Finite Group Databasebeta

← #192424

Subgroups — #192424

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34389C211yesmaximal, characteristic
2#21983C211yesmaximal, characteristic
3#21866C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#14356C211yesmaximal, characteristic
6#8872C212yesmaximal
7#8871C212yesmaximal
8#8868C212yesmaximal
9#8867C212yesmaximal
10#8021C211yesmaximal, characteristic
11#7916C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2381—21no—
16#2381—21no—
17#2368—22no—
18#2356—21no—
19#2356—21no—
20#2343—21no—
21#2343—21no—
22#2336—22no—
23#2329C2212yes—
24#2329C2212yes—
25#2320—21no—
26#2320—21no—
27#2283C2211yescharacteristic
28C2×#383C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—21no—
32#1960—21no—
33#1949—22no—
34#1872C2212yes—
35#1872C2212yes—
36#1853C2212yes—
37#1852C2212yes—
38#1846—21no—
39#1835—21no—
40#1805—21no—
41#1805—21no—
42#1503C418yescomplemented
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47#1444C2212yes—
48#1442C2212yes—
49#1441—22no—
50C4⋊#95—21no—
51#1434—21no—
52#1294C2212yes—
53#1294C2212yes—
54#1280C2212yes—
55#1280C2212yes—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×C22⋊Q8C2311yescharacteristic
66C2×C4×Q8—21no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431C2×C414yes—
79C2×#107—22no—
80C2×#106—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2311yescharacteristic
87C2×C22⋊C8C2311yescharacteristic
88C2×#97—21no—
89C2×C4×C8—21no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#389—22no—
95#389—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—22no—
109#385—22no—
110#383—21no—
111#383—21no—
112#383—21no—
113#383—21no—
114#383—21no—
115#383—21no—
116#383—21no—
117#382—21no—
118#382—21no—
119#382—21no—
120#382—21no—
121#382—21no—
122#382—21no—
123#382D411yescharacteristic
124#382D411yescharacteristic
125#381—21no—
126#380C2312yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C414yes—
133C4×C4⋊C4D412yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137#376—21no—
138#376—21no—
139#376—21no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95D411yescharacteristic
146C2×#95D411yescharacteristic
147C2×#95D412yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#328C2×C414yes—
156#328C2×C414yes—
157#328C2×C414yes—
158#328C2×C414yes—
159C22×Q8—21no—
160C22×Q8C2×D411yescharacteristic
161C22×Q8C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8C4⋊C414yescomplemented
176C4⋊Q8C4⋊C414yescomplemented
177#125—24no—
178#125—24no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8C22×C412yes—
186C22⋊Q8C22×C412yes—
187C22⋊Q8C2×Q812yes—
188C22⋊Q8C2×Q812yes—
189C4×Q8—24no—
190C4×Q8—24no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8C4⋊C414yescomplemented
194C4×Q8C4⋊C414yescomplemented
195#117—24no—
196#117—24no—
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—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—82no—
232C2×C42C22×C412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×Q812yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240#107—82no—
241#106—82no—
242C4⋊C8—24no—
243C4⋊C8—42no—
244#103—82no—
245#103—82no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8C22×C412yes—
251C22⋊C8C22×C412yes—
252#97—42no—
253C4×C8—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—44no—
279#95—41no—
280#95—41no—
281#95—41no—
282#95—41no—
283#95—82no—
284#95—82no—
285#95C22×C412yes—
286#95C22×C412yes—
287C24C22×D411yescharacteristic
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—24no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8C2×C4⋊C412yes—
297C2×Q8C2×C4⋊C412yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×Q811yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—88no—
382C4⋊C4C2×C4⋊C412yes—
383C4⋊C4C2×C4⋊C412yes—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C42—24no—
391C42—24no—
392C42—82no—
393C42—84no—
394C42—82no—
395C42—82no—
396C42—82no—
397C42—82no—
398C42C2×C4⋊C412yes—
399C42C2×C4⋊C412yes—
400C42C4×D412yes—
401C42C4×D412yes—
402C42C22⋊Q812yes—
403C42C22⋊Q812yes—
404C42#12312yes—
405C42#12312yes—
406C23—22no—
407C23—21no—
408C23—21no—
409C23C2×C4×D411yescharacteristic
410C23C2×#12011yescharacteristic
411C23C2×C22⋊Q811yescharacteristic
412C23#57712yes—
413C23#57812yes—
414Q8—42no—
415Q8—42no—
416Q8—81no—
417Q8—81no—
418Q8—84no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4#38512yes—
480C2×C4#38512yes—
481C2×C4#38512yes—
482C2×C4#38512yes—
483C2×C4C22×C4⋊C411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C8—42no—
488C8—42no—
489C8—82no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22C2×#38511yescharacteristic
499C22#294911yescharacteristic
500C22#295011yescharacteristic
501C22#301312yes—
502C22#301812yes—
503C22#302711yescharacteristic
504C22#302811yescharacteristic
505C22#308212yes—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—44no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—162no—
525C2—21no—
526C2—22no—
527C2—21no—
528C2#1843512yes—
529C2#1844512yes—
530C2#2086311yescharacteristic
531C2#2088011yescharacteristic
532C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
42. #1503C4semidirect
175. C4⋊Q8C4⋊C4semidirect
176. C4⋊Q8C4⋊C4semidirect
193. C4×Q8C4⋊C4semidirect
194. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#343892
#219832
#218662
C2×#15032
#143562
#88722
#88712
#88682
#88672
#80212
#79162