Finite Group Databasebeta

← #114098

Subgroups — #114098

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6C2×#1301C211yesmaximal, characteristic
7#7908C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3230—22no—
11#3230—22no—
12#3230—22no—
13#3230C414yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1989—22no—
24#1988—22no—
25#1988—22no—
26#1988—21no—
27#1987—21no—
28#1736—22no—
29C2×#340—22no—
30#1548—22no—
31#1548—22no—
32C2×#333—22no—
33#1481—22no—
34#1481—22no—
35C2×Q8⋊C8—22no—
36#1301—22no—
37#1280—22no—
38C23×Q8—21no—
39C22×M16—21no—
40C23×C8—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8D412yes—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D412yes—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#500—24no—
75#500—24no—
76#487—24no—
77#487—24no—
78#484—28no—
79#484—28no—
80#477—28no—
81#477—28no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431—21no—
87#431C2×C412yes—
88#431D412yes—
89#431D412yes—
90C2×#107—24no—
91#423—21no—
92#423—21no—
93#423—21no—
94#423—21no—
95C2×C4⋊C8—21no—
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×C4⋊C8D412yes—
105C2×#103—24no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D412yes—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C2×#95—21no—
115#340—24no—
116#340—28no—
117#340—24no—
118#333—28no—
119Q8⋊C8—28no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—82no—
129C22×Q8—82no—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—41no—
143C22×C8—41no—
144C22×C8—41no—
145C22×C8—42no—
146C22×C8—42no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8C22⋊C414yes—
169C4⋊Q8C22⋊C412yes—
170C4⋊Q8C22⋊C412yes—
171#125—22no—
172#125—22no—
173#125—41no—
174#125—41no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191#117—22no—
192#117—22no—
193#117—41no—
194#117—41no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4Q1614yes—
219C2×C22⋊C4—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—82no—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236#107—216no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—41no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249#103—216no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—41no—
253C22⋊C8C22⋊C414yes—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—41no—
258#95—41no—
259#95—41no—
260C24#12111yescharacteristic
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—44no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×Q8—82no—
277C2×Q8—82no—
278C2×Q8—82no—
279C2×Q8—82no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12411yescharacteristic
310C22×C4#12411yescharacteristic
311C22×C4#12711yescharacteristic
312C22×C4#12711yescharacteristic
313C22×C4C2×Q1612yes—
314C22×C4#13512yes—
315C22×C4#13712yes—
316C22×C4#13712yes—
317M16—82no—
318M16—81no—
319M16—81no—
320C2×C8—24no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8—81no—
330C2×C8—81no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42—82no—
361C42C2×C22⋊C411yescharacteristic
362C42C2×C22⋊C411yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42#12011yescharacteristic
366C42#12011yescharacteristic
367C42#12011yescharacteristic
368C42#12011yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12111yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12311yescharacteristic
376C42#12311yescharacteristic
377C23—22no—
378C23—21no—
379C23—21no—
380C23#39011yescharacteristic
381C23#39211yescharacteristic
382C23#39411yescharacteristic
383C23#46712yes—
384C23#47012yes—
385Q8—44no—
386Q8—44no—
387Q8—84no—
388Q8—82no—
389Q8—84no—
390Q8—82no—
391Q8—84no—
392Q8—84no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—84no—
420C2×C4#37911yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#39011yescharacteristic
426C2×C4#39211yescharacteristic
427C2×C4#39211yescharacteristic
428C2×C4#39211yescharacteristic
429C2×C4#39411yescharacteristic
430C2×C4#39411yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4C4×Q1612yes—
433C2×C4#44112yes—
434C2×C4#45112yes—
435C2×C4#45112yes—
436C2×C4#45212yes—
437C2×C4#45212yes—
438C2×C4#46212yes—
439C2×C4#46212yes—
440C2×C4#46412yes—
441C2×C4#46412yes—
442C2×C4#47112yes—
443C2×C4#47212yes—
444C2×C4#49411yescharacteristic
445C2×C4#49511yescharacteristic
446C2×C4#49711yescharacteristic
447C2×C4#49711yescharacteristic
448C8—44no—
449C8—82no—
450C8—82no—
451C8—82no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22#143811yescharacteristic
461C22#196911yescharacteristic
462C22#197111yescharacteristic
463C22#200112yes—
464C22#200912yes—
465C22#201811yescharacteristic
466C22#201911yescharacteristic
467C22#205612yes—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—84no—
475C4—82no—
476C4—84no—
477C4—82no—
478C4#187812yes—
479C4#187812yes—
480C4#190712yes—
481C4#201511yescharacteristic
482C4#201511yescharacteristic
483C4#201611yescharacteristic
484C4#201611yescharacteristic
485C4#205112yes—
486C2—22no—
487C2—21no—
488C2—21no—
489C2#1019012yes—
490C2#1019112yes—
491C2#1100511yescharacteristic
492C2#1102011yescharacteristic
493C2#1102411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#213122
C2×#19882
C2×#15482
C2×#14812
C2×#13012
#79082