Finite Group Databasebeta

← #115421

Subgroups — #115421

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2#21321C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#7893C211yesmaximal, characteristic
7C2×#1275C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#3104—22no—
11#3104—22no—
12#3104—22no—
13#3104C414yescomplemented
14C2×#484—21no—
15C2×#477—21no—
16C2×#462—21no—
17C2×#451—21no—
18C2×C4×Q16—22no—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2285C2211yescharacteristic
24#1993—21no—
25#1993—21no—
26#1992—22no—
27#1985—22no—
28#1985—22no—
29#1824—21no—
30#1803—22no—
31#1787—21no—
32C4×#103—22no—
33#1735—22no—
34C2×#336—22no—
35#1481—22no—
36#1481—22no—
37C2×Q8⋊C8—22no—
38#1279—22no—
39#1275—22no—
40C23×Q8D411yescharacteristic
41C22×Q16—22no—
42C23×C8—21no—
43C23×C8D412yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533C2×C412yes—
51#533D412yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
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×Q8C2×C412yes—
70C2×C4×Q8D412yes—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#484—24no—
78#484—24no—
79#477—24no—
80#477—24no—
81#462—24no—
82#462—24no—
83#451—24no—
84#451—24no—
85C4×Q16—28no—
86C4×Q16—28no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431C2×C412yes—
94#431D412yes—
95C2×#107—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D412yes—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8D412yes—
116C2×C4×C8D412yes—
117#384D412yes—
118#382—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123#340—24no—
124#340—24no—
125#340—24no—
126#336—28no—
127Q8⋊C8—28no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—44no—
136C22×Q8C22⋊C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4#4111yescharacteristic
142C2×Q16—28no—
143C2×Q16—48no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8Q1614yescomplemented
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C22⋊C412yes—
165C4⋊Q8C22⋊C412yes—
166#125—22no—
167#125—22no—
168#125—41no—
169#125—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202#117—22no—
203#117—22no—
204#117—41no—
205#117—41no—
206#117—42no—
207#117—42no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C412yes—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—41no—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251#107—28no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260#103—28no—
261#103—28no—
262#103—48no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267#97—41no—
268#97—41no—
269#97—41no—
270#97—41no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8C22⋊C414yes—
275#95—41no—
276#95—41no—
277C24#12111yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—44no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—44no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4#12411yescharacteristic
327C22×C4#12711yescharacteristic
328C22×C4C2×Q1612yes—
329C22×C4#13512yes—
330C22×C4#13712yes—
331C22×C4#13712yes—
332Q16—416no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—44no—
346C2×C8—82no—
347C2×C8—81no—
348C2×C8—81no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—82no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—81no—
382C42—81no—
383C42C2×C22⋊C411yescharacteristic
384C42C2×C22⋊C411yescharacteristic
385C42C4×D411yescharacteristic
386C42C4×D411yescharacteristic
387C42#12011yescharacteristic
388C42#12011yescharacteristic
389C42#12011yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12111yescharacteristic
393C42#12111yescharacteristic
394C42#12111yescharacteristic
395C42#12111yescharacteristic
396C42#12311yescharacteristic
397C42#12411yescharacteristic
398C42#12711yescharacteristic
399C23—22no—
400C23—22no—
401C23#38611yescharacteristic
402C23#39211yescharacteristic
403C23#39411yescharacteristic
404C23#46212yes—
405C23#46412yes—
406Q8—24no—
407Q8—24no—
408Q8—42no—
409Q8—44no—
410Q8—42no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414Q8—82no—
415Q8—82no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#37911yescharacteristic
445C2×C4#38611yescharacteristic
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#39011yescharacteristic
449C2×C4#39011yescharacteristic
450C2×C4#39211yescharacteristic
451C2×C4#39211yescharacteristic
452C2×C4#39211yescharacteristic
453C2×C4#39411yescharacteristic
454C2×C4#39411yescharacteristic
455C2×C4#39411yescharacteristic
456C2×C4C4×Q1612yes—
457C2×C4#44112yes—
458C2×C4#45112yes—
459C2×C4#45112yes—
460C2×C4#45212yes—
461C2×C4#45212yes—
462C2×C4#46212yes—
463C2×C4#46412yes—
464C2×C4#46712yes—
465C2×C4#47012yes—
466C2×C4#49411yescharacteristic
467C2×C4#49511yescharacteristic
468C2×C4#49711yescharacteristic
469C2×C4#49711yescharacteristic
470C8—44no—
471C8—44no—
472C8—82no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—24no—
477C22—24no—
478C22—21no—
479C22—21no—
480C22#143811yescharacteristic
481C22#187812yes—
482C22#201012yes—
483C22#201511yescharacteristic
484C22#201611yescharacteristic
485C22#201712yes—
486C22#205612yes—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—44no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—162no—
500C4#187812yes—
501C4#196911yescharacteristic
502C4#197111yescharacteristic
503C4#200112yes—
504C4#201511yescharacteristic
505C4#201611yescharacteristic
506C2—22no—
507C2—22no—
508C2#1003512yes—
509C2#1003712yes—
510C2#1099411yescharacteristic
511C2#1102411yescharacteristic
512C2#1104011yescharacteristic
subgroupG/H
centerC22×C4#127
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#392
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3104C4semidirect
157. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#213212
C2×#19852
#207342
C2×#14812
#78932
C2×#12752