Finite Group Databasebeta

← #380169

Subgroups — #380169

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1959C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1746C211yesmaximal, characteristic, complemented
4#14277C211yesmaximal, characteristic, complemented
5#14273C211yesmaximal, characteristic
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1959—24no—
29#1958C418yescomplemented
30#1746—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444—22no—
38#1442—22no—
39#1439—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#124—22no—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124C2×C414yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103C2×C418yes—
72C2×#102—24no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102C2×C418yes—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97D414yes—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8C2×C414yes—
84#391—42no—
85#391—42no—
86#390—42no—
87#390—42no—
88#389—42no—
89#389—42no—
90#384—22no—
91#384—22no—
92#384—22no—
93#384—22no—
94#382—22no—
95#382C2×C414yes—
96#376—22no—
97#376—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106C2×#95D412yes—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—24no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22×C412yes—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C22×C412yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C22×C414yes—
157C22×C8C2×D414yes—
158#124—28no—
159#124—28no—
160#124—44no—
161#124—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22×C414yes—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215#103—48no—
216#103—48no—
217#102—48no—
218#102—48no—
219#97—28no—
220C4×C8—28no—
221#95—28no—
222#95—24no—
223#95—24no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—44no—
228#95—82no—
229#95—82no—
230C24—22no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C4×D414yes—
327C22×C4#12114yes—
328C22×C4#12414yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4H(5,𝔽2)12yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C22⋊C4—416no—
369C22⋊C4—84no—
370C22⋊C4—84no—
371C22⋊C4—88no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42#11714yes—
377C42C4×D414yes—
378C42#12314yes—
379C42#12614yes—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—82no—
388C23—82no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23C2×C4×D411yescharacteristic
404C23C2×#12111yescharacteristic
405C23C2×#12411yescharacteristic
406C23#53511yescharacteristic
407C23#53511yescharacteristic
408C23#53511yescharacteristic
409C23#53611yescharacteristic
410C23#57612yes—
411C23#57712yes—
412C23#57712yes—
413C23#57812yes—
414Q8—44no—
415Q8—44no—
416Q8—82no—
417Q8—82no—
418D4—44no—
419D4—44no—
420D4—82no—
421D4—82no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#38818yes—
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4#51812yes—
480C2×C4#52012yes—
481C2×C4C2×#12311yescharacteristic
482C2×C4C2×#12611yescharacteristic
483C2×C4#53512yes—
484C2×C4#53712yes—
485C2×C4#54012yes—
486C2×C4#54012yes—
487C2×C4#54012yes—
488C2×C4#54012yes—
489C2×C4#54212yes—
490C2×C4#54212yes—
491C2×C4#54212yes—
492C2×C4#54212yes—
493C8—44no—
494C8—44no—
495C22—84no—
496C22—84no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—162no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—161no—
505C22—162no—
506C22C2×#38811yescharacteristic
507C22#232611yescharacteristic
508C22#232811yescharacteristic
509C22#261911yescharacteristic
510C22#262011yescharacteristic
511C22#264011yescharacteristic
512C22#264111yescharacteristic
513C22#295112yes—
514C22#295212yes—
515C22#306212yes—
516C22#306312yes—
517C22#313612yes—
518C22#313712yes—
519C22#322011yescharacteristic
520C22#322111yescharacteristic
521C22#322211yescharacteristic
522C22#322311yescharacteristic
523C22#322412yes—
524C22#322412yes—
525C22#322512yes—
526C22#322512yes—
527C22#324312yes—
528C22#324412yes—
529C4—28no—
530C4—84no—
531C4—81no—
532C4—81no—
533C4—82no—
534C4—82no—
535C4—161no—
536C4—161no—
537C4—162no—
538C4#231912yes—
539C4#232412yes—
540C4#262412yes—
541C4#264412yes—
542C2—162no—
543C2#1426011yescharacteristic
544C2#2108712yes—
545C2#2222711yescharacteristic
546C2#2222811yescharacteristic
547C2#2227412yes—
548C2#2981611yescharacteristic
549C2#2981711yescharacteristic
550C2#2985712yes—
551C2#3021112yes—
552C2#3022211yescharacteristic
553C2#3022311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1746C2semidirect
4. #14277C2semidirect
6. #11190C2semidirect
8. #11067C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
29. #1958C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19592
C2×#19582
C2×#17462
#142772
#142732
#111902
#111892
#110672
#110662
#108432
#108412