Finite Group Databasebeta

← #116039

Subgroups — #116039

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic
4#14687C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382—22no—
15C2×C43C2211yescharacteristic
16#1994—22no—
17#1994C414yescomplemented
18#1990—21no—
19#1990—21no—
20#1956—21no—
21#1956—21no—
22#1872—22no—
23#1872—22no—
24#1803—22no—
25#1803—22no—
26C2×#339—22no—
27C4⋊#95—22no—
28C4⋊#95—22no—
29#1296—22no—
30#1296—22no—
31#1276—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533Q812yes—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D412yes—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D412yes—
74C2×C4⋊C8D412yes—
75C2×#103—24no—
76C2×#103—24no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#382—48no—
90C2×#95—28no—
91C43—21no—
92C43—21no—
93C43C2×C412yes—
94C43D412yes—
95#340—24no—
96#340—24no—
97#339—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—42no—
109C2×M16—42no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C4⋊C412yes—
120C4⋊Q8C4⋊C412yes—
121#125—22no—
122#125—22no—
123#125—41no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—22no—
142#117—22no—
143#117—41no—
144#117—41no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C22⋊C4—41no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—41no—
186C2×C42C22⋊C414yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207#103—84no—
208#103—84no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C4×C8—41no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217#95—416no—
218C24#12011yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4C22⋊Q811yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12411yescharacteristic
263C22×C4#12411yescharacteristic
264C22×C4#12411yescharacteristic
265C22×C4#12711yescharacteristic
266C22×C4#13512yes—
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4#13712yes—
270M16—44no—
271M16—82no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—81no—
280C2×C8—81no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C42—28no—
310C42—28no—
311C42—24no—
312C42—24no—
313C42—44no—
314C42—44no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42C2×C4⋊C411yescharacteristic
320C42#11711yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42C22⋊Q811yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C42#12311yescharacteristic
333C42#12611yescharacteristic
334C42#12611yescharacteristic
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#38611yescharacteristic
340C23#39211yescharacteristic
341C23#39311yescharacteristic
342C23#45212yes—
343C23#45212yes—
344Q8—44no—
345Q8—44no—
346Q8—82no—
347Q8—82no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—28no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#38011yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38811yescharacteristic
385C2×C4#38811yescharacteristic
386C2×C4#39011yescharacteristic
387C2×C4#39411yescharacteristic
388C2×C4#39411yescharacteristic
389C2×C4#39611yescharacteristic
390C2×C4#39611yescharacteristic
391C2×C4#39711yescharacteristic
392C2×C4#39911yescharacteristic
393C2×C4#41612yes—
394C2×C4#41912yes—
395C2×C4#45312yes—
396C2×C4#45412yes—
397C2×C4#45512yes—
398C2×C4#45612yes—
399C2×C4#46412yes—
400C2×C4#46412yes—
401C2×C4#47912yes—
402C2×C4#47912yes—
403C2×C4#48512yes—
404C2×C4#48512yes—
405C2×C4#48812yes—
406C2×C4#49112yes—
407C2×C4#49511yescharacteristic
408C2×C4#49511yescharacteristic
409C2×C4#49711yescharacteristic
410C2×C4#49711yescharacteristic
411C8—82no—
412C8—82no—
413C8—82no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22#143911yescharacteristic
423C22#187312yes—
424C22#200211yescharacteristic
425C22#200311yescharacteristic
426C22#200411yescharacteristic
427C22#200511yescharacteristic
428C22#201511yescharacteristic
429C22#201511yescharacteristic
430C22#202612yes—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—48no—
436C4—82no—
437C4—82no—
438C4—162no—
439C4—162no—
440C4#180512yes—
441C4#185912yes—
442C4#186012yes—
443C4#195811yescharacteristic
444C4#196011yescharacteristic
445C4#203912yes—
446C4#204012yes—
447C4#207412yes—
448C4#209711yescharacteristic
449C4#209711yescharacteristic
450C2—22no—
451C2—22no—
452C2#1029111yescharacteristic
453C2#1029211yescharacteristic
454C2#1029611yescharacteristic
455C2#1029811yescharacteristic
456C2#1105211yescharacteristic
457C2#1107811yescharacteristic
458C2#1107911yescharacteristic
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
17. #1994C4semidirect

Maximal subgroups

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

subgroupindex
#218602
C2×#19942
#146872
#80432
C2×#12762