Finite Group Databasebeta

← #117105

Subgroups — #117105

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21401C211yesmaximal, characteristic
2C2×#1962C211yesmaximal, characteristic
3C2×#1566C211yesmaximal, characteristic, complemented
4C2×#1492C211yesmaximal, characteristic
5#10839C211yesmaximal, characteristic, complemented
6C4×#336C211yesmaximal, characteristic, complemented
7#7897C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×#104—22no—
12C22×C4×C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14#2285C2211yescharacteristic
15C2×C43C2211yescharacteristic
16#1990C411yescharacteristic, complemented
17#1990C411yescharacteristic, complemented
18#1962—22no—
19#1962—22no—
20#1957C411yescharacteristic, complemented
21#1957C411yescharacteristic, complemented
22#1893—22no—
23C4×#102—22no—
24C2×#339—22no—
25C2×#336—22no—
26#1566—22no—
27#1566—22no—
28C2×#334—22no—
29#1492—22no—
30#1492—22no—
31C2×D4⋊C8—22no—
32#1279—22no—
33C22×#41—21no—
34C22×M16—21no—
35C22×M16—21no—
36C23×C8—21no—
37C23×C8D412yes—
38#532—21no—
39#532—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532D412yes—
45#532Q812yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#124—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×#117—21no—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42D412yes—
57C22×C42C2311yescharacteristic
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423D412yes—
65#423D412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×#104—28no—
71C2×#104—44no—
72C2×#104—44no—
73C2×#102—22no—
74C2×#102—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2×C411yescharacteristic
86C2×C4×C8D412yes—
87C2×C4×C8D412yes—
88#384C2×C411yescharacteristic
89#384C2×C411yescharacteristic
90#382—21no—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C43—21no—
96C43—21no—
97C43—21no—
98C43—21no—
99#340—22no—
100#340—22no—
101#339—44no—
102#336—28no—
103#334—28no—
104D4⋊C8—28no—
105C2×#41—22no—
106C2×#41—22no—
107C2×#41—41no—
108C2×#41—41no—
109C2×#41—42no—
110C2×#41—42no—
111C2×#41—42no—
112C2×#41—42no—
113C22×Q8—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—41no—
118C23×C4—22no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4#4111yescharacteristic
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C22×C8—24no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—24no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8C22⋊C412yes—
143C22×C8C22⋊C412yes—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C4⋊C412yes—
148#127—21no—
149#127—21no—
150#127—21no—
151#127C4⋊C412yes—
152#124—22no—
153#124—22no—
154#124—41no—
155#124—41no—
156#121—22no—
157#121—22no—
158#121—22no—
159#121—22no—
160#121—41no—
161#121—41no—
162#121—41no—
163#121—41no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171C4×D4—41no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42SD1614yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q811yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220#104—416no—
221#102—44no—
222#102—44no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—22no—
228C4×C8—22no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—41no—
232C4×C8—41no—
233#95—41no—
234#95—41no—
235C24—41no—
236C24#12111yescharacteristic
237#41—44no—
238#41—44no—
239#41—44no—
240#41—44no—
241#41—82no—
242#41—82no—
243#41—82no—
244#41—82no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—82no—
264C2×D4—82no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4C22⋊Q811yescharacteristic
296C22×C4C22⋊Q811yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12411yescharacteristic
300C22×C4#12411yescharacteristic
301C22×C4#12611yescharacteristic
302C22×C4#12711yescharacteristic
303C22×C4C2×SD1612yes—
304C22×C4#13512yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4AGL(1,ℤ/8)12yes—
307M16—44no—
308M16—44no—
309M16—82no—
310M16—82no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—81no—
326C2×C8—81no—
327C2×C8—81no—
328C2×C8—81no—
329C2×C8#10414yescomplemented
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—82no—
345C42—24no—
346C42—24no—
347C42—44no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—42no—
352C42—42no—
353C42—81no—
354C42—81no—
355C42C2×C4⋊C411yescharacteristic
356C42#11711yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42#12011yescharacteristic
360C42#12011yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42C22⋊Q811yescharacteristic
364C42C22⋊Q811yescharacteristic
365C42C22⋊Q811yescharacteristic
366C42#12311yescharacteristic
367C42#12311yescharacteristic
368C42#12311yescharacteristic
369C42#12411yescharacteristic
370C42#12611yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—42no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23#39011yescharacteristic
383C23#39411yescharacteristic
384C23#39611yescharacteristic
385C23#46512yes—
386C23#46912yes—
387Q8—44no—
388Q8—82no—
389D4—44no—
390D4—44no—
391D4—44no—
392D4—84no—
393D4—84no—
394D4—82no—
395D4—82no—
396D4—82no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#38011yescharacteristic
430C2×C4#38511yescharacteristic
431C2×C4#38511yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38811yescharacteristic
434C2×C4#38811yescharacteristic
435C2×C4#39211yescharacteristic
436C2×C4#39311yescharacteristic
437C2×C4#39411yescharacteristic
438C2×C4#39611yescharacteristic
439C2×C4#39711yescharacteristic
440C2×C4#39911yescharacteristic
441C2×C4C2×#10412yes—
442C2×C4#42112yes—
443C2×C4C4×SD1612yes—
444C2×C4#44212yes—
445C2×C4#44312yes—
446C2×C4#44412yes—
447C2×C4#44912yes—
448C2×C4#45012yes—
449C2×C4#46012yes—
450C2×C4#46312yes—
451C2×C4#47612yes—
452C2×C4#47812yes—
453C2×C4#48112yes—
454C2×C4#48212yes—
455C2×C4#49211yescharacteristic
456C2×C4#49511yescharacteristic
457C2×C4#49611yescharacteristic
458C2×C4#49611yescharacteristic
459C8—24no—
460C8—44no—
461C8—42no—
462C8—82no—
463C8—82no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—161no—
476C22—161no—
477C22—161no—
478C22—161no—
479C22#143911yescharacteristic
480C22#196211yescharacteristic
481C22#196311yescharacteristic
482C22#196811yescharacteristic
483C22#197211yescharacteristic
484C22#197411yescharacteristic
485C22#197511yescharacteristic
486C22#203412yes—
487C22#206512yes—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—48no—
493C4—82no—
494C4—161no—
495C4—161no—
496C4#180612yes—
497C4#186412yes—
498C4#186612yes—
499C4#187112yes—
500C4#192612yes—
501C4#192912yes—
502C4#201211yescharacteristic
503C4#201411yescharacteristic
504C4#209211yescharacteristic
505C4#209411yescharacteristic
506C2—22no—
507C2—22no—
508C2—162no—
509C2#984311yescharacteristic
510C2#984611yescharacteristic
511C2#984911yescharacteristic
512C2#985211yescharacteristic
513C2#1105311yescharacteristic
514C2#1112111yescharacteristic
515C2#1112511yescharacteristic
subgroupG/H
centerC22×C4#127
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#390
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1566C2semidirect
5. #10839C2semidirect
6. C4×#336C2semidirect
7. #7897C2semidirect
16. #1990C4semidirect
17. #1990C4semidirect
20. #1957C4semidirect
21. #1957C4semidirect
329. C2×C8#104semidirect

Maximal subgroups

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

subgroupindex
#214012
C2×#19622
C2×#15662
C2×#14922
#108392
C4×#3362
#78972