Finite Group Databasebeta

← #194149

Subgroups — #194149

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2C4⋊C4×#41C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8985C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41—21no—
13C2×C4×#41C2211yescharacteristic
14#3039—24no—
15#3039C418yescomplemented
16C2×#462—21no—
17C2×#462C2212yes—
18C2×#461—21no—
19C2×#461C2212yes—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419—21no—
24#2351—21no—
25Q8×C4⋊C4C2212yes—
26#2349—22no—
27#2349—21no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4C2212yes—
30#2296—22no—
31#2296—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1990C2214yes—
37#1988—21no—
38#1988—21no—
39#1988—22no—
40#1988C2214yes—
41#1823—22no—
42C2×#340—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C2×#137—28no—
51C2×#137—44no—
52C2×#137—44no—
53C22×Q16—22no—
54C22×SD16—22no—
55C22×M16—21no—
56C23×C8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yescomplemented
64C2×C4⋊Q8C2312yes—
65C4×#41—22no—
66C4×#41—22no—
67C4×#41—44no—
68C4×#41—44no—
69C4×#41C2×C414yes—
70C4×#41D414yes—
71C2×C4×Q8—22no—
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—22no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D412yes—
88C22×C42—21no—
89C22×C42—22no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#462—28no—
93#462—24no—
94#462C2×C418yescomplemented
95#461—28no—
96#461—24no—
97#461C2×C418yescomplemented
98#431—22no—
99#431—22no—
100#423—22no—
101#423C2×C414yescomplemented
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109#419—28no—
110C2×#103—22no—
111C2×#103—22no—
112C2×C22⋊C8—21no—
113C2×C4×C8C2312yes—
114#389—24no—
115#389—22no—
116#389—22no—
117#389—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#384—22no—
122#384C2312yes—
123#382—22no—
124#382—21no—
125#380—22no—
126#380—21no—
127#380—22no—
128#380D412yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D412yes—
139C4×C4⋊C4C2312yes—
140C2×#95—22no—
141C2×#95—21no—
142#340—28no—
143#340—24no—
144#340—24no—
145Q8⋊C8—24no—
146D4⋊C8—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41C22⋊C414yes—
151C2×#41D814yes—
152C2×#41SD1614yes—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173#137—432no—
174C2×Q16—28no—
175C2×Q16—44no—
176C2×Q16—44no—
177C2×SD16—28no—
178C2×SD16—44no—
179C2×SD16—44no—
180C2×M16—24no—
181C2×M16—42no—
182C2×M16—42no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—44no—
187C22×C8—42no—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4⋊Q8C22×C414yes—
198C4×Q8—22no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—44no—
202C4×Q8—44no—
203C4×Q8C22⋊C414yes—
204C4×Q8C22×C412yes—
205C4×Q8C2×D412yes—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—22no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—416no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4C22⋊C414yes—
217C4×D4C22×C412yes—
218C4×D4C2×D412yes—
219#117—22no—
220#117—24no—
221#117—24no—
222#117—48no—
223#117—42no—
224#117—42no—
225#117C22⋊C414yes—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—48no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42C22×C412yes—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4112yes—
275C2×C42#4112yes—
276C2×C42C2411yescharacteristic
277C4⋊C8—24no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8C22×C414yes—
284#103—28no—
285#103—28no—
286C22⋊C8—42no—
287C4×C8—22no—
288C4×C8—22no—
289#95—24no—
290#95—24no—
291#95—21no—
292#95—22no—
293#95—21no—
294#95—22no—
295#95—22no—
296C24—21no—
297C24#12011yescharacteristic
298#41—28no—
299#41—28no—
300#41—28no—
301#41#102116yes—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—24no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8C2×C22⋊C412yes—
319C2×Q8C2×D812yes—
320C2×Q8C2×SD1612yes—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4C2×C22⋊C412yes—
328C2×D4C2×D812yes—
329C2×D4C2×SD1612yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4—84no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4C2×C22⋊C412yes—
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4#12011yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4#12312yes—
367C22×C4C2×D812yes—
368C22×C4C2×SD1612yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4#13712yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374Q16—416no—
375SD16—416no—
376M16—48no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—44no—
384C2×C8—84no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—84no—
408C4⋊C4C2×C22⋊C412yes—
409C4⋊C4C2×C22⋊C412yes—
410C4⋊C4#13714yescomplemented
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—24no—
415C22⋊C4—48no—
416C22⋊C4—44no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—24no—
421C42—48no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—44no—
426C42—42no—
427C42C2×C22⋊C412yes—
428C42C4×D412yes—
429C42C4×D412yes—
430C42#12112yes—
431C42#12112yes—
432C42#12312yes—
433C42C23×C411yescharacteristic
434C42C22×D411yescharacteristic
435C42C2×#4111yescharacteristic
436C42C2×#4111yescharacteristic
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23#38612yes—
445C23#44712yes—
446C23#44812yes—
447C23C2×#12011yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—22no—
451Q8—44no—
452Q8—44no—
453Q8—82no—
454Q8—82no—
455Q8—88no—
456Q8C2×#10214yes—
457D4—216no—
458D4—22no—
459D4—22no—
460D4—22no—
461D4C2×#10214yes—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—216no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—84no—
492C2×C4—88no—
493C2×C4—816no—
494C2×C4—88no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#38612yes—
498C2×C4#38612yes—
499C2×C4#38612yes—
500C2×C4C2×#10214yes—
501C2×C4#41714yes—
502C2×C4#44712yes—
503C2×C4#44812yes—
504C2×C4#48014yes—
505C2×C4#48114yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C2×C4C22×D811yescharacteristic
513C2×C4C22×SD1611yescharacteristic
514C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
515C2×C4C2×#13711yescharacteristic
516C2×C4C2×#13712yes—
517C2×C4#57512yes—
518C8—44no—
519C8—82no—
520C8—82no—
521C22—24no—
522C22—24no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22#187414yes—
531C22C2×#38611yescharacteristic
532C22C2×#44711yescharacteristic
533C22C2×#44811yescharacteristic
534C22#300712yes—
535C22#302212yes—
536C4—24no—
537C4—22no—
538C4—21no—
539C4—22no—
540C4—21no—
541C4—44no—
542C4—42no—
543C4—42no—
544C4—82no—
545C4—84no—
546C4—84no—
547C4#187414yes—
548C4C22×#10211yescharacteristic
549C4C2×#41711yescharacteristic
550C4C4×#13712yes—
551C4#303911yescharacteristic
552C4#304011yescharacteristic
553C4C2×#48011yescharacteristic
554C4C2×#48111yescharacteristic
555C2—24no—
556C2—44no—
557C2#1879812yes—
558C2#1880012yes—
559C2C2×#187411yescharacteristic
560C2#2092711yescharacteristic
561C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ 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
3. #21833C2semidirect
4. C2×#1988C2semidirect
6. #8987C2semidirect
7. #8985C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
15. #3039C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
94. #462C2×C4semidirect
97. #461C2×C4semidirect
101. #423C2×C4semidirect
410. C4⋊C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
C4⋊C4×#412
#218332
C2×#19882
C2×#14772
#89872
#89852
#80452
#79012