Finite Group Databasebeta

← #161870

Subgroups — #161870

Order 512.

865 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35198C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7719C212yesmaximal, complemented
6#7706C212yesmaximal, complemented
7#7699C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#542—21no—
13#2860C2212yes—
14#2858C2212yes—
15#2662—22no—
16#2551C411yescharacteristic
17#2548C411yescharacteristic
18#2541C411yescharacteristic
19#2538C411yescharacteristic
20#2491—21no—
21#2461—22no—
22#2452C412yes—
23#2445C412yes—
24#2426—21no—
25#2324—21no—
26C4×#127C2211yescharacteristic
27#2293C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1270C2211yescharacteristic
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2211yescharacteristic
52#1270C2212yes—
53#542—44no—
54#542—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×#127C2×C411yescharacteristic
58C2×#127C2×C412yes—
59C2×#126—22no—
60C2×#124—21no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×#117—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#396—22no—
85#396—22no—
86#395—22no—
87#394—22no—
88#394—22no—
89#394—22no—
90#393—22no—
91#392—22no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—22no—
96#390—22no—
97#390C2×C412yes—
98#390C2×C412yes—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2×C411yescharacteristic
102#390C2312yes—
103#390C2311yescharacteristic
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C411yescharacteristic
108#389C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#388—21no—
113#388—21no—
114#386—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#380—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—21no—
126C43C2311yescharacteristic
127C43C2312yes—
128#336—22no—
129#336—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—22no—
143C22×Q8C22⋊C411yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C411yescharacteristic
149C22×D4C22×C412yes—
150C23×C4—21no—
151C23×C4—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C4⋊Q8—44no—
159C4⋊Q8—82no—
160#127—24no—
161#127—82no—
162#126—44no—
163#126—44no—
164#124—44no—
165#124—44no—
166#124—44no—
167#124—82no—
168#124—82no—
169#123—48no—
170C22⋊Q8—48no—
171C22⋊Q8—44no—
172#121—44no—
173#121—44no—
174C4×Q8—24no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—42no—
178C4×D4—24no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—81no—
182C4×D4—81no—
183C4×D4—82no—
184#117—42no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C42—21no—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—82no—
237#97—44no—
238C4×C8—42no—
239C4×C8—42no—
240C4×C8—81no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24—21no—
250C24—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—44no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—81no—
270C2×D4—81no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—81no—
331C4⋊C4—82no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—81no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—24no—
353C42—22no—
354C42—22no—
355C42—21no—
356C42—22no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—42no—
361C42—41no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—41no—
366C42—42no—
367C42#13114yes—
368C42AGL(1,ℤ/8)12yes—
369C42AGL(1,ℤ/8)12yes—
370C23—41no—
371C23—42no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23C22×C22⋊C411yescharacteristic
385Q8—44no—
386Q8—164no—
387D4—82no—
388D4—84no—
389D4—84no—
390D4—84no—
391D4—164no—
392D4—164no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—84no—
431C2×C4—164no—
432C2×C4#40812yes—
433C2×C4#40812yes—
434C2×C4#41812yes—
435C2×C4#41812yes—
436C2×C4#42112yes—
437C2×C4C2×#13112yes—
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C8—42no—
443C8—81no—
444C8—82no—
445C8—81no—
446C8—81no—
447C8—82no—
448C22—81no—
449C22—82no—
450C22—82no—
451C22—81no—
452C22—81no—
453C22—82no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22—82no—
458C22—81no—
459C22—82no—
460C22—81no—
461C22—82no—
462C22C2×#40811yescharacteristic
463C22#288411yescharacteristic
464C22#288811yescharacteristic
465C22C2×#41811yescharacteristic
466C22#289711yescharacteristic
467C22C2×#42111yescharacteristic
468C22#290111yescharacteristic
469C4—22no—
470C4—22no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—22no—
476C4—41no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—81no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4#149614yes—
486C2—161no—
487C2—162no—
488C2#1739011yescharacteristic
489C2#1746811yescharacteristic
490C2C2×#149611yescharacteristic
491C2#1760211yescharacteristic
492C2#1763111yescharacteristic
493C2#1768111yescharacteristic
494C2#1775811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
3. #22018C2semidirect
4. #7726C2semidirect
5. #7719C2semidirect
6. #7706C2semidirect
7. #7699C2semidirect
8. #7587C2semidirect
9. #7574C2semidirect
10. #7569C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#351982
#228322
#220182
#77262
#77192
#77062
#76992
#75872
#75742
#75692
#75602