—21no—
40#1926—21no—
41#1926—21no—
42#1921—21no—
43#1877—21no—
44#1864—21no—
45#1835—21no—
46#1834C2211yescharacteristic
47C4×#107—21no—
48C4×#107—21no—
49C4×#106C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#103C2211yescharacteristic
54C4×#102—21no—
55C4×#102—21no—
56#1678C418yescomplemented
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×C82C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74#486—22no—
75#486C2×C414yes—
76#477—22no—
77#477C2×C414yescomplemented
78#460—22no—
79#460C2×C414yes—
80C8×Q8—22no—
81C8×Q8C2×C414yescomplemented
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106D411yescharacteristic
88C2×#106D411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103—21no—
95C2×#103—41no—
96C2×#103D411yescharacteristic
97C2×#103D411yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—41no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—41no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#387—41no—
121#387—41no—
122#386—41no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126#382—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340D411yescharacteristic
147#340D411yescharacteristic
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#334C2×C414yescomplemented
154#333C2×C414yescomplemented
155#331C2×C414yes—
156Q8⋊C8—22no—
157C82—22no—
158C22×Q8—21no—
159C22×Q8C2×D411yescharacteristic
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C23×C4—41no—
164C2×SD16—24no—
165C2×SD16—81no—
166C2×SD16—81no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180#127—21no—
181#127—21no—
182#127—21no—
183#127C22×C412yes—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8C22×C412yes—
187C4×Q8#4112yes—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42C2411yescharacteristic
220#107—24no—
221#107—42no—
222#107—42no—
223#107—42no—
224#106—22no—
225#106—22no—
226#106—81no—
227C4⋊C8—24no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8C22×C412yes—
234C4⋊C8C22×C412yes—
235C4⋊C8C22×C412yes—
236#103—24no—
237#103—22no—
238#103—22no—
239#103—82no—
240#102—24no—
241#102—24no—
242#102—24no—
243#102—42no—
244#102—42no—
245#102—81no—
246#102—81no—
247#102—82no—
248C4×C8—24no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—81no—
255C4×C8—81no—
256C4×C8C22×C412yes—
257C4×C8#4112yes—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24—41no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×Q8C4×D412yes—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—161no—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313SD16—84no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
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×C8—41no—
337C2×C8—41no—
338C2×C8C4×D412yes—
339C2×C8C4×D412yes—
340C2×C8#13512yes—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—161no—
368C4⋊C4C4×D412yes—
369C4⋊C4#13512yes—
370C4⋊C4#13512yes—
371C4⋊C4#13512yes—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—161no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—161no—
384C42—161no—
385C42—161no—
386C42C23×C411yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C422−1+411yescharacteristic
390C23—42no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23D4211yescharacteristic
398Q8—22no—
399Q8—22no—
400Q8—41no—
401Q8—41no—
402Q8—82no—
403Q8—161no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—42no—
408D4—82no—
409D4—81no—
410D4—81no—
411D4—81no—
412D4—81no—
413D4—161no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
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—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—164no—
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4#52011yescharacteristic
453C2×C4#54711yescharacteristic
454C2×C4#54811yescharacteristic
455C2×C4#55311yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×#13511yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57811yescharacteristic
464C8—22no—
465C8—22no—
466C8—22no—
467C8—42no—
468C8—41no—
469C8—42no—
470C8—42no—
471C8—42no—
472C8—81no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—161no—
477C22—161no—
478C22—161no—
479C22—161no—
480C22#236111yescharacteristic
481C22#328211yescharacteristic
482C22#328211yescharacteristic
483C22#328411yescharacteristic
484C22#328511yescharacteristic
485C22#329011yescharacteristic
486C22#329211yescharacteristic
487C4—21no—
488C4—21no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—161no—
499C4—161no—
500C4C4×#13511yescharacteristic
501C4C4×#13511yescharacteristic
502C4#294711yescharacteristic
503C4#294911yescharacteristic
504C4#333711yescharacteristic
505C4#334011yescharacteristic
506C4#336411yescharacteristic
507C4#336511yescharacteristic
508C2—162no—
509C2#1999611yescharacteristic
510C2#2000211yescharacteristic
511C2#2000511yescharacteristic
512C2#2001211yescharacteristic
513C2#2300011yescharacteristic
514C2#2302311yescharacteristic
515C2#2304611yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23063C2semidirect
4. #22739C2semidirect
7. #9613C2semidirect
8. #9568C2semidirect
11. #8974C2semidirect
12. #8927C2semidirect
14. C8×#106C2semidirect
15. C8×#103C2semidirect
56. #1678C4semidirect
77. #477C2×C4semidirect
81. C8×Q8C2×C4semidirect
153. #334C2×C4semidirect
154. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232152
#230952
#230632
#227392
C2×#16782
#98222
#96132
#95682
#95632
#89782
#89742
#89272
#89202
C8×#1062
C8×#1032