Finite Group Databasebeta

← #120052

Subgroups — #120052

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#10840C211yesmaximal, characteristic
4#8043C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#389—21no—
12#2285C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#376C2211yescharacteristic
16#1948—22no—
17#1853—22no—
18#1844C412yescomplemented
19#1834C412yescomplemented
20C2×#339—21no—
21C2×#339—21no—
22#1441—22no—
23#1441—22no—
24C4⋊#95—21no—
25#1296—22no—
26#1294—22no—
27#1286—22no—
28#1274—22no—
29C22×M16—21no—
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4D412yes—
36C22×C42D411yescharacteristic
37C22×C42D411yescharacteristic
38C22×C42C2311yescharacteristic
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431D412yes—
45#431Q812yes—
46#431Q812yes—
47C2×#106—24no—
48#423—21no—
49#423—21no—
50#423—21no—
51#423Q812yes—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8C2×C412yes—
60#389—42no—
61#389—42no—
62#384—21no—
63#384—21no—
64#384C2×C411yescharacteristic
65#384C2×C411yescharacteristic
66#384C2×C412yes—
67#384D412yes—
68#382—21no—
69#382—21no—
70#382—21no—
71#382—21no—
72#382—21no—
73#380—21no—
74#380—21no—
75C4×C4⋊C4—21no—
76C4×C4⋊C4—21no—
77#376—21no—
78#376—21no—
79#376—21no—
80#376—21no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88#340—24no—
89#339—42no—
90#339—42no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—22no—
99C2×M16—41no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110#117—41no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—82no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4C22⋊C414yes—
138C2×C22⋊C4—41no—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—41no—
150C2×C42SD1612yes—
151C2×C42SD1612yes—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q811yescharacteristic
156C2×C42C2×Q811yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164#106—84no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—41no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8C4⋊C414yescomplemented
181#95—22no—
182#95—22no—
183#95—44no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—82no—
193#95—82no—
194C24#12011yescharacteristic
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4C22⋊Q811yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4C4⋊Q811yescharacteristic
245C22×C4C4⋊Q811yescharacteristic
246C22×C4C4⋊Q811yescharacteristic
247C22×C4C2×SD1611yescharacteristic
248C22×C4C2×SD1611yescharacteristic
249C22×C4#13511yescharacteristic
250C22×C4#13511yescharacteristic
251C22×C4AGL(1,ℤ/8)11yescharacteristic
252C22×C4AGL(1,ℤ/8)11yescharacteristic
253C22×C4#13711yescharacteristic
254C22×C4#13711yescharacteristic
255M16—42no—
256M16—42no—
257M16—81no—
258M16—81no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—81no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—81no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—88no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—41no—
299C42—81no—
300C42—81no—
301C42C2×C4⋊C411yescharacteristic
302C42#11711yescharacteristic
303C42C4×Q811yescharacteristic
304C42C4×Q811yescharacteristic
305C42#12111yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42#12511yescharacteristic
312C42#12511yescharacteristic
313C42#12511yescharacteristic
314C42#12511yescharacteristic
315C42#12611yescharacteristic
316C42#12611yescharacteristic
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23#38611yescharacteristic
322C23#39411yescharacteristic
323C23#39511yescharacteristic
324C23#44811yescharacteristic
325C23#44911yescharacteristic
326C23#45011yescharacteristic
327C23#45211yescharacteristic
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4#38011yescharacteristic
379C2×C4#38711yescharacteristic
380C2×C4#38711yescharacteristic
381C2×C4#38911yescharacteristic
382C2×C4#38911yescharacteristic
383C2×C4#39111yescharacteristic
384C2×C4#39611yescharacteristic
385C2×C4#39611yescharacteristic
386C2×C4#39811yescharacteristic
387C2×C4#39811yescharacteristic
388C2×C4#39811yescharacteristic
389C2×C4#40011yescharacteristic
390C2×C4C4×SD1611yescharacteristic
391C2×C4C4×SD1611yescharacteristic
392C2×C4#44011yescharacteristic
393C2×C4#44011yescharacteristic
394C2×C4#46011yescharacteristic
395C2×C4#46111yescharacteristic
396C2×C4#46311yescharacteristic
397C2×C4#46411yescharacteristic
398C2×C4#46511yescharacteristic
399C2×C4#46511yescharacteristic
400C2×C4#46911yescharacteristic
401C2×C4#47011yescharacteristic
402C2×C4#47511yescharacteristic
403C2×C4#47611yescharacteristic
404C2×C4#47811yescharacteristic
405C2×C4#47911yescharacteristic
406C2×C4#48111yescharacteristic
407C2×C4#48211yescharacteristic
408C2×C4#48311yescharacteristic
409C2×C4#48511yescharacteristic
410C2×C4#49811yescharacteristic
411C2×C4#49911yescharacteristic
412C2×C4#50111yescharacteristic
413C2×C4#50111yescharacteristic
414C8—44no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22#144111yescharacteristic
430C22#187711yescharacteristic
431C22#188011yescharacteristic
432C22#190411yescharacteristic
433C22#190511yescharacteristic
434C22#191111yescharacteristic
435C22#191211yescharacteristic
436C22#203411yescharacteristic
437C22#203511yescharacteristic
438C22#205811yescharacteristic
439C22#205911yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—82no—
449C4—82no—
450C4—84no—
451C4—162no—
452C4—161no—
453C4—161no—
454C4#191911yescharacteristic
455C4#192011yescharacteristic
456C4#194111yescharacteristic
457C4#194411yescharacteristic
458C4#206511yescharacteristic
459C4#206611yescharacteristic
460C4#208411yescharacteristic
461C4#208511yescharacteristic
462C2—21no—
463C2—21no—
464C2—21no—
465C2—21no—
466C2#929511yescharacteristic
467C2#929611yescharacteristic
468C2#930111yescharacteristic
469C2#930211yescharacteristic
470C2#1121311yescharacteristic
471C2#1122011yescharacteristic
472C2#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. #1844C4semidirect
19. #1834C4semidirect
180. C4×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#142762
#108402
#80432
#80262
C2×#12862
C2×#12742