Finite Group Databasebeta

← #122174

Subgroups — #122174

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2665C211yesmaximal, characteristic
2#10854C211yesmaximal, characteristic
3C2×#1305C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#8019C211yesmaximal, characteristic
6#8015C211yesmaximal, characteristic
7C2×#1284C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10#2665—22no—
11#2665—22no—
12#2665—22no—
13#2665C414yescomplemented
14C2×#400—21no—
15C2×#396—21no—
16C2×#395—21no—
17C2×#393—21no—
18C2×#391—21no—
19C2×#387—21no—
20C2×#381C2211yescharacteristic
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#342—21no—
26C2×#342—21no—
27#1305—22no—
28#1297—22no—
29#1294—21no—
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1284—22no—
34C23×Q8—21no—
35C24×C4C2311yescharacteristic
36C22×M16—21no—
37C22×M16—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4Q812yescomplemented
43C22×C22⋊C4—21no—
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—21no—
50C22×C42—21no—
51#407—21no—
52#407—21no—
53#407—21no—
54#407—21no—
55#407—21no—
56#407—21no—
57#407Q812yescomplemented
58#407Q812yescomplemented
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63#400—24no—
64#400—24no—
65#396—24no—
66#396—24no—
67#395—24no—
68#395—24no—
69#393—24no—
70#393—24no—
71#391—24no—
72#391—24no—
73#387—24no—
74#387—24no—
75#381—21no—
76#381—21no—
77#381C2×C412yes—
78#381D412yes—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380C2×C412yes—
86#380D412yes—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92C4×C22⋊C4C2×C412yes—
93C4×C22⋊C4D412yes—
94C4×C22⋊C4Q812yescomplemented
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#342—42no—
108#342—42no—
109C25C2×D411yescharacteristic
110C22×Q8—24no—
111C22×Q8—44no—
112C22×Q8—82no—
113C22×Q8—81no—
114C22×Q8—81no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—41no—
123C23×C4—82no—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q811yescharacteristic
131C23×C4C2×Q811yescharacteristic
132C23×C4C2×Q811yescharacteristic
133C23×C4C2×Q811yescharacteristic
134C23×C4#4111yescharacteristic
135C23×C4#4111yescharacteristic
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—82no—
143C2×M16—82no—
144C22×C8—41no—
145C22×C8—41no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—41no—
189C2×C42—82no—
190C2×C42—82no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95C22⋊C412yes—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—42no—
224C24—42no—
225C24C2×C22⋊C411yescharacteristic
226C24#12011yescharacteristic
227C24#12011yescharacteristic
228C24C22⋊Q811yescharacteristic
229C24C22⋊Q811yescharacteristic
230C24#12411yescharacteristic
231C24C4⋊Q811yescharacteristic
232C2×Q8—48no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—88no—
236C2×Q8—88no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C4×Q811yescharacteristic
294C22×C4C4×Q811yescharacteristic
295C22×C4C4×Q811yescharacteristic
296C22×C4C4×Q811yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4C22⋊Q811yescharacteristic
307C22×C4C22⋊Q811yescharacteristic
308C22×C4C22⋊Q811yescharacteristic
309C22×C4#12411yescharacteristic
310C22×C4#12411yescharacteristic
311C22×C4#12411yescharacteristic
312C22×C4#12411yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4C4⋊Q811yescharacteristic
315C22×C4C4⋊Q811yescharacteristic
316C22×C4C4⋊Q811yescharacteristic
317C22×C4C4⋊Q811yescharacteristic
318C22×C4C4⋊Q811yescharacteristic
319M16—82no—
320M16—82no—
321M16—82no—
322M16—82no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—82no—
326C2×C8—82no—
327C4⋊C4—48no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—24no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—81no—
343C22⋊C4—81no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—82no—
351C42—82no—
352C42—82no—
353C42—82no—
354C23—22no—
355C23—22no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23#37911yescharacteristic
385C23#39111yescharacteristic
386C23#39111yescharacteristic
387C23#39311yescharacteristic
388C23#39311yescharacteristic
389C23#39511yescharacteristic
390C23#39511yescharacteristic
391C23#45311yescharacteristic
392C23#45411yescharacteristic
393C23#45511yescharacteristic
394C23#45611yescharacteristic
395Q8—816no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—162no—
463C2×C4—162no—
464C2×C4#39111yescharacteristic
465C2×C4#39111yescharacteristic
466C2×C4#39111yescharacteristic
467C2×C4#39111yescharacteristic
468C2×C4#39311yescharacteristic
469C2×C4#39311yescharacteristic
470C2×C4#39511yescharacteristic
471C2×C4#39511yescharacteristic
472C8—82no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22#144611yescharacteristic
507C22#179211yescharacteristic
508C22#179311yescharacteristic
509C22#202711yescharacteristic
510C22#202811yescharacteristic
511C22#203112yes—
512C22#203212yes—
513C22#206011yescharacteristic
514C22#206111yescharacteristic
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—44no—
524C4—82no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—84no—
530C4—162no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2—41no—
536C2—41no—
537C2—41no—
538C2—41no—
539C2#1038212yes—
540C2#1038312yes—
541C2#1138711yescharacteristic
542C2#1139211yescharacteristic
543C2#1139411yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2665C4semidirect
42. C22×C4⋊C4Q8semidirect
57. #407Q8semidirect
58. #407Q8semidirect
94. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26652
#108542
C2×#13052
C2×#12972
#80192
#80152
C2×#12842