Finite Group Databasebeta

← #223111

Subgroups — #223111

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic, complemented
3#20494C211yesmaximal, characteristic, complemented
4#9890C211yesmaximal, characteristic, complemented
5#9890C211yesmaximal, characteristic, complemented
6#9123C212yesmaximal, complemented
7#9120C212yesmaximal, complemented
8#8719C212yesmaximal, complemented
9#8579C211yesmaximal, characteristic, complemented
10C4×#329C211yesmaximal, characteristic, complemented
11#8033C212yesmaximal, complemented
12C2×#554C411yescharacteristic
13C2×#547C411yescharacteristic
14C2×#436—21no—
15C2×#432C2212yes—
16C2×C4×M16C2212yes—
17#2862C2211yescharacteristic
18#2850C2212yes—
19#2568C411yescharacteristic
20#2563C411yescharacteristic
21#2504C412yes—
22#2503C412yes—
23#2489—21no—
24#2489—21no—
25#2488—22no—
26#2488—22no—
27#2339—21no—
28#2339—21no—
29C4×#124C2211yescharacteristic
30C2×#382C2212yes—
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1939—21no—
34#1924—22no—
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1749C2212yes—
41C4×#97C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×#329C2211yescharacteristic
44C2×#329C2211yescharacteristic
45C2×#329C2212yes—
46#1295C2211yescharacteristic
47#1295C2212yes—
48#1295C2212yes—
49#1295C2212yes—
50#1295C2211yescharacteristic
51#1295C2212yes—
52#1295C2212yes—
53C22×#41—21no—
54C22×M16—22no—
55#554—44no—
56#547—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×#124C2311yescharacteristic
61C2×#123—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C412yes—
65C2×#121C2×C412yes—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C42C2312yes—
75#436—28no—
76#432—24no—
77#432—42no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C4×M16—24no—
85C4×M16—42no—
86C2×#97—21no—
87C2×#97—21no—
88C2×#97C2312yes—
89C2×#97C2312yes—
90C2×#97C2311yescharacteristic
91C2×#97C2311yescharacteristic
92C2×#97C2311yescharacteristic
93C2×#97C2311yescharacteristic
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97#400—22no—
98#400—22no—
99#400—22no—
100#397—22no—
101#397—22no—
102#393—22no—
103#391—21no—
104#391—21no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—21no—
109#391C2×C412yes—
110#391C2×C411yescharacteristic
111#390C2×C412yes—
112#390C2×C411yescharacteristic
113#388—22no—
114#388—22no—
115#388—21no—
116#388—21no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#382—22no—
122#382—22no—
123#382C2×C412yes—
124#382C2×C412yes—
125#382C2×C412yes—
126#382C2312yes—
127#382C2311yescharacteristic
128#381—22no—
129#381—22no—
130#380—21no—
131#380—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140C2×#95—22no—
141C2×#95—22no—
142C43C2311yescharacteristic
143#336—21no—
144#336—22no—
145#336—22no—
146#336—21no—
147#329—22no—
148#329—22no—
149#329—42no—
150C2×#41—44no—
151C2×#41—44no—
152C2×#41—44no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8C22×C411yescharacteristic
156C22×D4—22no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4C22×C412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C2×M16—28no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#125—44no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181#123—48no—
182C22⋊Q8—48no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185#121—44no—
186C4×Q8—42no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—24no—
235C4⋊C8—44no—
236C4⋊C8—42no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240#97—21no—
241#97—21no—
242#97—22no—
243#97—22no—
244#97—21no—
245#97—21no—
246#97—22no—
247#97—22no—
248#97—21no—
249#97—21no—
250#97—42no—
251#97—42no—
252#97#4112yescomplemented
253#97#4112yescomplemented
254C4×C8—24no—
255C4×C8—44no—
256C4×C8—42no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95C22⋊C412yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274C24—21no—
275C24C2×C22⋊C412yes—
276#41—816no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4#10414yes—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—416no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—22no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4#10014yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—44no—
375C22⋊C4—81no—
376C22⋊C4—81no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C422−1+411yescharacteristic
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23C2×#10414yes—
410C23C22×C22⋊C411yescharacteristic
411Q8—81no—
412Q8—81no—
413Q8—84no—
414Q8—162no—
415D4—28no—
416D4—81no—
417D4—81no—
418D4—88no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4—84no—
465C2×C4—164no—
466C2×C4C2×#10014yes—
467C2×C4C2×#10012yes—
468C2×C4C2×#10412yes—
469C2×C4#42112yes—
470C2×C4C4×#4111yescharacteristic
471C2×C4#52011yescharacteristic
472C2×C4#54611yescharacteristic
473C2×C4#54611yescharacteristic
474C2×C4#54711yescharacteristic
475C2×C4#54711yescharacteristic
476C8—24no—
477C8—42no—
478C8—42no—
479C8—42no—
480C8—44no—
481C8—82no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22#233911yescharacteristic
497C22C22×#10011yescharacteristic
498C22#288911yescharacteristic
499C22C22×#10411yescharacteristic
500C22C2×#42111yescharacteristic
501C22#290111yescharacteristic
502C22#290111yescharacteristic
503C4—21no—
504C4—21no—
505C4—24no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—41no—
511C4—41no—
512C4—42no—
513C4—42no—
514C4—41no—
515C4—81no—
516C4—82no—
517C4—81no—
518C4—81no—
519C4—82no—
520C4#152714yes—
521C2—44no—
522C2—161no—
523C2C2×#152711yescharacteristic
524C2#1765311yescharacteristic
525C2#1768211yescharacteristic
526C2#1768811yescharacteristic
527C2#2247911yescharacteristic
528C2#2250511yescharacteristic
529C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21270C2semidirect
3. #20494C2semidirect
4. #9890C2semidirect
5. #9890C2semidirect
6. #9123C2semidirect
7. #9120C2semidirect
8. #8719C2semidirect
9. #8579C2semidirect
10. C4×#329C2semidirect
11. #8033C2semidirect
252. #97#41semidirect
253. #97#41semidirect

Maximal subgroups

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

subgroupindex
#361642
#212702
#204942
#98902
#91232
#91202
#87192
#85792
C4×#3292
#80332