Finite Group Databasebeta

← #380065

Subgroups — #380065

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#2922C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5#20648C211yesmaximal, characteristic
6#11216C214yesmaximal, complemented
7#11192C214yesmaximal
8#10857C211yesmaximal, characteristic, complemented
9#10844C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477—21no—
15C2×#475—21no—
16C2×#476—21no—
17C2×#474—21no—
18#2922—24no—
19#2922—24no—
20C2×#433—22no—
21C2×#431—22no—
22C2×#428C2212yes—
23C2×#424C2212yes—
24C2×#423C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#417C2212yes—
27C2×#417C2212yes—
28C2×#408C2212yes—
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#1997—24no—
33#1990—22no—
34#1990—22no—
35#1803—21no—
36#1803—21no—
37#1802—21no—
38#1802—21no—
39C2×#343C2214yes—
40C2×#343C2214yes—
41C2×#341C2214yes—
42#1736C2212yes—
43C2×#340C2214yes—
44C22×#41C2311yescharacteristic
45C23×Q8—21no—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41C2×C414yes—
61C4×#41D414yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117—21no—
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#477—28no—
78#476—28no—
79#475—28no—
80#474—28no—
81#433—216no—
82#431—216no—
83#428—24no—
84#428C2×C418yes—
85C2×#107—22no—
86C2×#106—22no—
87#424—24no—
88#424—24no—
89#423—22no—
90#423—22no—
91#423—22no—
92#423C2×C414yes—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98#417—24no—
99#417—44no—
100#417C2×C418yes—
101C2×#103—22no—
102C2×#103—22no—
103C2×#102—22no—
104C2×#102—22no—
105#408—24no—
106#408—24no—
107C2×C22⋊C8—24no—
108C4×M16—22no—
109C4×M16D414yes—
110C2×#97—21no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—21no—
114C2×C4×C8D412yes—
115C2×C4×C8D412yes—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#343—28no—
121#343—44no—
122#341—44no—
123#340—28no—
124#340—22no—
125#340—22no—
126C2×#41—21no—
127C2×#41—21no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41C22×C412yes—
133C2×#41C2×D412yes—
134C22×Q8—24no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4C2411yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—24no—
154C2×M16—22no—
155C2×M16—24no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—24no—
162C2×M16—24no—
163C2×M16C22×C414yes—
164C2×M16C2×D414yes—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C2×D414yes—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8C22⋊C414yes—
186C4×Q8C22⋊C414yes—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4C22⋊C414yes—
194C4×D4C22⋊C414yes—
195#117—21no—
196#117—21no—
197#117—24no—
198#117—24no—
199#117C22×C412yes—
200#117C2×D412yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42C22×C412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241#107—28no—
242#106—28no—
243C4⋊C8—216no—
244C4⋊C8—24no—
245C4⋊C8—24no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248#103—28no—
249#103—44no—
250#102—28no—
251#102—44no—
252C22⋊C8—216no—
253#97—24no—
254C4×C8—24no—
255C4×C8—24no—
256C4×C8—24no—
257C24—21no—
258C24C2×#4111yescharacteristic
259#41—24no—
260#41—24no—
261#41—24no—
262#41—24no—
263#41—42no—
264#41—42no—
265#41—42no—
266#41—42no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—88no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—88no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8C2×C22⋊C412yes—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4C2×C22⋊C412yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4C4×D412yes—
318C22×C4C4×D412yes—
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C22×C42−1+412yes—
335M16—28no—
336M16—24no—
337M16—24no—
338M16—42no—
339M16—44no—
340M16—42no—
341M16—44no—
342M16—44no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4C2×C22⋊C412yes—
377C4⋊C4C2×C22⋊C412yes—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—24no—
386C42C2×C22⋊C412yes—
387C42C2×C22⋊C412yes—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12012yes—
391C42#12012yes—
392C42#12112yes—
393C42#12112yes—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C2×C4×D411yescharacteristic
406C23C2×#12111yescharacteristic
407C23C2×#12311yescharacteristic
408C23#54112yes—
409C23#54212yes—
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414Q8—88no—
415Q8—88no—
416D4—22no—
417D4—22no—
418D4—44no—
419D4—44no—
420D4—41no—
421D4—41no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—44no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—164no—
452C2×C4#38612yes—
453C2×C4#38612yes—
454C2×C4#38612yes—
455C2×C4#38612yes—
456C2×C4#47112yes—
457C2×C4#47112yes—
458C2×C4#47212yes—
459C2×C4#47312yes—
460C2×C4C22×C22⋊C411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4#51912yes—
463C2×C4#52012yes—
464C2×C4C2×#12011yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4#53511yescharacteristic
467C2×C4#53612yes—
468C2×C4#53611yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53611yescharacteristic
471C2×C4#53612yes—
472C2×C4#53612yes—
473C2×C4#53712yes—
474C2×C4#53712yes—
475C2×C4#53712yes—
476C2×C4#54112yes—
477C2×C4#54212yes—
478C8—44no—
479C8—44no—
480C8—44no—
481C8—82no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22C2×#38611yescharacteristic
498C22#232312yes—
499C22#262011yescharacteristic
500C22#262111yescharacteristic
501C22#262512yes—
502C22C2×#47111yescharacteristic
503C22C2×#47111yescharacteristic
504C22C2×#47211yescharacteristic
505C22C2×#47311yescharacteristic
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—162no—
513C4#190714yes—
514C4#190814yes—
515C4#231011yescharacteristic
516C4#231111yescharacteristic
517C4#232312yes—
518C4#261612yes—
519C4#262011yescharacteristic
520C4#262111yescharacteristic
521C2—22no—
522C2—22no—
523C2—82no—
524C2#1425811yescharacteristic
525C2C2×#190711yescharacteristic
526C2C2×#190811yescharacteristic
527C2#2985811yescharacteristic
528C2#2985911yescharacteristic
529C2#2988611yescharacteristic
530C2#2988711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
3. C2×#1997C2semidirect
4. C2×#1990C2semidirect
6. #11216C2semidirect
8. #10857C2semidirect
9. #10844C2semidirect
13. #3079C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#29222
C2×#19972
C2×#19902
#206482
#112162
#111922
#108572
#108442