Finite Group Databasebeta

← #379752

Subgroups — #379752

Order 512.

2296 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2#34044C211yesmaximal, characteristic
3#20911C212yesmaximal, complemented
4C2×#1785C211yesmaximal, characteristic
5#11162C212yesmaximal, complemented
6#11161C212yesmaximal, complemented
7#10978C212yesmaximal, complemented
8#10977C212yesmaximal, complemented
9#10733C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2212yes—
15C2×#419C2214yes—
16C22×#101C2212yes—
17C22×#100C2212yes—
18C2×#407C2212yes—
19#2432—24no—
20#2432C418yescomplemented
21C2×#394—22no—
22#2311—24no—
23#2310C2212yes—
24#2309C2212yes—
25#2308—22no—
26#2297—24no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#385—22no—
30#2280—22no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#1873—24no—
33#1872—24no—
34#1785—24no—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C22×#41—22no—
39C22×#41—22no—
40C22×#41—24no—
41C22×#41—44no—
42C22×#41—44no—
43C22×#41C2×C414yes—
44C22×#41D414yes—
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4D412yes—
48C23×D4C2311yescharacteristic
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C22×M16C2314yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—44no—
54C2×C4⋊Q8C2×C414yescomplemented
55C2×#124—22no—
56C2×#124—44no—
57C2×#124C2×C414yescomplemented
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8C2×C414yes—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—44no—
66C2×#117—44no—
67C2×#117D414yes—
68C2×#117C2314yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#419—44no—
82#419—44no—
83C2×#103—28no—
84C2×#101—24no—
85C2×#101—44no—
86C2×#100—24no—
87C2×#100—44no—
88#407—24no—
89#407—24no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#394—216no—
93#391—22no—
94#391—28no—
95#391C2×C414yescomplemented
96#390—22no—
97#390—28no—
98#390C2×C414yescomplemented
99#386—216no—
100#385—216no—
101#381—24no—
102#380—24no—
103#380—24no—
104#379—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4C2×C414yes—
109C2×#95—22no—
110C2×#95—21no—
111#340—28no—
112#339—44no—
113#339—44no—
114C25C2×D412yes—
115C25C2×D411yescharacteristic
116C2×#41—28no—
117C2×#41—28no—
118C2×#41—28no—
119C2×#41—28no—
120C2×#41—48no—
121C2×#41—48no—
122C2×#41—48no—
123C2×#41—48no—
124C2×#41—48no—
125C2×#41—48no—
126C2×#41—48no—
127C2×#41—48no—
128C2×#41—416no—
129C2×#41—416no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8C22×C412yes—
140C22×Q8C22×C412yes—
141C22×Q8C2×D412yes—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4C22⋊C418yes—
151C22×D4C22×C412yes—
152C22×D4C2×D412yes—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4C22⋊C418yes—
166C23×C4C22×C412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4#4114yes—
173C23×C4C2411yescharacteristic
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—24no—
179C2×M16—24no—
180C22×C8—24no—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183#124—48no—
184#124—48no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—48no—
194#117—44no—
195#117—44no—
196#117—48no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C2×D418yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—48no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—48no—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234#103—416no—
235#101—48no—
236#100—48no—
237C22⋊C8—44no—
238C22⋊C8—44no—
239#95—28no—
240#95—28no—
241#95—24no—
242#95—28no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—24no—
247C24—22no—
248C24—24no—
249C24—42no—
250C24—42no—
251C24—44no—
252C24—44no—
253C24C2×C22⋊C414yes—
254C24C2×C22⋊C412yes—
255C24#12012yes—
256C24#12312yes—
257C24C22×D411yescharacteristic
258#41—432no—
259#41—432no—
260#41—432no—
261#41—432no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×D4—216no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—48no—
292C2×D4—48no—
293C2×D4—416no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—28no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—416no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4C2×C22⋊C414yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C4×D414yes—
345C22×C4C4×D414yes—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12114yes—
349C22×C4#12114yes—
350C22×C4#12312yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)14yes—
356C22×C4H(5,𝔽2)14yes—
357M16—44no—
358M16—44no—
359M16—44no—
360M16—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C4⋊C4—24no—
365C4⋊C4—28no—
366C4⋊C4—28no—
367C4⋊C4—48no—
368C4⋊C4—416no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C22⋊C4—48no—
373C22⋊C4—416no—
374C22⋊C4—48no—
375C22⋊C4—416no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—48no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C42—24no—
382C42—24no—
383C42—48no—
384C42—42no—
385C42—42no—
386C42—44no—
387C42—44no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—416no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—84no—
412C23—84no—
413C23—82no—
414C23—82no—
415C23—88no—
416C23#38614yes—
417C23C22×C22⋊C411yescharacteristic
418C23C2×#12011yescharacteristic
419C23C2×#12311yescharacteristic
420C23#53412yes—
421C23#53412yes—
422Q8—48no—
423Q8—48no—
424Q8—48no—
425Q8—84no—
426D4—432no—
427D4—48no—
428D4—48no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—416no—
443C2×C4—416no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—48no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—816no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38612yes—
467C2×C4#38612yes—
468C2×C4C2×C4×D412yes—
469C2×C4#51812yes—
470C2×C4#51812yes—
471C2×C4C2×#12112yes—
472C2×C4#53414yes—
473C2×C4#53412yes—
474C2×C4#53412yes—
475C2×C4#53514yes—
476C2×C4#53814yes—
477C2×C4#53814yes—
478C8—84no—
479C22—24no—
480C22—24no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—48no—
489C22—44no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—44no—
494C22—82no—
495C22—82no—
496C22—84no—
497C22—84no—
498C22—88no—
499C22—88no—
500C22C2×#38611yescharacteristic
501C22#230811yescharacteristic
502C22#230811yescharacteristic
503C22#261312yes—
504C22#262212yes—
505C22#301812yes—
506C22#302212yes—
507C4—22no—
508C4—22no—
509C4—44no—
510C4—44no—
511C4—82no—
512C4—82no—
513C4—88no—
514C4—82no—
515C4#232114yes—
516C4#261714yes—
517C2—21no—
518C2—21no—
519C2—22no—
520C2—44no—
521C2—84no—
522C2#1425511yescharacteristic
523C2#2095211yescharacteristic
524C2#2095511yescharacteristic
525C2#2961514yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20911C2semidirect
5. #11162C2semidirect
6. #11161C2semidirect
7. #10978C2semidirect
8. #10977C2semidirect
9. #10733C2semidirect
20. #2432C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
57. C2×#124C2×C4semidirect
95. #391C2×C4semidirect
98. #390C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
#340442
#209112
C2×#17852
#111622
#111612
#109782
#109772
#107332