Finite Group Databasebeta

← #197828

Subgroups — #197828

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21409C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9228C212yesmaximal, complemented
7#9224C212yesmaximal, complemented
8#9220C212yesmaximal, complemented
9#9216C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic
11#7876C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3237—24no—
15#3237C418yescomplemented
16C2×#501—21no—
17C2×#491—21no—
18C2×#485—22no—
19C2×#477C2212yes—
20C2×#475C2212yes—
21#2962—24no—
22#2962—24no—
23C2×#446—22no—
24C2×#446C2212yes—
25C2×C8×Q8C2212yes—
26C2×#432C2211yescharacteristic
27C2×#424—21no—
28C2×#423C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C2×#405—21no—
31C2×C4×M16C2211yescharacteristic
32#2285C2211yescharacteristic
33#1990C2212yes—
34#1988—21no—
35#1981—21no—
36#1956C2212yes—
37#1894—21no—
38#1894—21no—
39#1894—22no—
40#1803C2212yes—
41#1803C2212yes—
42#1763—22no—
43#1550—24no—
44C2×#333C2211yescharacteristic
45C2×#333C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50#1277C2212yes—
51#1277C2212yes—
52C22×M16—21no—
53C22×M16D411yescharacteristic
54C22×M16D411yescharacteristic
55C23×C8—21no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533C2×C414yes—
62#533C2×C414yes—
63#533D414yes—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×#125—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8D412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—21no—
81C22×C42C2311yescharacteristic
82#501—28no—
83#491—28no—
84#485—216no—
85#477—24no—
86#477C2×C418yes—
87#475—24no—
88#475C2×C418yes—
89#446—216no—
90#446—24no—
91#446—24no—
92C8×Q8—24no—
93C8×Q8—24no—
94#432—22no—
95#432C2×C414yes—
96C2×#107—22no—
97C2×#106—22no—
98#424—28no—
99#423—22no—
100#423—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8D414yes—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—22no—
118C2×C22⋊C8—21no—
119#405—28no—
120C4×M16—22no—
121C4×M16—22no—
122C2×#97—22no—
123C2×#97—21no—
124C2×#97D412yes—
125C2×C4×C8—22no—
126C2×C4×C8C2312yes—
127#384C2312yes—
128#382—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#333—22no—
138#333—22no—
139#332—22no—
140#332—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C2×M16—82no—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—44no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C22⋊C414yes—
172C4⋊Q8C22⋊C414yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175#125—24no—
176#125—24no—
177#125—24no—
178#125—42no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8C22⋊C418yes—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8C22×C414yes—
196C4×Q8C2×D414yes—
197#117—24no—
198#117—24no—
199#117—24no—
200#117—42no—
201#117—82no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42C22×C412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42#4111yescharacteristic
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231#107—28no—
232#106—28no—
233C4⋊C8—28no—
234C4⋊C8—28no—
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—42no—
244C4⋊C8C22×C414yes—
245C4⋊C8C2×D412yes—
246C4⋊C8C2×D412yes—
247#103—28no—
248#103—28no—
249#103—44no—
250#103—44no—
251#103—82no—
252#103—82no—
253C22⋊C8—24no—
254#97—28no—
255#97—24no—
256#97—24no—
257C4×C8—28no—
258C4×C8—22no—
259C4×C8—22no—
260#95—41no—
261#95—41no—
262C24#12111yescharacteristic
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×Q8C2×C22⋊C414yes—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12012yes—
295C22×C4#12112yes—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12312yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303M16—28no—
304M16—44no—
305M16—84no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—48no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8#13714yes—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4C2×C22⋊C414yes—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—82no—
350C42—24no—
351C42—24no—
352C42—44no—
353C42—82no—
354C42C2×C22⋊C412yes—
355C42C4×D412yes—
356C42C4×D412yes—
357C42#12012yes—
358C42#12112yes—
359C42#12312yes—
360C42C23×C411yescharacteristic
361C42C22×D411yescharacteristic
362C42C22×D411yescharacteristic
363C42C2×#4111yescharacteristic
364C23—22no—
365C23—22no—
366C23—21no—
367C23—21no—
368C23#38612yes—
369C23C2×#12111yescharacteristic
370Q8—24no—
371Q8—24no—
372Q8—42no—
373Q8—42no—
374Q8—44no—
375Q8—84no—
376Q8—82no—
377Q8—82no—
378Q8—81no—
379Q8—81no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#38612yes—
409C2×C4#38612yes—
410C2×C4#38612yes—
411C2×C4C22×C22⋊C411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×#12011yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12311yescharacteristic
417C2×C4C2×#13712yes—
418C2×C4#57512yes—
419C2×C4#57712yes—
420C2×C4#57812yes—
421C8—24no—
422C8—44no—
423C8—42no—
424C8—42no—
425C8—42no—
426C8—81no—
427C8—81no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22C2×#38611yescharacteristic
434C22#303712yes—
435C22#306312yes—
436C22#306811yescharacteristic
437C22#306911yescharacteristic
438C4—21no—
439C4—21no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—44no—
445C4—82no—
446C4—82no—
447C4—162no—
448C4#289812yes—
449C4C4×#13712yes—
450C4#300411yescharacteristic
451C4#300511yescharacteristic
452C4#305711yescharacteristic
453C4#305911yescharacteristic
454C2—42no—
455C2#1937312yes—
456C2#1937512yes—
457C2#2092611yescharacteristic
458C2#2097411yescharacteristic
459C2#2100411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9228C2semidirect
7. #9224C2semidirect
8. #9220C2semidirect
9. #9216C2semidirect
15. #3237C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
C2×#29622
#218942
#214092
C2×#15502
#92282
#92242
#92202
#92162
#80442
#78762