Finite Group Databasebeta

← #116325

Subgroups — #116325

Order 512.

766 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1994C211yesmaximal, characteristic
2C2×#1518C211yesmaximal, characteristic
3C2×#1518C211yesmaximal, characteristic
4#14707C211yesmaximal, characteristic
5C2×#1286C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1276C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#389—21no—
13C2×#382C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×C43C2211yescharacteristic
16#1994—22no—
17#1994—22no—
18C2×#339—21no—
19C2×#339—21no—
20#1518—22no—
21#1518—22no—
22#1518—22no—
23#1518C414yescomplemented
24C2×#328—22no—
25C2×#328—22no—
26#1436—22no—
27#1436—22no—
28C4⋊#95—21no—
29#1286—22no—
30#1286—22no—
31#1276—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533D412yes—
41C2×C4⋊Q8Q811yescharacteristic
42C2×C4⋊Q8Q811yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431C2×C412yes—
64#431D412yes—
65#423—21no—
66#423—21no—
67#423—21no—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#389—42no—
78#389—42no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382C2×C412yes—
86#382D412yes—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95D412yes—
93C2×#95D412yes—
94C43—21no—
95C43—21no—
96C43—21no—
97C43—21no—
98#339—24no—
99#339—24no—
100#328—28no—
101#328—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—22no—
112C2×M16—41no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8C4⋊C412yescomplemented
127C4⋊Q8C4⋊C412yescomplemented
128#125—22no—
129#125—22no—
130#125—41no—
131#125—41no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148#117—22no—
149#117—22no—
150#117—41no—
151#117—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42SD1612yes—
194C2×C42Q1612yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q811yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8—41no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—41no—
232#95—41no—
233#95C22⋊C414yes—
234C24#12011yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—81no—
240C2×Q8—81no—
241C2×Q8—81no—
242C2×Q8—81no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
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×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22⋊Q811yescharacteristic
288C22×C4C22⋊Q811yescharacteristic
289C22×C4C22⋊Q811yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#12411yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12411yescharacteristic
294C22×C4#12711yescharacteristic
295C22×C4C2×SD1611yescharacteristic
296C22×C4C2×Q1611yescharacteristic
297C22×C4#13511yescharacteristic
298C22×C4#13511yescharacteristic
299C22×C4AGL(1,ℤ/8)11yescharacteristic
300C22×C4#13711yescharacteristic
301C22×C4#13711yescharacteristic
302C22×C4#13711yescharacteristic
303M16—42no—
304M16—42no—
305M16—81no—
306M16—81no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—81no—
311C2×C8—81no—
312C2×C8—81no—
313C2×C8—81no—
314C2×C8—81no—
315C2×C8—81no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—88no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42C2×C4⋊C411yescharacteristic
354C42#11711yescharacteristic
355C42C4×D411yescharacteristic
356C42C4×D411yescharacteristic
357C42#12111yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42C22⋊Q811yescharacteristic
361C42C22⋊Q811yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42#12311yescharacteristic
364C42#12311yescharacteristic
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C42#12611yescharacteristic
368C42#12611yescharacteristic
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23#38611yescharacteristic
374C23#39211yescharacteristic
375C23#39311yescharacteristic
376C23#45011yescharacteristic
377C23#45111yescharacteristic
378C23#45211yescharacteristic
379C23#45211yescharacteristic
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
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—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#38011yescharacteristic
435C2×C4#38511yescharacteristic
436C2×C4#38511yescharacteristic
437C2×C4#38811yescharacteristic
438C2×C4#38811yescharacteristic
439C2×C4#39011yescharacteristic
440C2×C4#39411yescharacteristic
441C2×C4#39411yescharacteristic
442C2×C4#39611yescharacteristic
443C2×C4#39611yescharacteristic
444C2×C4#39711yescharacteristic
445C2×C4#39911yescharacteristic
446C2×C4C4×SD1611yescharacteristic
447C2×C4C4×Q1611yescharacteristic
448C2×C4#44011yescharacteristic
449C2×C4#44111yescharacteristic
450C2×C4#44811yescharacteristic
451C2×C4#44911yescharacteristic
452C2×C4#45111yescharacteristic
453C2×C4#45211yescharacteristic
454C2×C4#46111yescharacteristic
455C2×C4#46211yescharacteristic
456C2×C4#46311yescharacteristic
457C2×C4#46411yescharacteristic
458C2×C4#46511yescharacteristic
459C2×C4#46711yescharacteristic
460C2×C4#46811yescharacteristic
461C2×C4#47011yescharacteristic
462C2×C4#47511yescharacteristic
463C2×C4#47711yescharacteristic
464C2×C4#47911yescharacteristic
465C2×C4#47911yescharacteristic
466C2×C4#48711yescharacteristic
467C2×C4#48811yescharacteristic
468C2×C4#48911yescharacteristic
469C2×C4#49111yescharacteristic
470C8—82no—
471C8—82no—
472C8—82no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22#143911yescharacteristic
486C22#187711yescharacteristic
487C22#187811yescharacteristic
488C22#200011yescharacteristic
489C22#200111yescharacteristic
490C22#202411yescharacteristic
491C22#202611yescharacteristic
492C22#203111yescharacteristic
493C22#203111yescharacteristic
494C22#203111yescharacteristic
495C22#203111yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—44no—
505C4—44no—
506C4—84no—
507C4—82no—
508C4—82no—
509C4—162no—
510C4#186311yescharacteristic
511C4#186511yescharacteristic
512C4#193311yescharacteristic
513C4#193511yescharacteristic
514C4#204211yescharacteristic
515C4#204311yescharacteristic
516C4#206611yescharacteristic
517C4#206811yescharacteristic
518C2—21no—
519C2—21no—
520C2—21no—
521C2—21no—
522C2#1036311yescharacteristic
523C2#1036511yescharacteristic
524C2#1036711yescharacteristic
525C2#1036911yescharacteristic
526C2#1106711yescharacteristic
527C2#1108511yescharacteristic
528C2#1109111yescharacteristic
subgroupG/H
center, socleC23#393
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
23. #1518C4semidirect
126. C4⋊Q8C4⋊C4semidirect
127. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19942
C2×#15182
#147072
C2×#12862
C2×#12762