Finite Group Databasebeta

← #198141

Subgroups — #198141

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#21845C211yesmaximal, characteristic
4#17584C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6#9240C218yesmaximal
7#8010C211yesmaximal, characteristic
8#7926C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3237—22no—
12#3237—22no—
13#3237—22no—
14#3237—22no—
15C2×#501C412yescomplemented
16C2×#491C412yescomplemented
17C2×#485C2214yes—
18C2×#477C412yescomplemented
19C2×#475C412yescomplemented
20#2986—22no—
21#2985—22no—
22#2984—22no—
23#2983—21no—
24#2982—21no—
25#2962—22no—
26#2960—22no—
27#2924C2214yes—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2283C2211yescharacteristic
33#1995—22no—
34#1986—22no—
35#1826C2214yes—
36#1518—22no—
37#1515—22no—
38#1508C2214yes—
39#1472C2214yes—
40#1470—21no—
41#1470—21no—
42#1461—22no—
43#1293C2214yes—
44#1282C2214yes—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533D412yes—
58#533D412yes—
59#533D412yes—
60#533D412yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125—21no—
66C2×#125C2×C412yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2312yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×#117—21no—
76C2×#117C2312yes—
77C22×C42C2311yescharacteristic
78#501—24no—
79#501—24no—
80#491—24no—
81#491—24no—
82#485—28no—
83#485—28no—
84#477—24no—
85#477—24no—
86#475—24no—
87#475—24no—
88#446—24no—
89#446—24no—
90#446—24no—
91#446—28no—
92C8×Q8—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95#433—28no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—24no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—24no—
109#431—21no—
110#431—21no—
111#431D411yescharacteristic
112#431D411yescharacteristic
113#431D412yes—
114#431D412yes—
115C2×#107C2×C414yes—
116C2×#106C2×C414yes—
117#424—24no—
118#423—21no—
119#423—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8C2×C412yes—
122C2×#103C2×C414yes—
123C2×#103C2×C414yes—
124#407C2312yes—
125#407C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128#405—24no—
129C4×M16—21no—
130C4×M16—21no—
131C2×#97C2×C412yes—
132C2×C4×C8—21no—
133#383—21no—
134#382—21no—
135#381—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4C2312yes—
139#376—21no—
140#339—28no—
141#328—28no—
142#324—24no—
143#324—24no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8C22×C412yes—
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4C2411yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C22×C8—22no—
154C22×C8C22×C414yes—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C22⋊C412yes—
168C4⋊Q8C22⋊C412yes—
169C4⋊Q8C22⋊C412yes—
170C4⋊Q8C22⋊C412yes—
171#125—22no—
172#125—22no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—41no—
177#125C22⋊C414yes—
178#125C22⋊C414yes—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211#117—22no—
212#117—22no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—41no—
217#117C2×D414yes—
218#117C2×D414yes—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C22×C411yescharacteristic
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
250#107—28no—
251#106—28no—
252C4⋊C8—28no—
253C4⋊C8—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263#103—28no—
264#103—28no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8C2×D412yes—
272C22⋊C8C2×D412yes—
273C22⋊C8C2×D414yes—
274#97—28no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—41no—
279C4×C8—28no—
280C4×C8—21no—
281C4×C8—22no—
282C4×C8—21no—
283C4×C8—22no—
284C4×C8—41no—
285#95—22no—
286#95—41no—
287#95—41no—
288#95—41no—
289C24C2×#4111yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—82no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328M16—44no—
329M16—44no—
330C2×C8—24no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—82no—
360C4⋊C4C2×C22⋊C414yes—
361C4⋊C4C2×C22⋊C414yes—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—44no—
367C42—24no—
368C42—24no—
369C42—44no—
370C42—42no—
371C42—42no—
372C42C2×C22⋊C411yescharacteristic
373C42C2×C22⋊C411yescharacteristic
374C42C2×C22⋊C411yescharacteristic
375C42C2×C22⋊C411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42#12011yescharacteristic
381C42#12011yescharacteristic
382C42#12011yescharacteristic
383C42#12011yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C23—22no—
389C23—41no—
390C23—41no—
391C23C2×C4×D411yescharacteristic
392C23C2×#12111yescharacteristic
393C23C2×#12311yescharacteristic
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—44no—
398Q8—82no—
399Q8—82no—
400Q8—82no—
401Q8—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
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—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4C22×C22⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×#12011yescharacteristic
442C2×C4C2×#12111yescharacteristic
443C8—42no—
444C8—42no—
445C8—44no—
446C8—44no—
447C8—82no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—82no—
457C22C2×#38611yescharacteristic
458C22#295512yes—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—44no—
470C4—44no—
471C4—82no—
472C4—82no—
473C4—162no—
474C4#190212yes—
475C4#288412yes—
476C2—22no—
477C2—41no—
478C2—41no—
479C2#2091711yescharacteristic
480C2#2098611yescharacteristic
481C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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×#501C4semidirect
16. C2×#491C4semidirect
18. C2×#477C4semidirect
19. C2×#475C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
#602502
#218452
#175842
#173852
#92402
#80102
#79262