Finite Group Databasebeta

← #198868

Subgroups — #198868

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3#17648C211yesmaximal, characteristic
4#17648C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic
6#9286C218yesmaximal, complemented
7#7965C211yesmaximal, characteristic
8#7860C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3231—22no—
12#3231—22no—
13#3231C414yes—
14#3231C2214yes—
15C2×#500C411yescharacteristic
16C2×#498C411yescharacteristic
17C2×#487C411yescharacteristic
18C2×#487C411yescharacteristic
19C2×#485—22no—
20C2×#479—22no—
21#2986—22no—
22#2985C2214yes—
23#2984—22no—
24#2983—21no—
25#2982—21no—
26#2962—22no—
27#2960—22no—
28#2924—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#1531C2214yes—
35#1531C2214yes—
36#1528—22no—
37#1528—22no—
38#1526—22no—
39#1526—22no—
40#1497C2214yes—
41#1493—22no—
42#1479—21no—
43#1479—21no—
44#1287C2214yes—
45#1275C2214yes—
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#533D412yes—
60#533D412yes—
61#533C2312yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
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#500—22no—
81#500C2×C414yes—
82#498—22no—
83#498C2×C414yes—
84#487—22no—
85#487—22no—
86#487C2×C414yes—
87#487C2×C414yes—
88#485—28no—
89#485—28no—
90#479—28no—
91#479—28no—
92#446—24no—
93#446—24no—
94#446—28no—
95#446—24no—
96C8×Q8—24no—
97C8×Q8—24no—
98C8×Q8—24no—
99#433—28no—
100#432—24no—
101#432—21no—
102#432D412yes—
103#432C2312yes—
104#432C2312yes—
105#431—24no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431C2×C412yes—
115#431D411yescharacteristic
116#431D411yescharacteristic
117#431C2312yes—
118#431C2312yes—
119C2×#107—22no—
120C2×#106—22no—
121#424—24no—
122#423—21no—
123#423—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×#103—22no—
127C2×#103—22no—
128#407—21no—
129#407—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132#405—24no—
133C4×M16—21no—
134C4×M16—21no—
135C4×M16—21no—
136C4×M16—21no—
137C4×M16—21no—
138C4×M16D412yes—
139C2×#97C2312yes—
140C2×C4×C8—21no—
141C2×C4×C8C2×C411yescharacteristic
142C2×C4×C8C2×C411yescharacteristic
143C2×C4×C8D412yes—
144#330—28no—
145#330—28no—
146Q8⋊C8—24no—
147Q8⋊C8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—24no—
156C2×M16—22no—
157C2×M16—42no—
158C2×M16—41no—
159C2×M16—41no—
160C2×M16—41no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C22⋊C412yes—
176C4⋊Q8C22⋊C412yes—
177C4⋊Q8C22×C412yes—
178C4⋊Q8C22×C412yes—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C22×C412yes—
181#125—22no—
182#125—22no—
183#125—22no—
184#125—22no—
185#125—41no—
186#125—41no—
187#125C22⋊C414yes—
188#125C2×D414yes—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—41no—
226#117—41no—
227#117C22⋊C414yes—
228#117C2×D414yes—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C412yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—41no—
243C2×C42—22no—
244C2×C42—41no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257#107—28no—
258#106—28no—
259C4⋊C8—28no—
260C4⋊C8—28no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—41no—
270#103—28no—
271#103—28no—
272C22⋊C8—22no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—22no—
280C22⋊C8—41no—
281#97—28no—
282#97—22no—
283#97—22no—
284#97—22no—
285#97C2×D414yes—
286C4×C8—28no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—41no—
292C4×C8—41no—
293C4×C8—41no—
294C4×C8C22×C412yes—
295C4×C8C22×C412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298C24#12311yescharacteristic
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—82no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12311yescharacteristic
334C22×C4#12311yescharacteristic
335C22×C4#12311yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339M16—44no—
340M16—44no—
341M16—44no—
342M16—82no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—44no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4C2×C22⋊C414yes—
378C4⋊C4C2×C22⋊C414yes—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C42—24no—
384C42—24no—
385C42—42no—
386C42—42no—
387C42C2×C22⋊C411yescharacteristic
388C42C2×C22⋊C411yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42#12011yescharacteristic
394C42#12011yescharacteristic
395C42#12111yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12111yescharacteristic
399C42C23×C411yescharacteristic
400C42C22×D411yescharacteristic
401C42C22×D411yescharacteristic
402C42C22×D411yescharacteristic
403C23—22no—
404C23—41no—
405C23—41no—
406C23#38611yescharacteristic
407C23#38611yescharacteristic
408C23C2×#12311yescharacteristic
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—82no—
417Q8—82no—
418Q8—82no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#44312yes—
453C2×C4#44412yes—
454C2×C4C22×C22⋊C411yescharacteristic
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×#12011yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C8—42no—
461C8—42no—
462C8—42no—
463C8—42no—
464C8—44no—
465C8—44no—
466C8—82no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—82no—
476C22#187612yes—
477C22C2×#38611yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—44no—
487C4—44no—
488C4—82no—
489C4—82no—
490C4#187612yes—
491C4#190212yes—
492C4#191212yes—
493C4C2×#44311yescharacteristic
494C4C2×#44411yescharacteristic
495C2—22no—
496C2—41no—
497C2—41no—
498C2C2×#187611yescharacteristic
499C2C2×#190211yescharacteristic
500C2C2×#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
C2×#32312
#602502
#176482
#174792
#92862
#79652
#78602