Finite Group Databasebeta

← #223000

Subgroups — #223000

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34943C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9072C212yesmaximal, complemented
7#8943C212yesmaximal, complemented
8#8711C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10C4×D4⋊C8C211yesmaximal, characteristic, complemented
11#7898C212yesmaximal, complemented
12C2×C8×D4—21no—
13C2×#432C2212yes—
14C22×C4×C8C2212yes—
15#2572—22no—
16#2571C411yescharacteristic
17#2563C411yescharacteristic
18#2563C411yescharacteristic
19#2537C411yescharacteristic
20#2495—21no—
21#2495C412yes—
22#2420—21no—
23#2420C412yes—
24#2398—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#121—21no—
27C4×#121C2212yes—
28#2283—21no—
29#2283C2212yes—
30C42×D4C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1924—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C23×C8—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#121—21no—
56C2×#121C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C22×C42—21no—
66C22×C42C2312yes—
67C8×D4—28no—
68#432—24no—
69#432—42no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—24no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—42no—
86C2×C4×C8C2312yes—
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#400—22no—
90#400—22no—
91#397—22no—
92#397—22no—
93#394—22no—
94#394—22no—
95#393—22no—
96#393—22no—
97#391—21no—
98#390—21no—
99#390C2×C412yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#384C2×C411yescharacteristic
109#383—22no—
110#383—22no—
111#382—21no—
112#382—21no—
113#382C2×C412yes—
114#382C2×C412yes—
115#381—22no—
116#381—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2312yes—
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140C43C2311yescharacteristic
141#336—22no—
142#336—21no—
143#336—22no—
144#336—21no—
145D4⋊C8—22no—
146D4⋊C8—22no—
147D4⋊C8—42no—
148C22×Q8—21no—
149C22×D4—22no—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C22×C8—21no—
159C22×C8—28no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8—44no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169#121—24no—
170#121—42no—
171#121—44no—
172C4×Q8—81no—
173C4×Q8—81no—
174C4×D4—24no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—82no—
188C4×D4—82no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—42no—
241C4⋊C8—81no—
242C4⋊C8#4112yes—
243C22⋊C8—28no—
244C22⋊C8—44no—
245C22⋊C8—82no—
246#97—44no—
247C4×C8—24no—
248C4×C8—21no—
249C4×C8—22no—
250C4×C8—21no—
251C4×C8—22no—
252C4×C8—21no—
253C4×C8—44no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8#4112yes—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273C24—21no—
274C24C2×C22⋊C412yes—
275C2×Q8—42no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—44no—
286C2×D4—44no—
287C2×D4—82no—
288C2×D4#10414yes—
289C2×D4#13112yes—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—81no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—416no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—22no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42#13112yes—
402C42#13112yes—
403C42#13512yes—
404C42#13512yes—
405C42C2×#4111yescharacteristic
406C42C2×#4111yescharacteristic
407C42H(5,𝔽2)11yescharacteristic
408C422−1+411yescharacteristic
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23C2×#10414yes—
423C23C22×C22⋊C411yescharacteristic
424Q8—84no—
425D4—22no—
426D4—22no—
427D4—28no—
428D4—41no—
429D4—41no—
430D4—88no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—44no—
450C2×C4—44no—
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—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—84no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#40812yes—
481C2×C4#40812yes—
482C2×C4#41612yes—
483C2×C4#41612yes—
484C2×C4C2×#10412yes—
485C2×C4C4×#4111yescharacteristic
486C2×C4#52011yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#54611yescharacteristic
489C2×C4#54711yescharacteristic
490C2×C4#54711yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4C2×#13111yescharacteristic
493C2×C4C2×#13111yescharacteristic
494C2×C4#56811yescharacteristic
495C2×C4C2×#13511yescharacteristic
496C2×C4C2×#13511yescharacteristic
497C2×C4#57611yescharacteristic
498C2×C4#57711yescharacteristic
499C8—24no—
500C8—42no—
501C8—44no—
502C8—42no—
503C8—81no—
504C8—82no—
505C22—24no—
506C22—24no—
507C22—24no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22#233911yescharacteristic
520C22C2×#40811yescharacteristic
521C22#288011yescharacteristic
522C22C2×#41611yescharacteristic
523C22#289711yescharacteristic
524C22C22×#10411yescharacteristic
525C22#290111yescharacteristic
526C4—22no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—41no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—81no—
535C4—82no—
536C4—81no—
537C4—82no—
538C4—81no—
539C4#149414yes—
540C4#296611yescharacteristic
541C4#296911yescharacteristic
542C4#299311yescharacteristic
543C4#299411yescharacteristic
544C4#330311yescharacteristic
545C4#330511yescharacteristic
546C4#334011yescharacteristic
547C4#334111yescharacteristic
548C2—44no—
549C2—82no—
550C2C2×#149411yescharacteristic
551C2#1748311yescharacteristic
552C2#1750811yescharacteristic
553C2#1751711yescharacteristic
554C2#2247411yescharacteristic
555C2#2248911yescharacteristic
556C2#2250511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. C4×C8×D4C2semidirect
4. #9870C2semidirect
5. #9864C2semidirect
6. #9072C2semidirect
7. #8943C2semidirect
8. #8711C2semidirect
9. #8563C2semidirect
10. C4×D4⋊C8C2semidirect
11. #7898C2semidirect

Maximal subgroups

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

subgroupindex
#349432
#227992
C4×C8×D42
#98702
#98642
#90722
#89432
#87112
#85632
C4×D4⋊C82
#78982