Finite Group Databasebeta

← #200730

Subgroups — #200730

Order 512.

953 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#60971C211yesmaximal, characteristic
2C2×#2987C211yesmaximal, characteristic
3#17672C211yesmaximal, characteristic
4#17596C211yesmaximal, characteristic
5#17584C211yesmaximal, characteristic
6#9416C218yesmaximal, complemented
7#7998C211yesmaximal, characteristic
8#7977C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3255—22no—
12#3229C412yescomplemented
13#3229C412yescomplemented
14#3209C412yescomplemented
15#3209C412yescomplemented
16#3195—22no—
17#3185C2214yes—
18#3113—22no—
19#3113—22no—
20#2987—22no—
21#2987—22no—
22#2987—22no—
23#2987C2214yes—
24C2×#446—22no—
25C2×#446—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#433—22no—
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31#2283C2211yescharacteristic
32#1537C2214yes—
33#1531—22no—
34#1528—22no—
35#1520—22no—
36#1518—22no—
37#1515C2214yes—
38#1515C2214yes—
39#1508—22no—
40#1506—22no—
41#1291C2214yes—
42#1288C2214yes—
43#533—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#533C2×C412yes—
54#533C2×C412yes—
55#533D412yes—
56#533D412yes—
57#533C2312yes—
58#533C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#125—21no—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8D412yes—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8D412yes—
73C2×#117—21no—
74C2×#117—21no—
75C22×C42C2311yescharacteristic
76#498C2×C414yes—
77#494C2×C414yes—
78#487C2×C414yes—
79#487C2×C414yes—
80#483—24no—
81#483—24no—
82#475—24no—
83#475—24no—
84#470—28no—
85#462—24no—
86#462—24no—
87#451—24no—
88#451—24no—
89#446—28no—
90#446—28no—
91#446—28no—
92#446—28no—
93#441—28no—
94#433—28no—
95#433—28no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#432D412yes—
105#432C2312yes—
106#432C2312yes—
107#432C2312yes—
108#431—21no—
109#431—21no—
110#431C2×C412yes—
111#431C2×C412yes—
112#423—21no—
113#423—21no—
114C2×C4⋊C8—24no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117#407—21no—
118#407—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8D412yes—
121C4×M16—21no—
122C4×M16C2312yes—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97D412yes—
126C2×C4×C8—22no—
127#383C2312yes—
128#382—21no—
129#381—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C22⋊C4—21no—
133#376—21no—
134#330—28no—
135#328—28no—
136#328—28no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8C22⋊C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144C2×Q16—28no—
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8C22⋊C412yes—
160C4⋊Q8C22⋊C412yes—
161C4⋊Q8C22×C412yes—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C22×C412yes—
164C4⋊Q8C22×C412yes—
165#125—22no—
166#125—22no—
167#125—22no—
168#125—22no—
169#125—22no—
170#125—22no—
171#125—41no—
172#125—41no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8C22⋊C414yes—
188C22⋊Q8C2×D414yes—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8C22⋊C414yes—
204C4×Q8C2×D414yes—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C42—22no—
230C2×C42—41no—
231C2×C42—41no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42C2411yescharacteristic
244#106—28no—
245C4⋊C8—216no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—41no—
254#103—28no—
255#103—28no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—41no—
263C22⋊C8C2×D414yes—
264#97—28no—
265#97—22no—
266#97—22no—
267#97—22no—
268#97—22no—
269#97—22no—
270#97C2×D414yes—
271C4×C8—28no—
272C4×C8—41no—
273C4×C8C22×C414yes—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—41no—
278C24#12111yescharacteristic
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—82no—
287C2×Q8—82no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8C2×C22⋊C414yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—81no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317Q16—88no—
318M16—82no—
319M16—82no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—82no—
350C4⋊C4C2×C22⋊C414yes—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—82no—
356C42—24no—
357C42—24no—
358C42—42no—
359C42—42no—
360C42—82no—
361C42C2×C22⋊C411yescharacteristic
362C42C2×C22⋊C411yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42#12011yescharacteristic
368C42#12011yescharacteristic
369C42#12111yescharacteristic
370C42#12111yescharacteristic
371C42#12311yescharacteristic
372C42#12311yescharacteristic
373C42C23×C411yescharacteristic
374C42C22×D411yescharacteristic
375C42C22×D411yescharacteristic
376C42C2×#4111yescharacteristic
377C23—22no—
378C23—41no—
379C23—41no—
380C23#38611yescharacteristic
381C23#38611yescharacteristic
382C23C2×#12111yescharacteristic
383Q8—44no—
384Q8—44no—
385Q8—82no—
386Q8—82no—
387Q8—82no—
388Q8—82no—
389Q8—82no—
390Q8—82no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
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—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—82no—
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×#12311yescharacteristic
432C8—44no—
433C8—44no—
434C8—44no—
435C8—82no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—82no—
445C22#190212yes—
446C22C2×#38611yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—44no—
459C4—44no—
460C4—82no—
461C4—82no—
462C4—162no—
463C4#190512yes—
464C2—22no—
465C2—41no—
466C2—41no—
467C2#2094711yescharacteristic
468C2C2×#190211yescharacteristic
469C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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. #9416C2semidirect
12. #3229C4semidirect
13. #3229C4semidirect
14. #3209C4semidirect
15. #3209C4semidirect

Maximal subgroups

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

subgroupindex
#609712
C2×#29872
#176722
#175962
#175842
#94162
#79982
#79772