Finite Group Databasebeta

← #117472

Subgroups — #117472

Order 512.

726 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3C2×#1556C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8033C211yesmaximal, characteristic
7#8022C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1993—22no—
16#1990—21no—
17#1990—21no—
18#1949—22no—
19#1853—21no—
20#1852—21no—
21#1847—21no—
22#1836—21no—
23#1736—22no—
24#1556—22no—
25#1556C414yescomplemented
26C2×#332—22no—
27#1446—22no—
28#1445—22no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31#1300—22no—
32#1295—22no—
33#1294—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#384D411yescharacteristic
78#384D411yescharacteristic
79#382—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88#376—21no—
89#376—21no—
90#376—21no—
91#376—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D412yes—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#332—28no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—41no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—82no—
110C22×Q8—82no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—24no—
119C2×M16—42no—
120C2×M16—41no—
121C2×M16—41no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—41no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141#117—44no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4C4⋊C414yes—
166C2×C22⋊C4—41no—
167C2×C42—24no—
168C2×C42—42no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C22×C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184#107—82no—
185#106—82no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—41no—
189C4⋊C8C22⋊C414yes—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—41no—
200C4×C8—41no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—84no—
213#95—84no—
214C24#12011yescharacteristic
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—82no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×Q8—84no—
226C2×Q8—84no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4#12311yescharacteristic
271C22×C4#12411yescharacteristic
272C22×C4#12711yescharacteristic
273M16—44no—
274M16—44no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—81no—
283C2×C8—81no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C22⋊C4—82no—
303C42—44no—
304C42—44no—
305C42—82no—
306C42C2×C22⋊C411yescharacteristic
307C42#11711yescharacteristic
308C42C4×D411yescharacteristic
309C42C4×D411yescharacteristic
310C42#12111yescharacteristic
311C42#12111yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42#12311yescharacteristic
317C42#12311yescharacteristic
318C42#12411yescharacteristic
319C42#12411yescharacteristic
320C42#12611yescharacteristic
321C42#12611yescharacteristic
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23#38511yescharacteristic
327C23#39211yescharacteristic
328C23#39411yescharacteristic
329C23#45412yes—
330C23#45612yes—
331Q8—84no—
332Q8—84no—
333Q8—84no—
334Q8—84no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#38011yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#39011yescharacteristic
389C2×C4#39311yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39611yescharacteristic
392C2×C4#39611yescharacteristic
393C2×C4#39711yescharacteristic
394C2×C4#39911yescharacteristic
395C2×C4#45512yes—
396C2×C4#45612yes—
397C8—44no—
398C8—44no—
399C8—82no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—21no—
407C22—21no—
408C22#143911yescharacteristic
409C22#185712yes—
410C22#200311yescharacteristic
411C22#200411yescharacteristic
412C22#200511yescharacteristic
413C22#200511yescharacteristic
414C22#200811yescharacteristic
415C22#200811yescharacteristic
416C22#203812yes—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—84no—
428C4—84no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4#188212yes—
434C4#202812yes—
435C2—22no—
436C2—22no—
437C2#1009211yescharacteristic
438C2#1009311yescharacteristic
439C2#1010011yescharacteristic
440C2#1010111yescharacteristic
441C2#1106411yescharacteristic
442C2#1112911yescharacteristic
443C2#1113611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#213242
C2×#15562
#142822
C2×#13002
#80332
#80222