Finite Group Databasebeta

← #369310

Subgroups — #369310

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#11064C214yesmaximal
7#10957C214yesmaximal
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3139—24no—
13#3139C418yescomplemented
14C2×#491—22no—
15C2×#487—21no—
16C2×#487—21no—
17C2×#419C2214yes—
18C2×#419C2214yes—
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2370—22no—
22#2369—21no—
23#2365—21no—
24#2355—21no—
25#2355—22no—
26Q8×C4⋊C4—21no—
27#2329—22no—
28#2317C2212yes—
29C4×C22⋊Q8—22no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32#2285C2211yescharacteristic
33#2274C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#1986—24no—
37#1960—22no—
38#1956—21no—
39#1956—21no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#339C2214yes—
43C2×#339C2214yes—
44#1445—22no—
45C4⋊#95—22no—
46C23×Q8C2312yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533D414yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42D412yes—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#491—216no—
80#487—28no—
81#487—28no—
82#419—28no—
83#419—28no—
84#419—44no—
85#419C2×C4116yes—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16—22no—
93C4×M16C2×C414yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#391—24no—
99#391—22no—
100#391—28no—
101#391—44no—
102#389—24no—
103#389—22no—
104#389—22no—
105#389—22no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#384—22no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#382—22no—
114#382D412yes—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C22⋊C4—24no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95—21no—
135C2×#95D412yes—
136#343—24no—
137#343—24no—
138#339—28no—
139#339—44no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8C22×C414yes—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16C22×C414yes—
170C2×M16#4114yes—
171C22×C8—22no—
172C22×C8—22no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8C22⋊C414yes—
184C4⋊Q8C22⋊C414yes—
185#125—24no—
186#125C22⋊C418yes—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—84no—
196#117—22no—
197#117—22no—
198#117—24no—
199#117—24no—
200#117C22×C414yes—
201#117C2×D414yes—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—44no—
248C2×C42C22×C412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4112yes—
255C2×C42#4112yes—
256#103—216no—
257#103—216no—
258#103—48no—
259#103—48no—
260#97—24no—
261#97—42no—
262C4×C8—24no—
263C4×C8—42no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—48no—
273#95—42no—
274#95—48no—
275C24C22×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—81no—
297C2×Q8—81no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
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×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4#12011yescharacteristic
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12414yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348M16—24no—
349M16—24no—
350M16—24no—
351M16—24no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—42no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4C2×C22⋊C414yes—
386C4⋊C4C2×C22⋊C414yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—24no—
392C42—24no—
393C42—44no—
394C42—44no—
395C42—44no—
396C42—44no—
397C42—82no—
398C42—84no—
399C42—84no—
400C42C2×C22⋊C412yes—
401C42C2×C22⋊C412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42#12112yes—
405C42#12112yes—
406C42#12312yes—
407C42#12312yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23C2×C4×D411yescharacteristic
413C23C2×#12011yescharacteristic
414C23C2×#12111yescharacteristic
415C23C2×#12411yescharacteristic
416C23#53211yescharacteristic
417C23#53511yescharacteristic
418C23#53611yescharacteristic
419Q8—48no—
420Q8—48no—
421Q8—48no—
422Q8—84no—
423Q8—84no—
424Q8—84no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
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—44no—
460C2×C4—44no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#38612yes—
472C2×C4#38612yes—
473C2×C4#38612yes—
474C2×C4#38612yes—
475C2×C4#45512yes—
476C2×C4#45612yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C4×#4112yes—
480C2×C4#52012yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4#52912yes—
484C2×C4#54212yes—
485C2×C4#54612yes—
486C2×C4#54812yes—
487C2×C4#54812yes—
488C2×C4#54812yes—
489C2×C4#55012yes—
490C2×C4#55012yes—
491C2×C4#55612yes—
492C2×C4#55912yes—
493C8—44no—
494C8—44no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22C2×#38611yescharacteristic
508C22C4×#12411yescharacteristic
509C22#232611yescharacteristic
510C22#243111yescharacteristic
511C22#243611yescharacteristic
512C22#254511yescharacteristic
513C22#254811yescharacteristic
514C22C2×#45511yescharacteristic
515C22C2×#45611yescharacteristic
516C22#302211yescharacteristic
517C22#302211yescharacteristic
518C4—22no—
519C4—22no—
520C4—22no—
521C4—44no—
522C4—44no—
523C4—44no—
524C4—84no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—84no—
529C4#188214yes—
530C4#234112yes—
531C4#236712yes—
532C4#242612yes—
533C4#244812yes—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2#1402611yescharacteristic
539C2C2×#188211yescharacteristic
540C2#2094411yescharacteristic
541C2#2562111yescharacteristic
542C2#2562311yescharacteristic
543C2#2591711yescharacteristic
544C2#2591911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#342132
C2×#19862
#212022
#141552
#110642
#109572
#108512
C4×#3432