Finite Group Databasebeta

← #188583

Subgroups — #188583

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34240C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic, complemented
3#20811C211yesmaximal, characteristic, complemented
4C2×#1533C211yesmaximal, characteristic
5C2×#1527C211yesmaximal, characteristic
6#8583C212yesmaximal, complemented
7#8581C212yesmaximal, complemented
8#8581C212yesmaximal, complemented
9#8579C212yesmaximal, complemented
10#8035C211yesmaximal, characteristic, complemented
11#8032C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2372—21no—
18#2372—21no—
19#2367C412yes—
20#2367C412yes—
21#2342—22no—
22Q8×C22⋊C4C412yes—
23D4×C22⋊C4C412yes—
24#2337—22no—
25#2312—21no—
26C4×#124C2212yes—
27C4×#124C2212yes—
28#2283C2211yescharacteristic
29#2273—21no—
30C4×#117C2211yescharacteristic
31#1939—22no—
32#1851—21no—
33#1851—21no—
34#1843—21no—
35#1832—21no—
36#1798—22no—
37#1767C2212yescomplemented
38#1767C2212yescomplemented
39#1767C2212yescomplemented
40#1767C2212yescomplemented
41#1533—24no—
42#1527—24no—
43C2×#329C2212yes—
44C2×#329C2212yes—
45C2×#329C2212yes—
46C2×#329C2212yes—
47#1295C2211yescharacteristic, complemented
48#1295C2211yescharacteristic, complemented
49#1295C2211yescharacteristic, complemented
50#1295C2211yescharacteristic, complemented
51#1295C2212yescomplemented
52#1295C2212yescomplemented
53C23×Q8C2×C411yescharacteristic
54C23×D4C2×C411yescharacteristic
55C22×M16—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×#124—22no—
59C2×#124—22no—
60C2×#124C2×C412yes—
61C2×#124D414yes—
62C2×#124Q814yes—
63C2×#124C2312yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2×C412yes—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×#117D412yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C4×M16—22no—
90C4×M16—22no—
91C2×#97C2312yes—
92C2×#97C2311yescharacteristic
93C2×#97C2311yescharacteristic
94C2×#97C2311yescharacteristic
95C2×#97C2311yescharacteristic
96C2×#97C2312yes—
97C2×C4×C8—21no—
98#391—22no—
99#391—22no—
100#391C2×C412yes—
101#391C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#383—21no—
117#382C2311yescharacteristic
118#382C2311yescharacteristic
119#381—21no—
120#380—22no—
121#380C2×C412yes—
122#379—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4C2×C412yes—
136C4×C22⋊C4D411yescharacteristic
137C4×C22⋊C4D411yescharacteristic
138#376—21no—
139#376D411yescharacteristic
140#376D411yescharacteristic
141C43C2312yes—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#329—24no—
147#329—24no—
148#329—24no—
149#329—24no—
150C25—21no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—81no—
154C22×Q8—81no—
155C22×Q8C4⋊C414yes—
156C22×Q8C22×C412yes—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—81no—
160C22×D4—81no—
161C22×D4C4⋊C414yes—
162C22×D4C22×C412yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—44no—
170C2×M16—42no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177#124—24no—
178#124—24no—
179#124—24no—
180#124—24no—
181#124—24no—
182#124—22no—
183#124—22no—
184#124C4⋊C418yes—
185#124C2×D414yes—
186#124C2×Q814yes—
187C4×Q8—82no—
188C4×Q8—82no—
189C4×D4—82no—
190C4×D4—82no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C22×C412yes—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×Q812yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—44no—
235C4⋊C8—44no—
236C4⋊C8—44no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243#97—22no—
244#97—21no—
245#97—21no—
246#97—21no—
247#97—22no—
248#97—21no—
249#97—22no—
250#97—21no—
251#97—21no—
252#97—21no—
253#97—21no—
254#97—22no—
255C4×C8—42no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95C22⋊C411yescharacteristic
267#95C22⋊C411yescharacteristic
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C411yescharacteristic
271#95C22⋊C411yescharacteristic
272C24—22no—
273C24—24no—
274C24—44no—
275C24—81no—
276C24—81no—
277C24—82no—
278C24C2×C22⋊C411yescharacteristic
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8C2×C4⋊C414yes—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4—84no—
295C2×D4—84no—
296C2×D4C2×C4⋊C414yes—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C4⋊C412yes—
321C22×C4#11712yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4C22⋊Q812yes—
325C22×C4C22⋊Q812yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—48no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—28no—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—42no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C42—24no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—82no—
372C42—82no—
373C42—82no—
374C42C2×C4⋊C412yes—
375C42#11712yes—
376C42#12112yes—
377C42#12112yes—
378C42C22⋊Q812yes—
379C42C22⋊Q812yes—
380C42C22×D411yescharacteristic
381C42C22×Q811yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—84no—
397C23—84no—
398C23—164no—
399C23—164no—
400C23#38012yes—
401C23C22×C22⋊C411yescharacteristic
402Q8—44no—
403Q8—44no—
404Q8—84no—
405D4—44no—
406D4—44no—
407D4—84no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—84no—
443C2×C4—164no—
444C2×C4—164no—
445C2×C4#38012yes—
446C2×C4#38012yes—
447C2×C4#38012yes—
448C2×C4#42114yes—
449C2×C4C22×C4⋊C411yescharacteristic
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×#12111yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C8—44no—
456C8—81no—
457C8—81no—
458C8—82no—
459C8—81no—
460C8—81no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—168no—
474C22C2×#38011yescharacteristic
475C22#288811yescharacteristic
476C22#288811yescharacteristic
477C22C2×#42112yes—
478C22#290112yes—
479C4—24no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—44no—
493C4—82no—
494C4—82no—
495C2—42no—
496C2—84no—
497C2#1767912yes—
498C2#1768112yes—
499C2#2063811yescharacteristic
500C2#2065411yescharacteristic
501C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21356C2semidirect
3. #20811C2semidirect
6. #8583C2semidirect
7. #8581C2semidirect
8. #8581C2semidirect
9. #8579C2semidirect
10. #8035C2semidirect
11. #8032C2semidirect
37. #1767C22semidirect
38. #1767C22semidirect
39. #1767C22semidirect
40. #1767C22semidirect
47. #1295C22semidirect
48. #1295C22semidirect
49. #1295C22semidirect
50. #1295C22semidirect
51. #1295C22semidirect
52. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#342402
#213562
#208112
C2×#15332
C2×#15272
#85832
#85812
#85792
#80352
#80322