Finite Group Databasebeta

← #120057

Subgroups — #120057

Order 512.

598 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#14296C211yesmaximal, characteristic
3#10851C211yesmaximal, characteristic
4#8033C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2285C2211yescharacteristic
12C2×#383C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1853—21no—
17#1852—21no—
18#1847C412yescomplemented
19#1836C412yescomplemented
20C2×#339—21no—
21C2×#339—21no—
22#1444—21no—
23#1442—21no—
24#1441—21no—
25#1441—21no—
26#1436—21no—
27#1436—21no—
28#1295—22no—
29#1294—21no—
30#1294—21no—
31#1288—22no—
32#1285—22no—
33C22×M16—21no—
34C2×#117—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42C2311yescharacteristic
43#432—21no—
44#432—21no—
45#432—21no—
46#432—21no—
47#432—21no—
48#432D412yes—
49#432Q812yes—
50#432Q812yes—
51C2×#107—22no—
52C2×#106—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16Q812yes—
61C2×#97—21no—
62C2×#97—21no—
63C2×#97C2×C412yes—
64C2×C4×C8—21no—
65#384C2×C411yescharacteristic
66#384C2×C411yescharacteristic
67#383—21no—
68#383—21no—
69#383—21no—
70#383—21no—
71#382—21no—
72#382—21no—
73#382—21no—
74#382C2×C412yes—
75#382D412yes—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97#340—22no—
98#340—22no—
99#339—42no—
100#339—42no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4—81no—
141C2×C4⋊C4—81no—
142C2×C4⋊C4—81no—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C22⋊C4—41no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167#107—82no—
168#106—82no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177#97—22no—
178#97—22no—
179#97—22no—
180#97—41no—
181#97—41no—
182#97C4⋊C414yescomplemented
183C4×C8—22no—
184C4×C8—41no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—42no—
194#95—42no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—81no—
204#95—81no—
205#95—81no—
206#95—81no—
207C24#12011yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4C22⋊Q811yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4C4⋊Q811yescharacteristic
276C22×C4C4⋊Q811yescharacteristic
277C22×C4C4⋊Q811yescharacteristic
278M16—24no—
279M16—42no—
280M16—42no—
281M16—42no—
282C2×C8—24no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—81no—
290C2×C8—81no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C42—42no—
310C42—42no—
311C42—81no—
312C42—81no—
313C42—81no—
314C42—81no—
315C42C2×C4⋊C411yescharacteristic
316C42#11711yescharacteristic
317C42C4×Q811yescharacteristic
318C42C4×Q811yescharacteristic
319C42#12111yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42C22⋊Q811yescharacteristic
322C42C22⋊Q811yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42#12511yescharacteristic
326C42#12511yescharacteristic
327C42#12511yescharacteristic
328C42#12511yescharacteristic
329C42#12611yescharacteristic
330C42#12611yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#38611yescharacteristic
336C23#39411yescharacteristic
337C23#39511yescharacteristic
338C23#45311yescharacteristic
339C23#45411yescharacteristic
340C23#45511yescharacteristic
341C23#45611yescharacteristic
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#38011yescharacteristic
419C2×C4#38711yescharacteristic
420C2×C4#38711yescharacteristic
421C2×C4#38911yescharacteristic
422C2×C4#38911yescharacteristic
423C2×C4#39111yescharacteristic
424C2×C4#39611yescharacteristic
425C2×C4#39611yescharacteristic
426C2×C4#39811yescharacteristic
427C2×C4#39811yescharacteristic
428C2×C4#39811yescharacteristic
429C2×C4#40011yescharacteristic
430C8—44no—
431C8—44no—
432C8—82no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22#144111yescharacteristic
446C22#188211yescharacteristic
447C22#188311yescharacteristic
448C22#190611yescharacteristic
449C22#190611yescharacteristic
450C22#191111yescharacteristic
451C22#191211yescharacteristic
452C22#203711yescharacteristic
453C22#203811yescharacteristic
454C22#206011yescharacteristic
455C22#206111yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—82no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C2—21no—
477C2—21no—
478C2—21no—
479C2—21no—
480C2#937911yescharacteristic
481C2#938011yescharacteristic
482C2#938111yescharacteristic
483C2#938211yescharacteristic
484C2#1121511yescharacteristic
485C2#1121711yescharacteristic
486C2#1122411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1847C4semidirect
19. #1836C4semidirect
182. #97C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#142962
#108512
#80332
#80182
C2×#12882
C2×#12852