Finite Group Databasebeta

← #233973

Subgroups — #233973

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10233C218yesmaximal, complemented
8#8039C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—21no—
16C2×#474—24no—
17C2×#474—24no—
18C2×#474—22no—
19C2×#474C2214yes—
20C2×#462C2214yescomplemented
21C2×#451—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×#107—21no—
25C22×C4⋊C8C2211yescharacteristic
26C22×#102—21no—
27#2848—22no—
28#2626—22no—
29#2580—22no—
30#2566—22no—
31#2563—21no—
32#2542—21no—
33#2538C2214yes—
34#2317—21no—
35#2309—21no—
36#2285C2211yescharacteristic
37#1990C2214yescomplemented
38#1987—22no—
39#1824—22no—
40#1481—24no—
41#1478—24no—
42C2×Q8⋊C8C2214yescomplemented
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×Q16—24no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8C2312yes—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85#494—28no—
86#487—28no—
87#474—216no—
88#474—28no—
89#474—28no—
90#462—28no—
91#462—28no—
92#451—216no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431D414yes—
97C2×#107—28no—
98C2×#107—24no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8D414yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103—24no—
106C2×#102—28no—
107C2×#102—24no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2312yes—
112#399—24no—
113#396—28no—
114#393—24no—
115#390—22no—
116#390—22no—
117#390—24no—
118#389—24no—
119#385—28no—
120#385—24no—
121#384—22no—
122#384C2312yes—
123#382—22no—
124#382—21no—
125#380—21no—
126#380—24no—
127#380D412yes—
128#379—24no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131#340—28no—
132Q8⋊C8—28no—
133D4⋊C8—28no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—84no—
141C22×Q8C2×D412yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4Q1614yescomplemented
149C22×D4C2×D412yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—84no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×Q16—216no—
161C2×Q16—88no—
162C22×C8—24no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—42no—
167C22×C8—42no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—24no—
171C4⋊Q8—22no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8C2×D414yes—
182#125—42no—
183#125—42no—
184C22⋊Q8—24no—
185C22⋊Q8—24no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—88no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42C2411yescharacteristic
249#107—216no—
250C4⋊C8—24no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—42no—
254C4⋊C8C2×D414yes—
255C4⋊C8C2×D414yes—
256#103—216no—
257#102—216no—
258C22⋊C8—24no—
259C22⋊C8—42no—
260C4×C8—24no—
261C4×C8—22no—
262C4×C8—22no—
263#95—28no—
264#95—22no—
265#95—22no—
266C24—22no—
267C24—24no—
268C24—42no—
269C24—42no—
270C24—82no—
271C24—82no—
272C24#12011yescharacteristic
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—28no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8—48no—
281C2×Q8—82no—
282C2×Q8—88no—
283C2×Q8—84no—
284C2×Q8—88no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—28no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4C2×Q1614yes—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—48no—
310C22×C4—44no—
311C22×C4—84no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4#12012yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12012yes—
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12712yes—
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332Q16—816no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—44no—
338C2×C8—48no—
339C2×C8—44no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—88no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—42no—
371C42—42no—
372C42—44no—
373C42#12012yes—
374C42#12012yes—
375C42#12112yes—
376C42#12112yes—
377C42#12112yes—
378C42#12712yes—
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C23—22no—
384C23—22no—
385C23—24no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—44no—
390C23—48no—
391C23—88no—
392C23—84no—
393C23—84no—
394C23—88no—
395C23#39212yes—
396C23#45112yes—
397C23#45212yes—
398C23C2×#12011yescharacteristic
399Q8—44no—
400Q8—48no—
401Q8—44no—
402Q8—84no—
403Q8—88no—
404Q8—84no—
405D4—24no—
406D4—24no—
407D4—42no—
408D4—48no—
409D4—42no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—88no—
428C2×C4—84no—
429C2×C4—88no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4#39212yes—
434C2×C4#39212yes—
435C2×C4#39212yes—
436C2×C4#45114yes—
437C2×C4#45112yes—
438C2×C4#45214yes—
439C2×C4#45212yes—
440C2×C4#46714yes—
441C2×C4#47014yes—
442C2×C4C2×#12011yescharacteristic
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12711yescharacteristic
448C2×C4C22×Q1611yescharacteristic
449C2×C4C2×#13511yescharacteristic
450C2×C4C2×#13711yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57614yes—
453C8—44no—
454C8—44no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—24no—
459C22—24no—
460C22—44no—
461C22—48no—
462C22—816no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22#200114yes—
467C22C2×#39211yescharacteristic
468C22C2×#45111yescharacteristic
469C22C2×#45211yescharacteristic
470C22#300812yes—
471C22#302212yes—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—82no—
477C4—82no—
478C4—84no—
479C4—82no—
480C4—84no—
481C4#200114yes—
482C4C2×#45111yescharacteristic
483C4C2×#45211yescharacteristic
484C4#300812yes—
485C4C2×#46711yescharacteristic
486C4C2×#47011yescharacteristic
487C4#305812yes—
488C2—24no—
489C2—84no—
490C2#1890512yes—
491C2#1892712yes—
492C2C2×#200111yescharacteristic
493C2#2338711yescharacteristic
494C2#2340811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3209C2semidirect
4. #21834C2semidirect
5. C2×#1481C2semidirect
7. #10233C2semidirect
8. #8039C2semidirect
20. C2×#462C22semidirect
37. #1990C22semidirect
42. C2×Q8⋊C8C22semidirect
44. #1296C22semidirect
148. C22×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4742
#351722
#218342
C2×#14812
C2×#14782
#102332
#80392