Finite Group Databasebeta

← #165080

Subgroups — #165080

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36270C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22736C211yesmaximal, characteristic, complemented
4#7712C212yesmaximal, complemented
5#7706C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7651C211yesmaximal, characteristic
9#7628C212yesmaximal, complemented
10#7594C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic
12C2×#552—21no—
13C2×#446—21no—
14#2860C2212yes—
15#2859C2211yescharacteristic
16#2601—22no—
17#2562—21no—
18#2558—22no—
19#2538C411yescharacteristic
20#2538C411yescharacteristic
21#2536—21no—
22#2506C412yes—
23#2445C412yes—
24#2354C411yescharacteristic
25#2352—21no—
26D4×C4⋊C4C411yescharacteristic
27C4×#127C2211yescharacteristic
28#2293C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1917—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1770—21no—
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1745C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53#552—44no—
54#552—44no—
55C2×#127C2×C411yescharacteristic
56C2×#126—22no—
57C2×#125—22no—
58C2×#125—21no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×#117—21no—
71C2×#117—22no—
72C22×C4⋊C4—21no—
73#446—44no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#400—22no—
85#398—22no—
86#396—22no—
87#396—22no—
88#394—22no—
89#394—22no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2311yescharacteristic
96#390C2311yescharacteristic
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2312yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#387—22no—
111#386—22no—
112#385—22no—
113#384C2×C411yescharacteristic
114#384C2×C411yescharacteristic
115#384C2×C411yescharacteristic
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C22⋊C4—22no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8C22⋊C411yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157#127—22no—
158#127—41no—
159#126—44no—
160#126—44no—
161#125—48no—
162#125—42no—
163#125—42no—
164#123—48no—
165C22⋊Q8—44no—
166C22⋊Q8—44no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—42no—
172C4×D4—24no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180#117—44no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—44no—
233C4⋊C8—42no—
234C4⋊C8—44no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238#97—44no—
239#97—44no—
240C4×C8—42no—
241C4×C8—42no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—21no—
252C24—21no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4#13112yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—22no—
353C42—21no—
354C42—22no—
355C42—21no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—42no—
366C42—41no—
367C42—44no—
368C42AGL(1,ℤ/8)12yes—
369C42AGL(1,ℤ/8)12yes—
370C42AGL(1,ℤ/8)12yes—
371C42#13712yes—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23C22×C22⋊C411yescharacteristic
387Q8—44no—
388D4—42no—
389D4—42no—
390D4—84no—
391D4—84no—
392D4—81no—
393D4—81no—
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—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—88no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4#41312yes—
435C2×C4#41712yes—
436C2×C4#41712yes—
437C2×C4#41812yes—
438C2×C4#41812yes—
439C2×C4C2×#13111yescharacteristic
440C2×C4#56811yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C8—42no—
446C8—42no—
447C8—81no—
448C8—82no—
449C8—81no—
450C8—82no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22#288011yescharacteristic
466C22#288411yescharacteristic
467C22C2×#41311yescharacteristic
468C22C2×#41811yescharacteristic
469C22C2×#41711yescharacteristic
470C22#290111yescharacteristic
471C22#290111yescharacteristic
472C4—21no—
473C4—21no—
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—41no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—82no—
488C4—82no—
489C4#155714yes—
490C2—82no—
491C2—161no—
492C2#1742311yescharacteristic
493C2#1751111yescharacteristic
494C2#1751211yescharacteristic
495C2#1762811yescharacteristic
496C2#1763111yescharacteristic
497C2#1768711yescharacteristic
498C2C2×#155711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
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. #22736C2semidirect
4. #7712C2semidirect
5. #7706C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
9. #7628C2semidirect
10. #7594C2semidirect

Maximal subgroups

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

subgroupindex
#362702
#228322
#227362
#77122
#77062
#76632
#76602
#76512
#76282
#75942
#75692