Finite Group Databasebeta

← #107019

Subgroups — #107019

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34192C211yesmaximal, characteristic
2C4×#419C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#8052C211yesmaximal, characteristic
5#7974C211yesmaximal, characteristic
6#7973C211yesmaximal, characteristic
7#7941C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#419—21no—
12C2×C4×M16C2211yescharacteristic
13#2356—22no—
14#2356C412yescomplemented
15#2351C412yescomplemented
16#2345—22no—
17#2345—21no—
18Q8×C22⋊C4—21no—
19#2320—22no—
20#2320—21no—
21#2283—21no—
22#2283—21no—
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25#2276—21no—
26C4×#117C2211yescharacteristic
27#1763—22no—
28C4×#103C414yescomplemented
29#1736—21no—
30#1491C414yescomplemented
31#1487—22no—
32#1479—21no—
33#1479—21no—
34#1297—22no—
35#1288—22no—
36#1288—22no—
37#1284—21no—
38#1284—21no—
39C23×Q8D411yescharacteristic
40C22×M16—21no—
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8Q812yes—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#423—21no—
70#423—21no—
71#419—28no—
72C2×#103—24no—
73#407—21no—
74#407—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16C2×C412yes—
81C2×#97—21no—
82C2×C4×C8C2×C412yes—
83#391—24no—
84#391—22no—
85#389—22no—
86#389—24no—
87#387—24no—
88#387—24no—
89#387—24no—
90#383—22no—
91#383—22no—
92#383D412yes—
93#383D412yes—
94#382—22no—
95#382—21no—
96#382—22no—
97#382C2×C412yes—
98#381—21no—
99#381D411yescharacteristic
100#381D411yescharacteristic
101#380—22no—
102#380—22no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4D412yes—
119#376—22no—
120#376—21no—
121#376—21no—
122#376—22no—
123#376—21no—
124C43—21no—
125#340—24no—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C22⋊C412yes—
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C4⋊C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—24no—
143C2×M16—22no—
144C2×M16—42no—
145C2×M16—41no—
146C2×M16—41no—
147C2×M16—41no—
148C22×C8—22no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—21no—
152C4×Q8—21no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×Q8C4212yes—
168C4×Q8C4212yes—
169C4×Q8C22⋊C412yes—
170C4×Q8C22⋊C412yes—
171C4×Q8C4⋊C412yes—
172C4×Q8C4⋊C412yes—
173C4×Q8C4⋊C412yes—
174C4×Q8C4⋊C412yes—
175#117—24no—
176#117—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C4⋊C412yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×Q811yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C4⋊C8—22no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232#103—216no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237#97—22no—
238#97—22no—
239#97—41no—
240#97—41no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—41no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—41no—
258#95C22⋊C412yes—
259#95C22⋊C414yes—
260#95C22⋊C412yes—
261C24#12011yescharacteristic
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—44no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8#9512yes—
277C2×Q8#9512yes—
278C2×Q8#10414yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4—82no—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4C22⋊Q811yescharacteristic
312C22×C4C22⋊Q811yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315M16—44no—
316M16—44no—
317M16—82no—
318M16—82no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C4⋊C4#9512yes—
344C4⋊C4#9512yes—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—48no—
358C42—44no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42—82no—
363C42—84no—
364C42—82no—
365C42C2×C4211yescharacteristic
366C42C2×C22⋊C411yescharacteristic
367C42C2×C4⋊C411yescharacteristic
368C42C2×C4⋊C411yescharacteristic
369C42#11711yescharacteristic
370C42#11711yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42C22⋊Q811yescharacteristic
378C42C22⋊Q811yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C23—22no—
382C23—22no—
383C23—41no—
384C23#37911yescharacteristic
385C23#38511yescharacteristic
386C23#38611yescharacteristic
387Q8—24no—
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8—44no—
392Q8—44no—
393Q8—42no—
394Q8#33914yes—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4C2×#9511yescharacteristic
439C2×C4C4×C22⋊C411yescharacteristic
440C2×C4C4×C22⋊C411yescharacteristic
441C2×C4#38011yescharacteristic
442C2×C4#38011yescharacteristic
443C2×C4#38111yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4#38511yescharacteristic
446C2×C4#38511yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4C2×#10412yes—
451C2×C4#42112yes—
452C8—44no—
453C8—82no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—44no—
463C22C22⋊#9511yescharacteristic
464C22#178912yes—
465C22#190512yes—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—24no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4—84no—
484C4C2×#33911yescharacteristic
485C4#173311yescharacteristic
486C4#182612yes—
487C2—22no—
488C2—22no—
489C2—41no—
490C2#1073311yescharacteristic
491C2#1078611yescharacteristic
492C2#1079211yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2356C4semidirect
15. #2351C4semidirect
28. C4×#103C4semidirect
30. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#341922
C4×#4192
#174492
#80522
#79742
#79732
#79412