Finite Group Databasebeta

← #223114

Subgroups — #223114

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36174C211yesmaximal, characteristic
2#21270C211yesmaximal, characteristic, complemented
3#20494C211yesmaximal, characteristic, complemented
4#9891C211yesmaximal, characteristic, complemented
5#9891C211yesmaximal, characteristic, complemented
6#9124C212yesmaximal, complemented
7#9120C212yesmaximal, complemented
8#8720C212yesmaximal, complemented
9#8579C211yesmaximal, characteristic, complemented
10C4×#329C211yesmaximal, characteristic, complemented
11#8035C212yesmaximal, complemented
12C2×#436—21no—
13C2×#432C2212yes—
14C2×C4×M16C2212yes—
15#2859C2211yescharacteristic
16#2850C2212yes—
17#2538C411yescharacteristic
18#2537C411yescharacteristic
19#2526—22no—
20#2518—22no—
21#2495—21no—
22#2490C412yes—
23#2489C412yes—
24#2420—21no—
25#2363—21no—
26#2355C411yescharacteristic
27#2354C411yescharacteristic
28#2346—21no—
29C4×#124C2211yescharacteristic
30#2283C2212yes—
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×M16—22no—
54C2×#125C2×C412yes—
55C2×#124C2311yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×#117—21no—
65C22×C42C2312yes—
66#436—28no—
67#432—24no—
68#432—42no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C4×M16—24no—
76C4×M16—42no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97C2312yes—
80C2×#97C2312yes—
81C2×#97C2311yescharacteristic
82C2×#97C2311yescharacteristic
83C2×#97C2311yescharacteristic
84C2×#97C2311yescharacteristic
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88#400—22no—
89#397—22no—
90#397—22no—
91#396—22no—
92#394—22no—
93#394—22no—
94#391C2×C412yes—
95#391C2×C411yescharacteristic
96#390—21no—
97#390—22no—
98#390C2×C412yes—
99#390C2×C411yescharacteristic
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—21no—
109#388—21no—
110#388—22no—
111#388—22no—
112#387—22no—
113#386—22no—
114#386—22no—
115#384C2×C411yescharacteristic
116#384C2×C411yescharacteristic
117#383—22no—
118#382—21no—
119#382—22no—
120#382C2312yes—
121#382C2311yescharacteristic
122#381—22no—
123#380—21no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140C43C2311yescharacteristic
141#336—21no—
142#336—22no—
143#336—22no—
144#336—21no—
145#329—22no—
146#329—22no—
147#329—42no—
148C22×Q8C22×C411yescharacteristic
149C22×D4—22no—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C2×M16—28no—
157C2×M16—44no—
158C2×M16—44no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165#125—82no—
166#124—21no—
167#124—21no—
168#124—21no—
169#124—21no—
170C22⋊Q8—82no—
171#121—44no—
172#121—82no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178#117—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—44no—
227C4⋊C8—42no—
228C22⋊C8—28no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231#97—21no—
232#97—21no—
233#97—22no—
234#97—22no—
235#97—21no—
236#97—21no—
237#97—22no—
238#97—22no—
239#97—21no—
240#97—21no—
241#97—42no—
242#97—42no—
243#97#4112yes—
244#97#4112yes—
245C4×C8—24no—
246C4×C8—44no—
247C4×C8—42no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—22no—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263C24—21no—
264C24C2×C22⋊C412yes—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—44no—
277C2×D4—82no—
278C2×D4—82no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—416no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4#10414yes—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—44no—
353C22⋊C4—81no—
354C22⋊C4—81no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—42no—
372C42—44no—
373C42—82no—
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C42H(5,𝔽2)11yescharacteristic
377C422−1+411yescharacteristic
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23C22×C22⋊C411yescharacteristic
393Q8—81no—
394Q8—81no—
395D4—44no—
396D4—81no—
397D4—81no—
398D4—164no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4#41312yes—
445C2×C4C2×#10414yes—
446C2×C4C2×#10412yes—
447C2×C4#42112yes—
448C2×C4C4×#4111yescharacteristic
449C2×C4#52011yescharacteristic
450C2×C4#54611yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4#54711yescharacteristic
454C8—24no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—44no—
459C8—82no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#233911yescharacteristic
475C22C2×#41311yescharacteristic
476C22#288811yescharacteristic
477C22C22×#10411yescharacteristic
478C22C2×#42111yescharacteristic
479C22#290111yescharacteristic
480C22#290111yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—42no—
491C4—42no—
492C4—41no—
493C4—82no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—82no—
498C4#152914yes—
499C2—82no—
500C2—161no—
501C2C2×#152911yescharacteristic
502C2#1765511yescharacteristic
503C2#1768111yescharacteristic
504C2#1768711yescharacteristic
505C2#2248311yescharacteristic
506C2#2250511yescharacteristic
507C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
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. #9891C2semidirect
5. #9891C2semidirect
6. #9124C2semidirect
7. #9120C2semidirect
8. #8720C2semidirect
9. #8579C2semidirect
10. C4×#329C2semidirect
11. #8035C2semidirect

Maximal subgroups

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

subgroupindex
#361742
#212702
#204942
#98912
#91242
#91202
#87202
#85792
C4×#3292
#80352