Finite Group Databasebeta

← #222998

Subgroups — #222998

Order 512.

978 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#34941C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3C4×C8×D4C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9071C212yesmaximal, complemented
7#8942C212yesmaximal, complemented
8#8710C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10C4×Q8⋊C8C211yesmaximal, characteristic, complemented
11#7897C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14C2×#423C2212yes—
15C22×C4×C8C2212yes—
16#2574—22no—
17#2567C411yescharacteristic
18#2563C411yescharacteristic
19#2552C411yescharacteristic
20#2538C411yescharacteristic
21#2508—21no—
22#2497C412yes—
23#2454—21no—
24#2423C412yes—
25#2396—22no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×#125—21no—
28C4×C22⋊Q8C2212yes—
29#2293—21no—
30#2285C2212yes—
31C42×Q8C2211yescharacteristic
32C42×D4C2211yescharacteristic
33#1939—21no—
34#1851—22no—
35#1842C2211yescharacteristic
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117—22no—
66C22×C42C2312yes—
67#436—44no—
68C8×D4—28no—
69#423—24no—
70#423—42no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C22⋊C8—22no—
79C2×C22⋊C8—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#399—22no—
91#398—22no—
92#398—22no—
93#396—22no—
94#396—22no—
95#395—22no—
96#393—22no—
97#391—21no—
98#391C2×C412yes—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2×C412yes—
108#388—22no—
109#388—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#384C2×C412yes—
117#382—21no—
118#382—22no—
119#380—22no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4C2312yes—
135C4×C4⋊C4C2311yescharacteristic
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140C43—21no—
141C43C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146Q8⋊C8—22no—
147Q8⋊C8—22no—
148Q8⋊C8—42no—
149C22×Q8—21no—
150C22×Q8C22×C411yescharacteristic
151C22×D4C22⋊C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—82no—
158C22×C8—21no—
159C22×C8—28no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—44no—
166C22×C8—44no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169#125—44no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—41no—
180C4×Q8—81no—
181C4×Q8—81no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4SD1614yes—
186C4×D4Q1614yes—
187#117—82no—
188#117—82no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2411yescharacteristic
238C4⋊C8—24no—
239C4⋊C8—21no—
240C4⋊C8—22no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—44no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8#4112yes—
248C22⋊C8—28no—
249C22⋊C8—44no—
250C4×C8—24no—
251C4×C8—21no—
252C4×C8—22no—
253C4×C8—21no—
254C4×C8—22no—
255C4×C8—21no—
256C4×C8—44no—
257C4×C8—42no—
258C4×C8—42no—
259C4×C8#4112yes—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274C24C2×C22⋊C412yes—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—42no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×Q8#13112yes—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4#10314yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
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—24no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×SD1614yes—
315C22×C4C2×Q1614yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319M16—88no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—22no—
326C2×C8—416no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—81no—
363C4⋊C4—82no—
364C4⋊C4#10314yes—
365C4⋊C4#10414yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4#10318yes—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—22no—
384C42—41no—
385C42—42no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—81no—
395C42#13112yes—
396C42#13112yes—
397C42C2×SD1612yes—
398C42C2×Q1612yes—
399C42C2×#4111yescharacteristic
400C42C2×#4111yescharacteristic
401C42H(5,𝔽2)11yescharacteristic
402C422−1+411yescharacteristic
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23C2×#10314yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—22no—
412Q8—22no—
413Q8—41no—
414Q8—41no—
415Q8—84no—
416D4—28no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4#40812yes—
462C2×C4#40812yes—
463C2×C4C2×#10312yes—
464C2×C4C2×#10314yes—
465C2×C4C2×#10312yes—
466C2×C4C2×#10414yes—
467C2×C4C2×#10412yes—
468C2×C4C4×#4111yescharacteristic
469C2×C4#52011yescharacteristic
470C2×C4#54611yescharacteristic
471C2×C4#54611yescharacteristic
472C2×C4#54711yescharacteristic
473C2×C4#54711yescharacteristic
474C2×C4C2×#13111yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4C2×#13111yescharacteristic
477C2×C4#56811yescharacteristic
478C2×C4C22×SD1611yescharacteristic
479C2×C4C22×Q1611yescharacteristic
480C2×C4#57611yescharacteristic
481C2×C4#57711yescharacteristic
482C8—24no—
483C8—42no—
484C8—44no—
485C8—42no—
486C8—81no—
487C8—82no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22#233911yescharacteristic
496C22C2×#40811yescharacteristic
497C22#288011yescharacteristic
498C22C22×#10311yescharacteristic
499C22#289711yescharacteristic
500C22C22×#10411yescharacteristic
501C22#290111yescharacteristic
502C4—22no—
503C4—22no—
504C4—24no—
505C4—21no—
506C4—21no—
507C4—24no—
508C4—41no—
509C4—42no—
510C4—42no—
511C4—81no—
512C4—82no—
513C4—81no—
514C4—82no—
515C4—81no—
516C4#149314yes—
517C4#296611yescharacteristic
518C4#296911yescharacteristic
519C4#299311yescharacteristic
520C4#299411yescharacteristic
521C4#329511yescharacteristic
522C4#329911yescharacteristic
523C4#333511yescharacteristic
524C4#333811yescharacteristic
525C2—44no—
526C2C2×#149311yescharacteristic
527C2#1748311yescharacteristic
528C2#1750911yescharacteristic
529C2#1752011yescharacteristic
530C2#2247411yescharacteristic
531C2#2248711yescharacteristic
532C2#2250511yescharacteristic
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×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9870C2semidirect
5. #9864C2semidirect
6. #9071C2semidirect
7. #8942C2semidirect
8. #8710C2semidirect
9. #8564C2semidirect
10. C4×Q8⋊C8C2semidirect
11. #7897C2semidirect

Maximal subgroups

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

subgroupindex
#349412
#226502
C4×C8×D42
#98702
#98642
#90712
#89422
#87102
#85642
C4×Q8⋊C82
#78972