Finite Group Databasebeta

← #192422

Subgroups — #192422

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34380C211yesmaximal, characteristic
2#21941C211yesmaximal, characteristic
3#21765C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#14353C211yesmaximal, characteristic
6#8871C214yesmaximal, complemented
7#8868C214yesmaximal, complemented
8#8020C211yesmaximal, characteristic
9#7915C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2378—22no—
14#2360—22no—
15#2344—21no—
16#2342—22no—
17Q8×C22⋊C4—21no—
18#2329C2214yes—
19#2323—22no—
20#2316—21no—
21#2313—21no—
22#2280C2211yescharacteristic
23C2×#380—22no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1894—22no—
28#1872C2214yes—
29#1854—22no—
30#1853C2212yes—
31#1852C2212yes—
32#1823—21no—
33#1808—21no—
34#1807—21no—
35#1787—21no—
36C2×#328C418yescomplemented
37C2×#328C2212yes—
38C2×#328C2212yes—
39#1444—22no—
40#1444C2214yes—
41#1294C2214yes—
42#1280C2212yes—
43#1280C2212yes—
44C23×Q8D411yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C22×M16—21no—
48C23×C8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yes—
54C2×C22⋊Q8D412yes—
55C2×C22⋊Q8Q814yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—22no—
58C2×#117—22no—
59C2×#117D412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4D412yes—
68C2×#107—22no—
69C2×#106—22no—
70C2×#103—24no—
71#407—22no—
72#407—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8C2×C414yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2311yescharacteristic
79C2×C22⋊C8C2311yescharacteristic
80#391—22no—
81#391—24no—
82#389—24no—
83#389—22no—
84#389—24no—
85#388—24no—
86#388—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#385—24no—
91#381—22no—
92#381—21no—
93#380—22no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#380—21no—
102#380—48no—
103#380D411yescharacteristic
104#380D411yescharacteristic
105#380C2312yes—
106#379—21no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—24no—
116C2×#95C2×C414yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119#340—22no—
120#340—22no—
121#340—24no—
122#328C2×C418yes—
123#328C2×C418yes—
124C25C2×D411yescharacteristic
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—82no—
128C22×Q8C4⋊C414yescomplemented
129C22×Q8C2×D412yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—82no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D414yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×Q812yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4C2411yescharacteristic
151C2×M16—42no—
152C2×M16—82no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—82no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8C4⋊C418yescomplemented
166C22⋊Q8C22×C414yes—
167C22⋊Q8C2×Q814yes—
168C4×Q8—84no—
169#117—82no—
170#117—84no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4C4⋊C414yescomplemented
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—24no—
209#107—82no—
210#106—82no—
211#103—84no—
212C22⋊C8—24no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—42no—
218C22⋊C8C22×C412yes—
219C22⋊C8C22×C412yes—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—48no—
232#95—41no—
233#95—41no—
234#95C22×C414yes—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—42no—
240C24C4×D412yes—
241C24#12011yescharacteristic
242C24#12011yescharacteristic
243C24C22⋊Q812yes—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×Q8C2×C4⋊C414yes—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—216no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—84no—
287C22×C4—82no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—88no—
291C22×C4C2×C4⋊C412yes—
292C22×C4C2×C4⋊C412yes—
293C22×C4C4×D412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4C22⋊Q812yes—
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309M16—84no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—44no—
315C2×C8—42no—
316C2×C8—84no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—84no—
330C4⋊C4—84no—
331C4⋊C4—816no—
332C4⋊C4C2×C4⋊C414yes—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—28no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C42—44no—
346C42—84no—
347C42—84no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23#38512yes—
369C23#38512yes—
370C23#45312yes—
371C23#45612yes—
372C23C2×C4×D411yescharacteristic
373C23C2×#12011yescharacteristic
374C23C2×C22⋊Q811yescharacteristic
375C23#57712yes—
376C23#57812yes—
377Q8—44no—
378Q8—48no—
379Q8—82no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—84no—
419C2×C4—816no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C8—42no—
434C8—42no—
435C8—82no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—22no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—24no—
445C22—22no—
446C22—22no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—44no—
452C22—48no—
453C22—44no—
454C22—42no—
455C22—42no—
456C22#185914yes—
457C22C2×#38511yescharacteristic
458C22#294911yescharacteristic
459C22#295011yescharacteristic
460C22#301312yes—
461C22C2×#45311yescharacteristic
462C22C2×#45611yescharacteristic
463C22#302612yes—
464C22#308212yes—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—22no—
471C4—48no—
472C4—44no—
473C4—44no—
474C4—82no—
475C4—84no—
476C4—82no—
477C4—162no—
478C2—21no—
479C2—22no—
480C2—24no—
481C2—21no—
482C2—42no—
483C2#1842012yes—
484C2#1842312yes—
485C2C2×#185911yescharacteristic
486C2#2088011yescharacteristic
487C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8871C2semidirect
7. #8868C2semidirect
36. C2×#328C4semidirect
128. C22×Q8C4⋊C4semidirect
165. C22⋊Q8C4⋊C4semidirect
188. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#343802
#219412
#217652
C22×#3282
#143532
#88712
#88682
#80202
#79152