Finite Group Databasebeta

← #164108

Subgroups — #164108

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35257C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic
3#22473C211yesmaximal, characteristic
4#7725C212yesmaximal, complemented
5#7711C212yesmaximal, complemented
6#7685C212yesmaximal, complemented
7#7671C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#544C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×C8×Q8—21no—
15C2×#436—21no—
16#2857C2212yes—
17#2851C2212yes—
18#2629C412yes—
19#2628C412yes—
20#2454—21no—
21#2453—21no—
22#2452—21no—
23#2451—21no—
24#2450—22no—
25#2449—22no—
26#2317C411yescharacteristic
27#2311C411yescharacteristic
28#2311C411yescharacteristic
29#2296C2211yescharacteristic
30C2×#389C2211yescharacteristic
31#1984—21no—
32#1954—22no—
33#1954—21no—
34#1870—22no—
35#1817C2212yes—
36#1797C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C22×M16—22no—
55#544—44no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C412yes—
58C2×#125—21no—
59C2×#125C2×C412yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C4×#41—44no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D412yes—
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2312yes—
78C8×Q8—44no—
79#436—44no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—22no—
93#400—22no—
94#399—22no—
95#398—22no—
96#398—22no—
97#396—22no—
98#396—22no—
99#395—22no—
100#393—22no—
101#391—21no—
102#391—21no—
103#391—22no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—22no—
109#389—21no—
110#389—21no—
111#389C2×C411yescharacteristic
112#389C2×C412yes—
113#389C2×C411yescharacteristic
114#389C2311yescharacteristic
115#389C2312yes—
116#389C2311yescharacteristic
117#387—22no—
118#387—22no—
119#386—22no—
120#386—22no—
121#385—22no—
122#385—22no—
123#385—21no—
124#385—21no—
125#385—22no—
126#384C2×C412yes—
127#382C2×C412yes—
128#380—22no—
129#380C2×C412yes—
130#380C2×C412yes—
131#380C2×C412yes—
132#380C2×C412yes—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C2×#95—21no—
136#336—22no—
137#336—22no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—42no—
147C22×Q8—22no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22⋊C412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—82no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170#125—44no—
171#125—44no—
172C22⋊Q8—48no—
173C22⋊Q8—82no—
174C22⋊Q8—82no—
175C4×Q8—24no—
176C4×Q8—42no—
177C4×D4—24no—
178C4×D4—44no—
179C4×D4—42no—
180C4×D4—44no—
181#117—24no—
182#117—24no—
183#117—44no—
184#117—42no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42—44no—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—44no—
234C4⋊C8—42no—
235C22⋊C8—42no—
236C22⋊C8—44no—
237C22⋊C8—42no—
238C22⋊C8—44no—
239C22⋊C8—81no—
240C22⋊C8—81no—
241C22⋊C8—82no—
242C4×C8—42no—
243C4×C8—44no—
244C4×C8—42no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257C24C2×C22⋊C412yes—
258C24C2×C22⋊C411yescharacteristic
259#41—416no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—44no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8#10114yes—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#13114yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315M16—88no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4#9914yes—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—22no—
370C42—24no—
371C42—22no—
372C42—24no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—42no—
377C42—42no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23#40812yes—
392C23C22×C22⋊C411yescharacteristic
393Q8—28no—
394Q8—164no—
395D4—48no—
396D4—44no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—84no—
445C2×C4—162no—
446C2×C4#40812yes—
447C2×C4C2×#9914yes—
448C2×C4C2×#9912yes—
449C2×C4C2×#10114yes—
450C2×C4C2×#10112yes—
451C2×C4C2×#13112yes—
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C8—42no—
457C8—42no—
458C8—81no—
459C8—82no—
460C8—81no—
461C8—82no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22C2×#40811yescharacteristic
477C22C22×#9911yescharacteristic
478C22C22×#10111yescharacteristic
479C22#289711yescharacteristic
480C22#289811yescharacteristic
481C22#290111yescharacteristic
482C22#290111yescharacteristic
483C4—21no—
484C4—21no—
485C4—24no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—24no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—82no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4#147214yes—
500C2—42no—
501C2—82no—
502C2C2×#147211yescharacteristic
503C2#1746911yescharacteristic
504C2#1748311yescharacteristic
505C2#1760311yescharacteristic
506C2#1760611yescharacteristic
507C2#1769211yescharacteristic
508C2#1777711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7725C2semidirect
5. #7711C2semidirect
6. #7685C2semidirect
7. #7671C2semidirect
8. #7598C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#352572
#225302
#224732
#77252
#77112
#76852
#76712
#75982
#75762
#75732
#75642