Finite Group Databasebeta

← #198349

Subgroups — #198349

Order 512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#475C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#21799C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17449C211yesmaximal, characteristic
6#9252C218yesmaximal, complemented
7#8002C211yesmaximal, characteristic
8#7935C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#475—21no—
12C2×#475—21no—
13C2×#475—22no—
14C2×#475—21no—
15C2×#475—21no—
16C2×#475—22no—
17C2×#475C414yes—
18C2×#475C2214yes—
19#2984—22no—
20#2984—22no—
21#2983—22no—
22#2983—21no—
23#2982—21no—
24Q8×M16—22no—
25Q8×M16C2214yes—
26#2920—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C22×#106C412yes—
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C412yes—
34#1966—22no—
35#1966C2214yes—
36#1784—22no—
37#1556—22no—
38#1549—22no—
39#1546C2214yes—
40#1491—22no—
41#1487C2214yes—
42#1479—21no—
43#1479—21no—
44#1292C2214yes—
45#1283C2214yes—
46C23×Q8—21no—
47C23×Q8C2×C411yescharacteristic
48C23×C8C2×C412yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2312yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D412yes—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D412yes—
74C2×C4×Q8C2312yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#475—24no—
82#475—24no—
83#475—24no—
84#475—24no—
85#475—24no—
86#475—24no—
87#475—24no—
88#475—24no—
89#446—28no—
90#446—24no—
91#446—24no—
92#446—28no—
93C8×Q8—24no—
94C8×Q8—24no—
95#432—24no—
96#432—24no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432D412yes—
105#431—24no—
106#431—24no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431D412yes—
114#431C2312yes—
115C2×#106—24no—
116C2×#106—22no—
117C2×#106—22no—
118C2×#106C2×C418yes—
119#423—24no—
120#423—21no—
121#423D412yes—
122#423C2312yes—
123#423C2312yes—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8C2×C412yes—
129C2×C4⋊C8C2312yes—
130C2×#103—22no—
131C2×#103—24no—
132C2×#103—22no—
133C2×#103C2×C418yes—
134C2×#101—28no—
135#407—21no—
136#407—21no—
137#407—21no—
138#407D412yes—
139C4×M16—24no—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16C2312yes—
144#332—28no—
145Q8⋊C8—24no—
146Q8⋊C8—24no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—42no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—84no—
159C22×Q8C22⋊C411yescharacteristic
160C22×Q8C22⋊C412yes—
161C22×Q8C22⋊C411yescharacteristic
162C22×Q8C22×C412yes—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—41no—
172C2×M16C2×D414yes—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8C22×C414yes—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—21no—
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8C22⋊C412yes—
210C4×Q8C22⋊C412yes—
211C4×Q8C22⋊C412yes—
212C4×Q8C22⋊C412yes—
213C4×Q8C2×D412yes—
214C4×Q8C2×D412yes—
215C4×Q8C2×D412yes—
216C4×Q8C2×D412yes—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊C412yes—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C412yes—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264#106—28no—
265#106—28no—
266C4⋊C8—28no—
267C4⋊C8—28no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—41no—
279C4⋊C8C2×D414yes—
280#103—28no—
281#103—28no—
282#101—88no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—41no—
289C22⋊C8—41no—
290C22⋊C8—41no—
291#97—28no—
292#97—22no—
293#97—22no—
294#97—41no—
295#97—41no—
296C4×C8—28no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—22no—
300C4×C8—41no—
301C24C4×D411yescharacteristic
302C2×Q8—22no—
303C2×Q8—21no—
304C2×Q8—22no—
305C2×Q8—21no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—44no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—48no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8#10114yescomplemented
327C2×Q8C2×C22⋊C412yes—
328C2×Q8C2×C22⋊C412yes—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4—84no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4C4×D411yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4#12111yescharacteristic
358C22×C4#12111yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12311yescharacteristic
361C22×C4#12311yescharacteristic
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—28no—
366M16—44no—
367M16—44no—
368M16—82no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4C2×C22⋊C412yes—
398C4⋊C4C2×C22⋊C412yes—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×C22⋊C411yescharacteristic
408C42C2×C22⋊C411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42#12011yescharacteristic
412C42#12011yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12311yescharacteristic
418C42#12311yescharacteristic
419C42C22×D411yescharacteristic
420C42C22×D411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C23—22no—
424C23—22no—
425C23—22no—
426C23#38611yescharacteristic
427C23#38611yescharacteristic
428C23C2×C4×D411yescharacteristic
429Q8—22no—
430Q8—24no—
431Q8—22no—
432Q8—22no—
433Q8—44no—
434Q8—44no—
435Q8—42no—
436Q8—84no—
437Q8—84no—
438Q8—84no—
439Q8—84no—
440Q8C2×#10114yes—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—84no—
474C2×C4—84no—
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4C2×#10112yes—
482C2×C4#41312yes—
483C2×C4C22×C22⋊C411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×#12011yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12111yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4#57714yes—
490C8—44no—
491C8—44no—
492C8—44no—
493C8—82no—
494C8—82no—
495C22—28no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22#190112yes—
503C22#190212yes—
504C22C2×#38611yescharacteristic
505C22#294912yes—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—24no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—84no—
519C4#190112yes—
520C4C22×#10111yescharacteristic
521C4C2×#41311yescharacteristic
522C4#301012yes—
523C4#304312yes—
524C2—22no—
525C2—22no—
526C2—22no—
527C2#2093611yescharacteristic
528C2C2×#190111yescharacteristic
529C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9252C2semidirect
326. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C22×#4752
#602482
#217992
#177422
#174492
#92522
#80022
#79352