Finite Group Databasebeta

← #198882

Subgroups — #198882

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60865C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17676C211yesmaximal, characteristic
4#17672C211yesmaximal, characteristic
5C2×#1497C211yesmaximal, characteristic
6#9286C218yesmaximal, complemented
7#7970C211yesmaximal, characteristic
8#7863C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3237—22no—
12#3236—22no—
13#3231C2214yes—
14#3230C412yes—
15#3229C412yes—
16#3165C412yes—
17#3139C412yes—
18#3095—22no—
19#3078—22no—
20#2986—22no—
21#2985C2214yes—
22#2984—22no—
23#2983—21no—
24#2982—21no—
25#2962—22no—
26#2960—22no—
27#2924—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#1537—22no—
34#1534—22no—
35#1531—22no—
36#1531C2214yes—
37#1531C2214yes—
38#1528—22no—
39#1497—21no—
40#1497—21no—
41#1497C2214yes—
42C2×Q8⋊C8—22no—
43#1287C2214yes—
44#1275C2214yes—
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#533C2×C412yes—
56#533C2×C412yes—
57#533D412yes—
58#533D412yes—
59#533C2312yes—
60#533C2312yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125—21no—
66C2×#125D412yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×#117D412yes—
77C22×C42C2311yescharacteristic
78#501—24no—
79#500C2×C414yes—
80#498C2×C414yes—
81#491—24no—
82#487C2×C414yes—
83#487C2×C414yes—
84#485—28no—
85#484—24no—
86#483—24no—
87#479—28no—
88#477—24no—
89#475—24no—
90#446—24no—
91#446—24no—
92#446—28no—
93#446—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96C8×Q8—24no—
97#433—28no—
98#432—24no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432D412yes—
109#432C2312yes—
110#432C2312yes—
111#431—24no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2×C412yes—
117#431D411yescharacteristic
118#431D411yescharacteristic
119#431C2312yes—
120#431C2312yes—
121#428—24no—
122#424—24no—
123#423—21no—
124#423—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128#419—24no—
129#407—21no—
130#407—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134#405—24no—
135C4×M16C2×C412yes—
136C4×M16D412yes—
137C2×#97—21no—
138C2×#97C2312yes—
139C2×C4×C8D412yes—
140#330—28no—
141#330—28no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4#4111yescharacteristic
151C2×M16—22no—
152C2×M16—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—41no—
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⋊Q8C22⋊C412yes—
167C4⋊Q8C22⋊C412yes—
168C4⋊Q8C22×C412yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172#125—22no—
173#125—22no—
174#125—22no—
175#125—22no—
176#125—41no—
177#125—41no—
178#125C22⋊C414yes—
179#125C2×D414yes—
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—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
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—22no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—41no—
217#117—41no—
218#117C22⋊C414yes—
219#117C2×D414yes—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C412yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C42—22no—
235C2×C42—41no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42C2411yescharacteristic
248#107—28no—
249#106—28no—
250C4⋊C8—28no—
251C4⋊C8—28no—
252C4⋊C8—22no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8—41no—
263#103—28no—
264#103—28no—
265C22⋊C8—22no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—41no—
274C22⋊C8—41no—
275C22⋊C8—41no—
276#97—28no—
277#97—22no—
278#97—22no—
279#97—22no—
280#97—22no—
281#97—41no—
282#97C2×D414yes—
283C4×C8—28no—
284C4×C8—22no—
285C4×C8—22no—
286C4×C8C22×C414yes—
287C4×C8C2×D412yes—
288C4×C8C2×D412yes—
289C24#12311yescharacteristic
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—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4#12011yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—44no—
328M16—44no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—82no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
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⋊C4C2×C22⋊C414yes—
361C4⋊C4C2×C22⋊C414yes—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C42—24no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42C2×C22⋊C411yescharacteristic
371C42C2×C22⋊C411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42C23×C411yescharacteristic
383C42C22×D411yescharacteristic
384C42C22×D411yescharacteristic
385C42C22×D411yescharacteristic
386C23—22no—
387C23—41no—
388C23—41no—
389C23#38611yescharacteristic
390C23#38611yescharacteristic
391C23C2×#12311yescharacteristic
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—44no—
396Q8—44no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
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—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4C22×C22⋊C411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C8—42no—
439C8—42no—
440C8—44no—
441C8—44no—
442C8—82no—
443C8—82no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—82no—
453C22#191212yes—
454C22C2×#38611yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—44no—
466C4—44no—
467C4—82no—
468C4—82no—
469C4#190212yes—
470C4#295512yes—
471C2—22no—
472C2—41no—
473C2—41no—
474C2#2091711yescharacteristic
475C2C2×#190211yescharacteristic
476C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#123
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9286C2semidirect

Maximal subgroups

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

subgroupindex
#608652
#602502
#176762
#176722
C2×#14972
#92862
#79702
#78632