Finite Group Databasebeta

← #224769

Subgroups — #224769

Order 512.

926 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
1C4×#556C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal
5#9673C211yesmaximal, characteristic, complemented
6#9645C212yesmaximal
7#9637C211yesmaximal, characteristic, complemented
8#8566C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic
10#8175C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#556C411yescharacteristic
13#2373—22no—
14#2367C412yes—
15#2365C411yescharacteristic
16#2362—21no—
17#2356C412yes—
18#2355C411yescharacteristic
19#2354C411yescharacteristic
20#2347—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#126—22no—
23C4×#124C2212yes—
24C4×C22⋊Q8—21no—
25C42×Q8C2212yes—
26C42×Q8C2211yescharacteristic
27C42×D4C2211yescharacteristic
28C4×#117—21no—
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1842C2211yescharacteristic
35#1831C2212yes—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#556—24no—
54#556—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2312yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C22×C42—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#391C2×C412yes—
87#391C2×C412yes—
88#390C2×C412yes—
89#389—21no—
90#389—21no—
91#389—21no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#383—22no—
104#382C2×C412yes—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130#376—22no—
131C43—22no—
132C43C2311yescharacteristic
133C43C2312yes—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141#336—22no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144Q8⋊C8—42no—
145Q8⋊C8—42no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148D4⋊C8—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8M1614yes—
163#126—216no—
164#124—24no—
165#124—24no—
166C22⋊Q8—28no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4M1614yes—
188#117—28no—
189#117—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2411yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—42no—
243C4⋊C8—44no—
244C4⋊C8—42no—
245C4⋊C8—81no—
246C4⋊C8#4112yes—
247#97—82no—
248C4×C8—21no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8#4112yes—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—21no—
261#95—21no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271C24—21no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8C2×M1614yes—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4C2×M1612yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—81no—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C4⋊C4—28no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—81no—
353C4⋊C4C2×M1612yes—
354C4⋊C4#13112yes—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—82no—
363C42—28no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—21no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—21no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42—42no—
379C42—42no—
380C42—44no—
381C42—44no—
382C42—42no—
383C42—42no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—82no—
389C42C2×M1612yes—
390C42C2×M1612yes—
391C42#13512yes—
392C42#13512yes—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—22no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23C22×C22⋊C411yescharacteristic
405Q8—24no—
406Q8—24no—
407Q8—42no—
408Q8—42no—
409Q8—44no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4#40712yes—
457C2×C4#40712yes—
458C2×C4#41612yes—
459C2×C4#41612yes—
460C2×C4#42112yes—
461C2×C4C4×#4111yescharacteristic
462C2×C4#52011yescharacteristic
463C2×C4#54811yescharacteristic
464C2×C4#54811yescharacteristic
465C2×C4#55011yescharacteristic
466C2×C4#55011yescharacteristic
467C2×C4C22×M1611yescharacteristic
468C2×C4C22×M1611yescharacteristic
469C2×C4C2×#13111yescharacteristic
470C2×C4#56811yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4#57711yescharacteristic
474C2×C4#57811yescharacteristic
475C8—42no—
476C8—42no—
477C8—42no—
478C8—82no—
479C8—81no—
480C8—82no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22#234111yescharacteristic
489C22C2×#40711yescharacteristic
490C22#288011yescharacteristic
491C22C2×#41611yescharacteristic
492C22#289811yescharacteristic
493C22C2×#42111yescharacteristic
494C22#290111yescharacteristic
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—44no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—81no—
508C4—82no—
509C4#148814yes—
510C4#296511yescharacteristic
511C4#297011yescharacteristic
512C4#298911yescharacteristic
513C4#299011yescharacteristic
514C4#336011yescharacteristic
515C4#336411yescharacteristic
516C4#339311yescharacteristic
517C4#339411yescharacteristic
518C2—82no—
519C2C2×#148811yescharacteristic
520C2#1745411yescharacteristic
521C2#1750811yescharacteristic
522C2#1752511yescharacteristic
523C2#2253211yescharacteristic
524C2#2254111yescharacteristic
525C2#2255111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. #21339C2semidirect
5. #9673C2semidirect
7. #9637C2semidirect
8. #8566C2semidirect
10. #8175C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5562
#228252
#213392
#96762
#96732
#96452
#96372
#85662
#85632
#81752
C4×D4⋊C82