Finite Group Databasebeta

← #106985

Subgroups — #106985

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#117C211yesmaximal, characteristic
2C2×#1487C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#10856C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6#7957C211yesmaximal, characteristic
7#7924C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2355—22no—
14#2355—21no—
15Q8×C4⋊C4—21no—
16#2345—22no—
17#2345C412yes—
18Q8×C22⋊C4C412yes—
19#2295—22no—
20#2295—21no—
21#2285—21no—
22#2285C2211yescharacteristic
23C42×Q8—21no—
24C4×#117—21no—
25C4×#117C2211yescharacteristic
26C2×#344—21no—
27C2×#344—21no—
28#1487—21no—
29#1487—21no—
30#1487—21no—
31#1487—21no—
32#1487—22no—
33#1487C414yescomplemented
34C2×Q8⋊C8—22no—
35C2×Q8⋊C8C414yescomplemented
36#1303—22no—
37#1286—22no—
38#1282—22no—
39C23×Q8C2×C411yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C2×C4×Q8—22no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8D411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×Q8Q812yes—
57C2×#117—21no—
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431C2×C412yes—
72#423—21no—
73#423—21no—
74#423—21no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423C2×C412yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2×C412yes—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2×C412yes—
90#391—22no—
91#391—24no—
92#389—24no—
93#389—22no—
94#387—24no—
95#387—24no—
96#387—24no—
97#384—22no—
98#384—21no—
99#382—22no—
100#382D412yes—
101#380—21no—
102#380—22no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119#376—22no—
120#376—21no—
121C43—22no—
122C43—21no—
123#344—42no—
124#344—42no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128Q8⋊C8—24no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8C4212yes—
134C22×Q8C22⋊C411yescharacteristic
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C4⋊C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—22no—
143C2×M16—41no—
144C2×M16—41no—
145C2×M16—41no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—41no—
151C2×M16—41no—
152C2×M16—41no—
153C2×M16—41no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—41no—
157C22×C8—41no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—48no—
175C4×Q8C22⋊C412yes—
176C4×Q8C22⋊C412yes—
177C4×Q8C22⋊C412yes—
178C4×Q8C22⋊C412yes—
179C4×Q8C4⋊C412yes—
180C4×Q8C4⋊C412yes—
181C4×Q8C4⋊C412yes—
182C4×Q8C4⋊C412yes—
183#117—21no—
184#117—21no—
185#117—21no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—41no—
190#117—41no—
191#117—41no—
192#117—41no—
193#117SD1612yes—
194#117Q1612yes—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C4212yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C4⋊C412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×Q811yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C22⋊C8—22no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—41no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95C22⋊C414yes—
275C24C2×C22⋊C411yescharacteristic
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8#9512yes—
290C2×Q8#9512yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4C2×C4211yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C4⋊C411yescharacteristic
310C22×C4#11711yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4C22⋊Q811yescharacteristic
323C22×C4C22⋊Q811yescharacteristic
324C22×C4C2×SD1611yescharacteristic
325C22×C4C2×Q1611yescharacteristic
326C22×C4#13511yescharacteristic
327C22×C4#13511yescharacteristic
328C22×C4AGL(1,ℤ/8)11yescharacteristic
329C22×C4#13711yescharacteristic
330C22×C4#13711yescharacteristic
331C22×C4#13711yescharacteristic
332M16—42no—
333M16—42no—
334M16—81no—
335M16—81no—
336M16—81no—
337M16—81no—
338M16—81no—
339M16—81no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—82no—
347C2×C8—81no—
348C2×C8—81no—
349C2×C8—82no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4#9512yes—
373C4⋊C4#9512yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4#10314yes—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—48no—
391C42—44no—
392C42—41no—
393C42—42no—
394C42—41no—
395C42—42no—
396C42—82no—
397C42—84no—
398C42C2×C22⋊C411yescharacteristic
399C42C2×C22⋊C411yescharacteristic
400C42C2×C4⋊C411yescharacteristic
401C42C2×C4⋊C411yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42C22⋊Q811yescharacteristic
409C42C22⋊Q811yescharacteristic
410C42#12311yescharacteristic
411C42#12311yescharacteristic
412C42#12311yescharacteristic
413C42#12311yescharacteristic
414C23—41no—
415C23—41no—
416C23C4×C22⋊C411yescharacteristic
417C23#37911yescharacteristic
418C23#38011yescharacteristic
419C23C2×#10311yescharacteristic
420C23#41611yescharacteristic
421C23#41711yescharacteristic
422C23#41911yescharacteristic
423Q8—24no—
424Q8—22no—
425Q8—22no—
426Q8—22no—
427Q8—44no—
428Q8—44no—
429Q8—42no—
430Q8#34414yescomplemented
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
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—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4C2×#9511yescharacteristic
465C2×C4C4×C22⋊C411yescharacteristic
466C2×C4#38011yescharacteristic
467C2×C4#38111yescharacteristic
468C2×C4#38511yescharacteristic
469C2×C4#38511yescharacteristic
470C2×C4#38511yescharacteristic
471C2×C4#38511yescharacteristic
472C2×C4#38611yescharacteristic
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4C4×SD1611yescharacteristic
477C2×C4C4×SD1611yescharacteristic
478C2×C4C4×Q1611yescharacteristic
479C2×C4C4×Q1611yescharacteristic
480C2×C4#44011yescharacteristic
481C2×C4#44011yescharacteristic
482C2×C4#44111yescharacteristic
483C2×C4#44111yescharacteristic
484C2×C4#44811yescharacteristic
485C2×C4#44811yescharacteristic
486C2×C4#44911yescharacteristic
487C2×C4#44911yescharacteristic
488C2×C4#45111yescharacteristic
489C2×C4#45111yescharacteristic
490C2×C4#45211yescharacteristic
491C2×C4#45211yescharacteristic
492C2×C4#46011yescharacteristic
493C2×C4#46211yescharacteristic
494C2×C4#46411yescharacteristic
495C2×C4#46411yescharacteristic
496C2×C4#47511yescharacteristic
497C2×C4#47711yescharacteristic
498C2×C4#47911yescharacteristic
499C2×C4#47911yescharacteristic
500C8—82no—
501C8—82no—
502C8—82no—
503C8—82no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22C22⋊#9511yescharacteristic
515C22C4×#10311yescharacteristic
516C22#176311yescharacteristic
517C22#178711yescharacteristic
518C22#178811yescharacteristic
519C22#180311yescharacteristic
520C22#180511yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—24no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—84no—
535C4C2×#34412yes—
536C4#186311yescharacteristic
537C4#186311yescharacteristic
538C4#186511yescharacteristic
539C4#186511yescharacteristic
540C4#187811yescharacteristic
541C4#187811yescharacteristic
542C4#188011yescharacteristic
543C4#188011yescharacteristic
544C2—21no—
545C2—21no—
546C2—21no—
547C2—21no—
548C2—21no—
549C2—21no—
550C2#1073811yescharacteristic
551C2#1078311yescharacteristic
552C2#1078911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #1487C4semidirect
35. C2×Q8⋊C8C4semidirect
430. Q8#344semidirect

Maximal subgroups

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

subgroupindex
Q8×#1172
C2×#14872
#108562
#80952
#79572
#79242