Finite Group Databasebeta

← #368939

Subgroups — #368939

Order 512.

1184 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
1C2×#3165C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21329C211yesmaximal, characteristic
5#21313C211yesmaximal, characteristic
6#11214C214yesmaximal
7#10904C214yesmaximal
8C4⋊#338C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—22no—
13#3165—22no—
14#3165—22no—
15#3165—22no—
16C2×#501C414yescomplemented
17C2×#500C412yescomplemented
18C2×#498C412yescomplemented
19Q8×M16—22no—
20Q8×M16—22no—
21#2962—22no—
22#2962—22no—
23#2961—22no—
24#2960—22no—
25C2×#428C2214yes—
26#2905C2212yes—
27C2×#423C2211yescharacteristic
28C2×#423C2211yescharacteristic
29#2873C2212yes—
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1997—22no—
34#1997—22no—
35#1995—22no—
36#1994—22no—
37#1988—22no—
38#1987—22no—
39C2×#344C2212yes—
40C2×#344C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2214yes—
44#1733C2214yes—
45C2×#338C2214yes—
46C23×Q8—21no—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533D412yes—
57#533D412yes—
58#533D412yes—
59#533D412yes—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2×C412yes—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×#125C2×C412yes—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8—21no—
79C2×#117C2312yes—
80C2×#117C2312yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2312yes—
83C22×C42C2311yescharacteristic
84#501—28no—
85#501—28no—
86#500—24no—
87#500—24no—
88#498—24no—
89#498—24no—
90#446—24no—
91#446—24no—
92#446—28no—
93#446—28no—
94C8×Q8—24no—
95C8×Q8—24no—
96#428—28no—
97#428—28no—
98C2×#107C2×C418yes—
99C2×#106C2×C418yes—
100#424—24no—
101#424—24no—
102#423—24no—
103#423—21no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8D412yes—
114C2×C4⋊C8D412yes—
115#405—24no—
116#405—24no—
117C4×M16—24no—
118C4×M16—21no—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16—21no—
123C4×M16—21no—
124C4×M16D412yes—
125C4×M16D412yes—
126C2×#97—21no—
127C2×#97C2×C412yes—
128C2×C4×C8—21no—
129C2×C4×C8C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#344—24no—
137#344Q818yescomplemented
138#343—24no—
139#343—24no—
140#340—28no—
141#339—28no—
142#338—28no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—84no—
152C23×C4—22no—
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C23×C4C2411yescharacteristic
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—24no—
172C2×M16—24no—
173C2×M16C2×D412yes—
174C2×M16C2×D412yes—
175C2×M16C2×Q814yes—
176C2×M16C2×Q814yes—
177C2×M16#4114yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8C22×C414yes—
182C22×C8C2×D414yes—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202C4⋊Q8—41no—
203C4⋊Q8C22⋊C412yes—
204C4⋊Q8C22⋊C412yes—
205C4⋊Q8C22⋊C412yes—
206C4⋊Q8C22⋊C412yes—
207#125—22no—
208#125—22no—
209#125C22⋊C414yes—
210#125C22⋊C414yes—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227#117—22no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117C2×D414yes—
233#117C2×D414yes—
234#117C2×Q814yes—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C22×C414yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272#107—216no—
273#106—216no—
274C4⋊C8—28no—
275C4⋊C8—28no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284#97—28no—
285#97—22no—
286#97—22no—
287#97—22no—
288#97—22no—
289C4×C8—28no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—22no—
294#95—41no—
295#95—41no—
296C24C2×#4111yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—84no—
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q812yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C42−1+412yes—
343C22×C42−1+412yes—
344M16—22no—
345M16—24no—
346M16—24no—
347M16—24no—
348M16—24no—
349M16—24no—
350M16—22no—
351M16—22no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—44no—
356M16—42no—
357M16—42no—
358M16—44no—
359M16C4⋊Q814yes—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—28no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—44no—
370C2×C8C4×Q818yes—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4C2×C22⋊C414yes—
392C4⋊C4C2×C22⋊C414yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—42no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—42no—
401C42—82no—
402C42—82no—
403C42C2×C22⋊C411yescharacteristic
404C42C2×C22⋊C411yescharacteristic
405C42C2×C22⋊C411yescharacteristic
406C42C2×C22⋊C411yescharacteristic
407C42C4×D411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42#12011yescharacteristic
412C42#12011yescharacteristic
413C42#12011yescharacteristic
414C42#12011yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12111yescharacteristic
419C23—41no—
420C23—41no—
421C23C2×C4×D411yescharacteristic
422C23C2×#12111yescharacteristic
423C23C2×#12311yescharacteristic
424C23#53312yes—
425C23#54112yes—
426Q8—44no—
427Q8—44no—
428Q8—82no—
429Q8—82no—
430Q8—84no—
431Q8—84no—
432Q8—84no—
433Q8—84no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4C22×C22⋊C411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×Q812yes—
468C2×C4#51912yes—
469C2×C4C2×#12011yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×C4⋊Q811yescharacteristic
472C2×C4#53211yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#54712yes—
476C2×C4#54812yes—
477C2×C4#54812yes—
478C2×C4D4×Q812yes—
479C2×C4D4×Q812yes—
480C2×C4D4×Q812yes—
481C2×C4#55612yes—
482C2×C4#55712yes—
483C8—28no—
484C8—24no—
485C8—44no—
486C8—42no—
487C8—82no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22C2×#38611yescharacteristic
497C22C4×C4⋊Q811yescharacteristic
498C22#232711yescharacteristic
499C22#242912yes—
500C22#244612yes—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—44no—
508C4—44no—
509C4—84no—
510C4—84no—
511C4#190212yes—
512C4#190312yes—
513C4Q8×C22⋊C412yes—
514C4#236512yes—
515C4#243211yescharacteristic
516C4#243411yescharacteristic
517C4#254511yescharacteristic
518C4#254611yescharacteristic
519C2—22no—
520C2—21no—
521C2—21no—
522C2—22no—
523C2#1402411yescharacteristic
524C2C2×#190211yescharacteristic
525C2C2×#190311yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#501C4semidirect
17. C2×#500C4semidirect
18. C2×#498C4semidirect
137. #344Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#601922
C2×#19972
#213292
#213132
#112142
#109042
C4⋊#3382
C4×#3442