Finite Group Databasebeta

← #229539

Subgroups — #229539

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23223C211yesmaximal, characteristic
2#23101C211yesmaximal, characteristic
3#22971C211yesmaximal, characteristic, complemented
4#22743C211yesmaximal, characteristic, complemented
5C2×#1698C211yesmaximal, characteristic
6#9822C211yesmaximal, characteristic
7#9619C211yesmaximal, characteristic, complemented
8#9576C211yesmaximal, characteristic, complemented
9#9572C211yesmaximal, characteristic
10#8979C211yesmaximal, characteristic
11#8974C211yesmaximal, characteristic, complemented
12#8927C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic
14#8282C211yesmaximal, characteristic, complemented
15#8207C211yesmaximal, characteristic, complemented
16C2×#486C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#460C2211yescharacteristic
19C2×#446C2211yescharacteristic
20#2369C2211yescharacteristic
21#2366C2211yescharacteristic
22#2355C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1982C2211yescharacteristic
27#1973—21no—
28#1957C2211yescharacteristic
29#1957C2211yescharacteristic
30#1956C2211yescharacteristic
31#1949—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1943—21no—
38#1936—21no—
39#1925—21no—
40#1925—21no—
41#1918—21no—
42#1877—21no—
43#1865—21no—
44#1862—21no—
45#1836—21no—
46#1834C2211yescharacteristic
47#1776—21no—
48C4×#107C2211yescharacteristic
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51#1770C2211yescharacteristic
52#1763—21no—
53#1762—21no—
54#1762—21no—
55C4×#103C2211yescharacteristic
56#1698C418yescomplemented
57C2×#335C2211yescharacteristic
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×Q8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75#486—22no—
76#486C2×C414yes—
77#477—22no—
78#477C2×C414yescomplemented
79#460—22no—
80#460C2×C414yes—
81#446—22no—
82#446C2×C414yescomplemented
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107D411yescharacteristic
87C2×#107D411yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—41no—
97C2×#103D411yescharacteristic
98C2×#103D411yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—41no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—41no—
116#390—21no—
117#390—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—41no—
122#387—41no—
123#386—41no—
124#384—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#382—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#335C2×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×Q8—42no—
187C4×Q8—81no—
188C4×Q8C22×C412yes—
189C4×Q8#4112yes—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—81no—
193C4×D4—81no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C2411yescharacteristic
224#107—24no—
225#107—41no—
226#107—41no—
227#107—42no—
228#107—81no—
229#106—42no—
230#106—81no—
231C4⋊C8—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C22×C412yes—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240#103—24no—
241#103—22no—
242#103—41no—
243#103—82no—
244#102—24no—
245#102—24no—
246#102—24no—
247#102—42no—
248#102—42no—
249#102—81no—
250#102—81no—
251#102—82no—
252#97—24no—
253#97—81no—
254#97—81no—
255#97—81no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8C22×C412yes—
261C4×C8#4112yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24—41no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—81no—
280C2×Q8—81no—
281C2×Q8C4×D412yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
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—41no—
295C2×D4—41no—
296C2×D4—81no—
297C2×D4—81no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—161no—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315SD16—84no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
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×C8—41no—
339C2×C8—41no—
340C2×C8C4×D412yes—
341C2×C8C4×D412yes—
342C2×C8#13712yes—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—161no—
374C4⋊C4—161no—
375C4⋊C4C4×D412yes—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—161no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—82no—
388C42—161no—
389C42—161no—
390C42C23×C411yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C422−1+411yescharacteristic
394C23—42no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23D4211yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406Q8—82no—
407Q8—161no—
408D4—42no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—82no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—161no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
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—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—164no—
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4#52011yescharacteristic
457C2×C4#54711yescharacteristic
458C2×C4#54811yescharacteristic
459C2×C4#55311yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57611yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—22no—
469C8—22no—
470C8—22no—
471C8—42no—
472C8—41no—
473C8—42no—
474C8—42no—
475C8—42no—
476C8—81no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22#236111yescharacteristic
485C22#327311yescharacteristic
486C22#327511yescharacteristic
487C22#327711yescharacteristic
488C22#327811yescharacteristic
489C22#329011yescharacteristic
490C22#329211yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—42no—
495C4—82no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4—161no—
504C4#294311yescharacteristic
505C4C4×#13711yescharacteristic
506C4#294611yescharacteristic
507C4#294711yescharacteristic
508C4#334611yescharacteristic
509C4#334811yescharacteristic
510C4#337111yescharacteristic
511C4#337711yescharacteristic
512C2—162no—
513C2#2028711yescharacteristic
514C2#2029911yescharacteristic
515C2#2030811yescharacteristic
516C2#2032011yescharacteristic
517C2#2300611yescharacteristic
518C2#2304111yescharacteristic
519C2#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. #22971C2semidirect
4. #22743C2semidirect
7. #9619C2semidirect
8. #9576C2semidirect
11. #8974C2semidirect
12. #8927C2semidirect
14. #8282C2semidirect
15. #8207C2semidirect
56. #1698C4semidirect
78. #477C2×C4semidirect
82. #446C2×C4semidirect
153. #335C2×C4semidirect
154. #333C2×C4semidirect

Maximal subgroups

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

subgroupindex
#232232
#231012
#229712
#227432
C2×#16982
#98222
#96192
#95762
#95722
#89792
#89742
#89272
#89212
#82822
#82072