Finite Group Databasebeta

← #193289

Subgroups — #193289

Order 512.

990 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#34371C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#21987C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7898C211yesmaximal, characteristic
11#7898C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2381—22no—
17#2365C2211yescharacteristic
18#2365C2211yescharacteristic
19#2365C2211yescharacteristic
20#2365C2211yescharacteristic
21#2345—21no—
22#2345—21no—
23Q8×C22⋊C4—21no—
24Q8×C22⋊C4—21no—
25#2331—21no—
26#2331—21no—
27#2327—21no—
28#2327—21no—
29#2313—22no—
30C2×#384C2211yescharacteristic
31#2283C2211yescharacteristic
32#2283C2211yescharacteristic
33#1990—22no—
34#1990C2212yes—
35#1990C2212yes—
36#1948C2212yes—
37#1948C2212yes—
38#1947C2212yes—
39#1947C2212yes—
40#1853—21no—
41#1853—21no—
42#1852—21no—
43#1852—21no—
44#1847—21no—
45#1847—21no—
46#1836—21no—
47#1836—21no—
48C2×#340—22no—
49#1556C418yescomplemented
50C2×#332C2212yes—
51C2×#332C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C23×Q8—21no—
57C23×C8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8Q814yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#432C2×C414yes—
80#432C2×C414yes—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95#391—22no—
96#391—22no—
97#391—22no—
98#391—22no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2312yes—
123#383—21no—
124#383—21no—
125#382D411yescharacteristic
126#382D411yescharacteristic
127#382D411yescharacteristic
128#382D411yescharacteristic
129#381—21no—
130#381—21no—
131#380—22no—
132#380—22no—
133#380D411yescharacteristic
134#380D411yescharacteristic
135#380D411yescharacteristic
136#380D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C22⋊C4—22no—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—21no—
149#376—21no—
150#376—21no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—24no—
156#340—24no—
157#332C2×C418yescomplemented
158#332C2×C418yescomplemented
159C22×Q8—24no—
160C22×Q8—22no—
161C22×Q8—24no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—82no—
165C22×Q8—82no—
166C23×C4—22no—
167C23×C4—41no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C4⋊C418yes—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C2×Q812yes—
190C4⋊Q8C2×Q812yes—
191C4×Q8—82no—
192C4×Q8—82no—
193C4×Q8—82no—
194C4×Q8—82no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C4⋊C418yes—
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42C22×C412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42C2411yescharacteristic
246#107—82no—
247#107—82no—
248#106—82no—
249#106—82no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8C22×C414yes—
253C4⋊C8C22×C414yes—
254C22⋊C8—24no—
255C22⋊C8—24no—
256#97—24no—
257#97—24no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—42no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C24#12011yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—28no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—82no—
296C2×Q8—82no—
297C2×Q8—88no—
298C2×Q8—88no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4C2×C4⋊C412yes—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4#12312yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—48no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4C2×C4⋊C414yes—
373C4⋊C4C2×C4⋊C414yes—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—82no—
383C42—82no—
384C42—82no—
385C42—82no—
386C42C2×C4⋊C412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42C22⋊Q812yes—
390C42C22⋊Q812yes—
391C42#12312yes—
392C42C23×C411yescharacteristic
393C42C22×Q811yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23#38512yes—
401C23#45412yes—
402C23#45612yes—
403C23C2×#12011yescharacteristic
404Q8—44no—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408Q8—88no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—88no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—164no—
458C2×C4#38512yes—
459C2×C4#38512yes—
460C2×C4#38512yes—
461C2×C4C22×C4⋊C411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×#12311yescharacteristic
467C2×C4#57612yes—
468C2×C4#57712yes—
469C2×C4#57712yes—
470C2×C4#57812yes—
471C8—44no—
472C8—44no—
473C8—44no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22#185714yes—
481C22C2×#38511yescharacteristic
482C22#300811yescharacteristic
483C22#300911yescharacteristic
484C22#301011yescharacteristic
485C22#301111yescharacteristic
486C22C2×#45411yescharacteristic
487C22C2×#45611yescharacteristic
488C4—21no—
489C4—21no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—44no—
494C4—44no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—162no—
504C4#294612yes—
505C4#294712yes—
506C4#308112yes—
507C4#308212yes—
508C2—24no—
509C2#1869411yescharacteristic
510C2#1869511yescharacteristic
511C2#1869911yescharacteristic
512C2#1870011yescharacteristic
513C2C2×#185711yescharacteristic
514C2#2087611yescharacteristic
515C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8926C2semidirect
7. #8925C2semidirect
8. #8923C2semidirect
9. #8922C2semidirect
49. #1556C4semidirect
157. #332C2×C4semidirect
158. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343712
#219872
C2×#19902
C2×#15562
#89262
#89252
#89232
#89222
#78982