Finite Group Databasebeta

← #198900

Subgroups — #198900

Order 512.

965 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60865C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic
3C2×#1531C211yesmaximal, characteristic
4C2×#1531C211yesmaximal, characteristic
5#17496C211yesmaximal, characteristic
6#9286C218yesmaximal, complemented
7#7967C211yesmaximal, characteristic
8#7861C211yesmaximal, 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#2987—22no—
21#2985C2214yes—
22#2984—22no—
23#2983—22no—
24#2962—22no—
25#2961—22no—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C22×#97C2211yescharacteristic
32#1531—22no—
33#1531—22no—
34#1531C2214yes—
35#1531C2214yes—
36C2×#330—22no—
37C2×#330—22no—
38#1497C2214yes—
39#1491—22no—
40#1481—22no—
41#1287C2214yes—
42#1275C2214yes—
43C23×C8—21no—
44#533—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—
53#533—21no—
54#533C2×C412yes—
55#533C2×C412yes—
56#533D412yes—
57#533D412yes—
58#533C2312yes—
59#533C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125—21no—
65C2×#125D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×#117—21no—
75C2×#117D412yes—
76C22×C42C2311yescharacteristic
77#501—24no—
78#500C2×C414yes—
79#498C2×C414yes—
80#491—24no—
81#487C2×C414yes—
82#487C2×C414yes—
83#485—28no—
84#484—24no—
85#483—24no—
86#479—28no—
87#477—24no—
88#475—24no—
89#446—28no—
90#446—28no—
91#446—28no—
92#446—24no—
93C8×Q8—24no—
94#433—28no—
95#432—24no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432D412yes—
106#432C2312yes—
107#432C2312yes—
108#431—24no—
109#431C2×C412yes—
110#431D412yes—
111#431C2312yes—
112#431C2312yes—
113#428—24no—
114#424—24no—
115#423—21no—
116#423—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119#419—24no—
120#407—21no—
121#407—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124#405—24no—
125C4×M16C2×C412yes—
126C4×M16D412yes—
127C2×#97—21no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97D412yes—
134C2×#97C2312yes—
135#330—28no—
136#330—28no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—44no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—41no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
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⋊Q8C22⋊C412yes—
164C4⋊Q8C22⋊C412yes—
165C4⋊Q8C22×C412yes—
166C4⋊Q8C22×C412yes—
167C4⋊Q8C22×C412yes—
168C4⋊Q8C22×C412yes—
169#125—22no—
170#125—22no—
171#125—22no—
172#125—22no—
173#125—41no—
174#125—41no—
175#125C22⋊C414yes—
176#125C2×D414yes—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
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—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C4×Q8—22no—
194C4×Q8—22no—
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—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—41no—
214#117—41no—
215#117C22⋊C414yes—
216#117C2×D414yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
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—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—41no—
231C2×C42—22no—
232C2×C42—41no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C2×C42C2411yescharacteristic
245#107—28no—
246#106—28no—
247C4⋊C8—28no—
248C4⋊C8—28no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—41no—
257#103—28no—
258#103—28no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267#97—28no—
268#97—22no—
269#97—22no—
270#97—22no—
271#97—22no—
272#97—22no—
273#97—22no—
274#97—41no—
275#97—41no—
276#97—41no—
277#97C2×D414yes—
278C4×C8—28no—
279C4×C8C22×C414yes—
280C4×C8C2×D414yes—
281C24#12311yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—44no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—82no—
293C2×Q8—82no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—44no—
320M16—44no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—82no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4C2×C22⋊C414yes—
355C4⋊C4C2×C22⋊C414yes—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—24no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42C2×C22⋊C411yescharacteristic
365C42C2×C22⋊C411yescharacteristic
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42C4×D411yescharacteristic
369C42C4×D411yescharacteristic
370C42#12011yescharacteristic
371C42#12011yescharacteristic
372C42#12111yescharacteristic
373C42#12111yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42C23×C411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C23—22no—
381C23—22no—
382C23—41no—
383C23#38611yescharacteristic
384C23#38611yescharacteristic
385C23C2×#12311yescharacteristic
386Q8—44no—
387Q8—44no—
388Q8—44no—
389Q8—82no—
390Q8—82no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38611yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4C22×C22⋊C411yescharacteristic
427C2×C4C2×C4×D411yescharacteristic
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C8—44no—
433C8—44no—
434C8—44no—
435C8—44no—
436C8—82no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—44no—
445C22#190212yes—
446C22#191212yes—
447C22C2×#38611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—44no—
460C4—44no—
461C4—82no—
462C4—82no—
463C4#295612yes—
464C2—22no—
465C2—22no—
466C2—41no—
467C2#2091811yescharacteristic
468C2C2×#190211yescharacteristic
469C2C2×#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
#602512
C2×#15312
#174962
#92862
#79672
#78612