Finite Group Databasebeta

← #198596

Subgroups — #198596

Order 512.

1035 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#9270C214yesmaximal, complemented
7#9268C214yesmaximal, complemented
8C2×#1301C211yesmaximal, characteristic
9#7932C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3230—21no—
14#3230C2212yes—
15#3209—21no—
16#3209C2212yes—
17#3189C414yes—
18#3170—22no—
19#3104—22no—
20#3104C414yes—
21#2962—22no—
22#2962—22no—
23#2962—22no—
24#2962C2214yes—
25C2×#446—21no—
26C2×#446—22no—
27C2×C8×Q8—21no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#424—21no—
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C2×#405—21no—
34C2×C4×M16C2211yescharacteristic
35#1998—22no—
36#1997C2212yes—
37#1997C2212yes—
38#1742—21no—
39#1742—21no—
40#1550C2214yes—
41#1549—21no—
42#1548—21no—
43#1542—21no—
44#1542—21no—
45#1542—22no—
46#1542—22no—
47C2×#333C2212yes—
48C2×#332C2212yes—
49#1301C2214yes—
50#1283C2214yes—
51C22×M16—21no—
52C22×M16D411yescharacteristic
53C22×M16D411yescharacteristic
54C23×C8—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533C2×C412yes—
68#533D412yes—
69#533D412yes—
70#533C2312yes—
71C2×C4⋊Q8D411yescharacteristic
72C2×C4⋊Q8D411yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125—21no—
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8C2×C412yes—
80C2×C22⋊Q8C2×C412yes—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8C2×C412yes—
84C2×C4×Q8D412yes—
85C2×#117—21no—
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#500—22no—
89#494—22no—
90#487—22no—
91#487—22no—
92#484—24no—
93#484C2×C418yes—
94#477—24no—
95#477—24no—
96#467—28no—
97#462—24no—
98#462—24no—
99#451—24no—
100#451C2×C418yes—
101#446—28no—
102#446—24no—
103#446—28no—
104#446—24no—
105C8×Q8—24no—
106C8×Q8—24no—
107C4×Q16—28no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431C2×C412yes—
115#431C2312yes—
116#424—24no—
117#424—24no—
118#423—21no—
119#423—21no—
120#423—21no—
121#423C2312yes—
122C2×C4⋊C8—22no—
123C2×C4⋊C8—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8D411yescharacteristic
131C2×C4⋊C8D411yescharacteristic
132C2×C4⋊C8D412yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C22⋊C8—21no—
136C2×C22⋊C8C2×C412yes—
137#405—24no—
138#405—24no—
139C4×M16—21no—
140C4×M16—21no—
141C4×M16D412yes—
142C4×M16C2312yes—
143C2×#97—22no—
144C2×#97—21no—
145C2×C4×C8—22no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149#343—22no—
150#343—22no—
151#343—22no—
152#343—22no—
153#333—24no—
154#333—22no—
155#333—22no—
156#332—24no—
157#332—22no—
158#332—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8C22⋊C412yes—
162C22×Q8C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C2×Q16—28no—
167C2×M16—21no—
168C2×M16—22no—
169C2×M16—21no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—41no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16Q1614yescomplemented
177C2×M16C2×D412yes—
178C2×M16C2×D412yes—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—41no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C22⋊C412yes—
203C4⋊Q8C22⋊C412yes—
204C4⋊Q8C2×D412yes—
205C4⋊Q8C2×D412yes—
206#125—22no—
207#125—22no—
208#125—22no—
209#125—22no—
210#125—41no—
211#125—41no—
212#125—41no—
213#125—41no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8C22⋊C414yes—
229C22⋊Q8C22×C414yes—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8—41no—
243C4×Q8—41no—
244C4×Q8C22⋊C414yes—
245C4×Q8C2×D414yes—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—41no—
251#117—41no—
252#117—41no—
253#117—41no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22⋊C412yes—
265C2×C4⋊C4C22×C412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42#4111yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283C4⋊C8—28no—
284C4⋊C8—28no—
285C4⋊C8—22no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—22no—
295C4⋊C8—41no—
296C4⋊C8—41no—
297C4⋊C8C2×D412yes—
298C4⋊C8C2×D412yes—
299#103—28no—
300#103—28no—
301C22⋊C8—22no—
302C22⋊C8—41no—
303C22⋊C8—41no—
304C22⋊C8C22×C414yes—
305#97—28no—
306#97—22no—
307#97—22no—
308C4×C8—28no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—41no—
314C4×C8—41no—
315C24C4×D411yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—82no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8C2×C22⋊C414yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C4×D411yescharacteristic
338C22×C4C4×D411yescharacteristic
339C22×C4C4×D411yescharacteristic
340C22×C4C4×D411yescharacteristic
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12111yescharacteristic
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12311yescharacteristic
348C22×C4#12311yescharacteristic
349C22×C4C2×Q1612yes—
350C22×C4#13512yes—
351C22×C4#13712yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356Q16—88no—
357M16—22no—
358M16—24no—
359M16—22no—
360M16—22no—
361M16—42no—
362M16—42no—
363M16—42no—
364M16—82no—
365M16C2×Q1614yes—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—44no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—82no—
381C2×C8—82no—
382C2×C8—82no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4C2×C22⋊C414yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C42—24no—
412C42—24no—
413C42—42no—
414C42—42no—
415C42C2×C22⋊C411yescharacteristic
416C42C2×C22⋊C411yescharacteristic
417C42C4×D411yescharacteristic
418C42C4×D411yescharacteristic
419C42#12011yescharacteristic
420C42#12011yescharacteristic
421C42#12111yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12311yescharacteristic
426C42#12311yescharacteristic
427C42C22×D411yescharacteristic
428C42C22×D411yescharacteristic
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C23—41no—
432C23—41no—
433C23#38611yescharacteristic
434C23#38611yescharacteristic
435C23C4×Q1612yes—
436C23#44112yes—
437C23C2×C4×D411yescharacteristic
438Q8—44no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—82no—
443Q8—82no—
444Q8—82no—
445Q8—82no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38611yescharacteristic
467C2×C4#38611yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4#38611yescharacteristic
471C2×C4#38611yescharacteristic
472C2×C4C4×Q1612yes—
473C2×C4#44112yes—
474C2×C4#45112yes—
475C2×C4#45112yes—
476C2×C4#45212yes—
477C2×C4#45212yes—
478C2×C4#46212yes—
479C2×C4#46212yes—
480C2×C4#46412yes—
481C2×C4#46412yes—
482C2×C4C22×C22⋊C411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4C22×Q1611yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4C2×#13711yescharacteristic
491C2×C4C2×#13711yescharacteristic
492C8—24no—
493C8—44no—
494C8—42no—
495C8—42no—
496C8—42no—
497C8—82no—
498C8—82no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—42no—
506C22—42no—
507C22#187812yes—
508C22#187812yes—
509C22C2×#38611yescharacteristic
510C22C2×C4×Q1611yescharacteristic
511C22C2×#44111yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—44no—
520C4—44no—
521C4—82no—
522C4—82no—
523C4#187812yes—
524C4#187812yes—
525C4C2×#45111yescharacteristic
526C4C2×#45211yescharacteristic
527C4C2×#46211yescharacteristic
528C4C2×#46411yescharacteristic
529C4#306712yes—
530C2—22no—
531C2—22no—
532C2—21no—
533C2—21no—
534C2C2×#187811yescharacteristic
535C2#2100311yescharacteristic
536C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9270C2semidirect
7. #9268C2semidirect
176. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
C2×#29622
#213372
#177082
C2×#15422
#92702
#92682
C2×#13012
#79322