Finite Group Databasebeta

← #198276

Subgroups — #198276

Order 512.

1047 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#9250C214yesmaximal, complemented
7#9248C214yesmaximal, complemented
8C2×#1292C211yesmaximal, characteristic
9C2×#1283C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3249—22no—
13#3249—22no—
14#3249—22no—
15#3249—22no—
16C2×#477C412yes—
17C2×#477C2212yes—
18C2×#467C414yes—
19C2×#462C412yes—
20C2×#462C2212yes—
21Q8×M16—22no—
22Q8×M16C2214yes—
23#2962—22no—
24#2962—22no—
25#2961—22no—
26#2960—22no—
27C2×#431C2211yescharacteristic
28#2905—21no—
29C2×#423C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×C4⋊C8C2211yescharacteristic
32#2873—21no—
33C2×C4×M16C2211yescharacteristic
34#1998—22no—
35#1997C2212yes—
36#1997C2212yes—
37#1742—21no—
38#1742—21no—
39#1550—22no—
40#1549C2212yes—
41#1548C2212yes—
42#1542—22no—
43#1493—22no—
44#1487C2214yes—
45#1481—22no—
46#1292C2214yes—
47#1283C2214yes—
48C22×Q16C2×C414yes—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16D411yescharacteristic
52C22×M16D411yescharacteristic
53C23×C8C2×C412yes—
54#533—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#533D412yes—
67#533D412yes—
68#533D412yes—
69#533D412yes—
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×C4⋊Q8C2×C411yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#125—21no—
75C2×#125—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2×C412yes—
79C2×C22⋊Q8C2×C412yes—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2×C412yes—
83C2×C4×Q8C2312yes—
84C2×#117—21no—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#477—24no—
88#477—24no—
89#477—24no—
90#477—24no—
91#467—28no—
92#467—28no—
93#462—24no—
94#462—24no—
95#462—24no—
96#462—24no—
97#446—24no—
98#446—28no—
99#446—24no—
100#446—28no—
101C8×Q8—24no—
102C8×Q8—24no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431C2312yes—
107C2×#107C2×C414yes—
108#424—24no—
109#424—24no—
110#423—24no—
111#423—21no—
112#423—21no—
113#423—21no—
114#423—21no—
115#423—21no—
116#423D412yes—
117#423C2312yes—
118#423C2312yes—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2×C412yes—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2×C414yes—
131C2×#103C2×C414yes—
132C2×C22⋊C8—21no—
133#405—24no—
134#405—24no—
135C4×M16—24no—
136C4×M16—21no—
137C4×M16—21no—
138C4×M16D412yes—
139C4×M16C2312yes—
140C2×#97—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143#343—22no—
144#343—22no—
145#343—22no—
146#343—22no—
147#333—24no—
148#332—24no—
149Q8⋊C8—28no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8C22×C412yes—
153C22×Q8C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C2×Q16—28no—
158C2×Q16—84no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16Q1614yescomplemented
173C2×M16C2×D412yes—
174C2×M16C2×D412yes—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8—82no—
183C22×C8C22×C414yes—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22⋊C412yes—
197C4⋊Q8C22⋊C412yes—
198C4⋊Q8C22⋊C412yes—
199C4⋊Q8C22⋊C412yes—
200#125—22no—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—41no—
205#125—41no—
206#125—41no—
207#125—41no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—41no—
219C22⋊Q8—41no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8C22⋊C414yes—
223C22⋊Q8C22⋊C414yes—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8C2×D414yes—
239C4×Q8C2×D414yes—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—41no—
245#117—41no—
246#117—41no—
247#117—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#107—28no—
277C4⋊C8—28no—
278C4⋊C8—28no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—41no—
290C4⋊C8—41no—
291C4⋊C8—41no—
292C4⋊C8C2×D414yes—
293#103—28no—
294#103—28no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297#97—28no—
298#97—22no—
299#97—22no—
300C4×C8—28no—
301C4×C8—22no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—41no—
305C24C4×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8C2×C22⋊C414yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—81no—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4#12311yescharacteristic
342C22×C4C2×Q1612yes—
343C22×C4#13512yes—
344C22×C4#13712yes—
345C22×C4#13712yes—
346C22×C4C23×C411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4111yescharacteristic
349Q16—88no—
350M16—22no—
351M16—24no—
352M16—22no—
353M16—22no—
354M16—42no—
355M16—42no—
356M16—42no—
357M16—44no—
358M16—42no—
359M16—42no—
360M16—82no—
361M16C2×Q1614yes—
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—82no—
373C2×C8—82no—
374C2×C8—82no—
375C2×C8—82no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4C2×C22⋊C414yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C42—24no—
405C42—24no—
406C42—42no—
407C42—42no—
408C42C2×C22⋊C411yescharacteristic
409C42C2×C22⋊C411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42#12011yescharacteristic
413C42#12011yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12311yescharacteristic
419C42#12311yescharacteristic
420C42C22×D411yescharacteristic
421C42C22×D411yescharacteristic
422C42C22×D411yescharacteristic
423C42C2×#4111yescharacteristic
424C23—41no—
425C23—41no—
426C23#38611yescharacteristic
427C23#38611yescharacteristic
428C23C4×Q1612yes—
429C23#44112yes—
430C23C2×C4×D411yescharacteristic
431Q8—24no—
432Q8—24no—
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#38611yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4C4×Q1612yes—
471C2×C4#44112yes—
472C2×C4#45112yes—
473C2×C4#45112yes—
474C2×C4#45212yes—
475C2×C4#45212yes—
476C2×C4#46212yes—
477C2×C4#46212yes—
478C2×C4#46412yes—
479C2×C4#46412yes—
480C2×C4C22×C22⋊C411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4C2×#12011yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12311yescharacteristic
486C2×C4C22×Q1611yescharacteristic
487C2×C4C2×#13511yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C2×C4C2×#13711yescharacteristic
490C8—24no—
491C8—44no—
492C8—42no—
493C8—44no—
494C8—82no—
495C8—82no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—42no—
503C22—42no—
504C22#187812yes—
505C22#187812yes—
506C22C2×#38611yescharacteristic
507C22C2×C4×Q1611yescharacteristic
508C22C2×#44111yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—44no—
518C4—82no—
519C4—82no—
520C4#187812yes—
521C4#187812yes—
522C4#288811yescharacteristic
523C4#288911yescharacteristic
524C4C2×#45111yescharacteristic
525C4C2×#45211yescharacteristic
526C4C2×#46211yescharacteristic
527C4C2×#46411yescharacteristic
528C2—22no—
529C2—22no—
530C2—21no—
531C2—21no—
532C2C2×#187811yescharacteristic
533C2#2099211yescharacteristic
534C2#2099511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9250C2semidirect
7. #9248C2semidirect
172. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#601922
#213372
#177082
#174862
#92502
#92482
C2×#12922
C2×#12832