Finite Group Databasebeta

← #198395

Subgroups — #198395

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21332C211yesmaximal, characteristic
5#17575C211yesmaximal, characteristic
6#9257C218yesmaximal
7#7981C211yesmaximal, characteristic
8#7924C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—22no—
12#3230—22no—
13#3230—22no—
14#3230—22no—
15C2×#500C412yescomplemented
16C2×#487C412yescomplemented
17C2×#484C2214yes—
18C2×#477C414yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2369—22no—
22#2365—22no—
23#2345C2214yes—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30#2274—21no—
31C4×#117C2211yescharacteristic
32#1996—22no—
33#1996C2214yes—
34#1990—22no—
35#1987—22no—
36#1824C2214yes—
37#1740—21no—
38#1740—21no—
39#1520C2214yes—
40#1518—22no—
41#1503—22no—
42#1289C2214yes—
43#1282C2214yes—
44C23×Q8—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8D412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8C2312yes—
70C2×C4×Q8—24no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C42C2311yescharacteristic
81#500—24no—
82#500—24no—
83#487—24no—
84#487—24no—
85#484—28no—
86#484—28no—
87#477—28no—
88#477—28no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97C2×#107C2×C418yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2×C412yes—
100C2×#103C2×C418yes—
101C2×C22⋊C8C2312yes—
102C2×C22⋊C8C2312yes—
103C2×C4×C8—21no—
104C2×C4×C8C2×C412yes—
105#391—24no—
106#391—28no—
107#389—24no—
108#389—24no—
109#389—24no—
110#387—28no—
111#384—21no—
112#384—21no—
113#382—21no—
114#382D412yes—
115#380—21no—
116#380—24no—
117#380—21no—
118#380C2312yes—
119#380C2312yes—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4C2312yes—
132#376—21no—
133C43—21no—
134#342—24no—
135#342—24no—
136#340—28no—
137#328—28no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—84no—
146C22×Q8C22×C412yes—
147C23×C4—22no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8C22×C414yes—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8C22⋊C414yes—
174C4⋊Q8C22⋊C414yes—
175C4⋊Q8C22⋊C414yes—
176C4⋊Q8C22⋊C414yes—
177#125—41no—
178#125—41no—
179#125—41no—
180#125—41no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—88no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117Q1614yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4C2×D412yes—
241C2×C22⋊C4C2×D412yes—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—41no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262#107—216no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267#103—216no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—41no—
275C4×C8—41no—
276#95—28no—
277#95—28no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—41no—
283#95C2×D414yes—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—48no—
292C2×Q8—44no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—82no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—84no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C2×C8—24no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—88no—
366C4⋊C4—82no—
367C4⋊C4C2×C22⋊C418yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4C2×Q1614yes—
378C42—24no—
379C42—44no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—44no—
385C42—88no—
386C42—82no—
387C42C2×C22⋊C411yescharacteristic
388C42C2×C22⋊C411yescharacteristic
389C42C2×C22⋊C411yescharacteristic
390C42C2×C22⋊C411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42C4×D411yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42#12311yescharacteristic
400C42#12311yescharacteristic
401C42#12311yescharacteristic
402C42#12311yescharacteristic
403C23—41no—
404C23—41no—
405C23C2×C4×D411yescharacteristic
406C23C2×#12011yescharacteristic
407C23C2×#12111yescharacteristic
408C23C22×Q1611yescharacteristic
409C23C2×#13511yescharacteristic
410C23C2×#13711yescharacteristic
411C23C2×#13711yescharacteristic
412Q8—48no—
413Q8—48no—
414Q8—44no—
415Q8—84no—
416Q8—84no—
417Q8—82no—
418Q8—82no—
419Q8—82no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4C4×Q1612yes—
461C2×C4C4×Q1612yes—
462C2×C4#44112yes—
463C2×C4#44112yes—
464C2×C4#45112yes—
465C2×C4#45112yes—
466C2×C4#45212yes—
467C2×C4#45212yes—
468C2×C4#46212yes—
469C2×C4#46212yes—
470C2×C4#46412yes—
471C2×C4#46412yes—
472C2×C4C22×C22⋊C411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12311yescharacteristic
476C8—44no—
477C8—82no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22C2×#38611yescharacteristic
489C22C2×C4×Q1611yescharacteristic
490C22C2×#44111yescharacteristic
491C22C2×#45111yescharacteristic
492C22C2×#45211yescharacteristic
493C22C2×#46211yescharacteristic
494C22C2×#46411yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—44no—
502C4—48no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—162no—
508C4#187812yes—
509C4#187812yes—
510C4#187812yes—
511C4#187812yes—
512C4#190412yes—
513C4#288412yes—
514C2—21no—
515C2—21no—
516C2—21no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2C2×#187811yescharacteristic
521C2#2098611yescharacteristic
522C2C2×#190411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#500C4semidirect
16. C2×#487C4semidirect
18. C2×#477C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#342162
#218342
#213322
#175752
#92572
#79812
#79242