Finite Group Databasebeta

← #200796

Subgroups — #200796

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic
3#17748C211yesmaximal, characteristic
4#17734C211yesmaximal, characteristic
5#17672C211yesmaximal, characteristic
6#9420C214yesmaximal, complemented
7#9420C214yesmaximal, complemented
8#7992C211yesmaximal, characteristic
9C2×#1291C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#533C2211yescharacteristic
12#3166—22no—
13#3166—22no—
14#3165C412yes—
15#3165C412yes—
16#3165C2212yes—
17#3165C2212yes—
18#3162—22no—
19#3160C414yes—
20#2987C2214yes—
21#2985—22no—
22#2984—22no—
23#2983—22no—
24#2962—22no—
25#2961—22no—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31#2911—22no—
32C2×C4×M16C2211yescharacteristic
33#1559—22no—
34#1558C2212yes—
35#1558C2212yes—
36#1556C2212yes—
37#1556C2212yes—
38#1554—22no—
39#1550—22no—
40#1542—22no—
41#1537C2214yes—
42#1531—22no—
43#1528—22no—
44#1291C2214yes—
45#1291C2214yes—
46C22×M16—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#533C2×C412yes—
58#533C2×C412yes—
59#533C2×C412yes—
60#533D412yes—
61#533D412yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×#125—21no—
68C2×#125D412yes—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117—21no—
78C2×#117D412yes—
79C22×C42C2311yescharacteristic
80#501—24no—
81#501—24no—
82#501—24no—
83#501—24no—
84#500C2×C414yes—
85#500C2×C414yes—
86#499—28no—
87#498C2×C414yes—
88#498C2×C414yes—
89#446—28no—
90#446—24no—
91#446—28no—
92#446—28no—
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#432D412yes—
104#432C2312yes—
105#432C2312yes—
106#432C2312yes—
107#432C2312yes—
108#431—24no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431D412yes—
113#428—24no—
114#428—24no—
115#428—24no—
116#428—24no—
117#427—28no—
118#424—24no—
119#423—21no—
120#423D412yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8D412yes—
123#407—21no—
124#407—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127#405—24no—
128C4×M16—21no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C4×M16C2×C412yes—
133C4×M16C2312yes—
134C2×#97—21no—
135C2×#97—21no—
136C2×#97—21no—
137C2×C4×C8C2×C412yes—
138#333—24no—
139#333—24no—
140#332—24no—
141#332—24no—
142#330—28no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4#4111yescharacteristic
149C23×C4#4111yescharacteristic
150C2×M16—28no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—41no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—41no—
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⋊Q8—21no—
168C4⋊Q8C22⋊C412yes—
169C4⋊Q8C22⋊C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174#125—22no—
175#125—22no—
176#125—22no—
177#125—22no—
178#125—41no—
179#125—41no—
180#125C22⋊C414yes—
181#125C2×D414yes—
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—22no—
193C22⋊Q8—22no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—41no—
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—22no—
209C4×Q8—22no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—41no—
219#117—41no—
220#117C22⋊C414yes—
221#117C2×D414yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C42—22no—
237C2×C42—41no—
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×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42C2411yescharacteristic
250#107—28no—
251#107—28no—
252#106—28no—
253#106—28no—
254C4⋊C8—28no—
255C4⋊C8—28no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8C2×D414yes—
263C4⋊C8C2×D414yes—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—41no—
271C22⋊C8—41no—
272#97—28no—
273#97—22no—
274#97—22no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—22no—
279#97—22no—
280#97—41no—
281C4×C8—28no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8C22×C414yes—
286C24#12311yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—82no—
298C2×Q8—82no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12311yescharacteristic
320C22×C4#12311yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324M16—28no—
325M16—28no—
326M16—44no—
327M16—82no—
328C2×C8—28no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—44no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—82no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4C2×C22⋊C414yes—
359C4⋊C4C2×C22⋊C414yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—24no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42C2×C22⋊C411yescharacteristic
369C42C2×C22⋊C411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42#12011yescharacteristic
375C42#12011yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42C23×C411yescharacteristic
381C42C22×D411yescharacteristic
382C42C22×D411yescharacteristic
383C42C22×D411yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23#38611yescharacteristic
388C23#38611yescharacteristic
389C23C2×#12311yescharacteristic
390Q8—44no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395Q8—82no—
396Q8—82no—
397Q8—82no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
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—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38611yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#45614yes—
430C2×C4C22×C22⋊C411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C8—28no—
437C8—44no—
438C8—44no—
439C8—82no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—82no—
450C22#188612yes—
451C22C2×#38611yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—44no—
463C4—44no—
464C4—82no—
465C4—82no—
466C4#188612yes—
467C4C2×#45612yes—
468C2—22no—
469C2—41no—
470C2—41no—
471C2C2×#188611yescharacteristic
472C2#2099511yescharacteristic
473C2#2099511yescharacteristic
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. #9420C2semidirect
7. #9420C2semidirect

Maximal subgroups

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

subgroupindex
#606632
#602512
#177482
#177342
#176722
#94202
#79922
C2×#12912