Finite Group Databasebeta

← #497606

Subgroups — #497606

Order 512.

795 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39613C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13303C212yesmaximal, complemented
4#13303C212yesmaximal, complemented
5#13301C212yesmaximal, complemented
6#12803C211yesmaximal, characteristic
7#12803C211yesmaximal, characteristic
8#12801C211yesmaximal, characteristic
9#12632C211yesmaximal, characteristic
10#12632C211yesmaximal, characteristic
11C4×#355C211yesmaximal, characteristic
12C2×#557C2211yescharacteristic
13C2×#554C411yescharacteristic
14Q8×M16—22no—
15#2859C411yescharacteristic
16#2729—22no—
17#2704—22no—
18#2660C411yescharacteristic
19#2660C411yescharacteristic
20#2568C2211yescharacteristic
21#2566C411yescharacteristic
22#2566C411yescharacteristic
23#2565C2211yescharacteristic
24#2565C2211yescharacteristic
25#2563C411yescharacteristic
26#2473—22no—
27#2465—22no—
28#2326C2211yescharacteristic
29#2326C2211yescharacteristic
30C4×#124C2211yescharacteristic
31#2295C411yescharacteristic
32C2×#355C2211yescharacteristic
33C2×#355C2211yescharacteristic
34C2×#355C2211yescharacteristic
35C2×#355C2211yescharacteristic
36#1998—22no—
37#1998—22no—
38#1997—22no—
39#1801C2212yes—
40#1801C2212yes—
41#1801C2212yes—
42C4×#101C2212yes—
43C4×#101C2212yes—
44C4×#101C2212yes—
45#1394C2212yes—
46#1394C2212yes—
47#1394C2212yes—
48#1394C2212yes—
49#1394C2212yes—
50#1394C2212yes—
51#557—42no—
52#557—42no—
53#554—42no—
54#554—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8D412yes—
58C2×#125C2×C411yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×#125C2×C411yescharacteristic
61C2×#125C2×C411yescharacteristic
62C2×#125D412yes—
63C2×#125D412yes—
64C2×#124C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D412yes—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C22×C4⋊C4—21no—
77#446—42no—
78#446—42no—
79#446—42no—
80C8×Q8—42no—
81#423—22no—
82#423—22no—
83#423—22no—
84C2×#101C2312yes—
85C2×#101C2312yes—
86C2×#101C2312yes—
87C2×#101C2312yes—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91#400—22no—
92#398—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#396—22no—
97#396—22no—
98#395—22no—
99#395—22no—
100#393—22no—
101#391—21no—
102#391—22no—
103#391C2311yescharacteristic
104#391C2311yescharacteristic
105#391C2311yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389—22no—
110#389—22no—
111#389—21no—
112#388—22no—
113#388—22no—
114#388—22no—
115#387—21no—
116#387—21no—
117#387—22no—
118#387—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#382C2×C411yescharacteristic
132#380—22no—
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C22⋊C4—22no—
137#376C2×C411yescharacteristic
138#376C2×C411yescharacteristic
139C43C2×C411yescharacteristic
140#355—22no—
141#355—22no—
142#355—22no—
143#355Q814yescomplemented
144#343—22no—
145#343—22no—
146#343—22no—
147#343—22no—
148#343—22no—
149#343—22no—
150C22×Q8—21no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22⋊C412yes—
153C22×Q8C2411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166#125—41no—
167#125—41no—
168#125—42no—
169#125—42no—
170#125—42no—
171#125—42no—
172#125—42no—
173#125—41no—
174#125—41no—
175#125—42no—
176#124—22no—
177#124C2×Q812yes—
178#124#4112yes—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—41no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—42no—
196C4×D4—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253#101—22no—
254#101—22no—
255#101—22no—
256#101—22no—
257#101—22no—
258#101—22no—
259#101—22no—
260#101C2×Q814yes—
261#97—42no—
262#97—42no—
263#97—42no—
264C4×C8—42no—
265C4×C8—42no—
266C4×C8—42no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282C24—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—82no—
290C2×Q8—81no—
291C2×Q8—82no—
292C2×Q8C22×Q811yescharacteristic
293C2×Q8C2×#4111yescharacteristic
294C2×Q82−1+412yes—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4C4×Q812yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D412yes—
324M16—24no—
325M16—42no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—42no—
330M16—42no—
331M16—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—81no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—81no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—81no—
368C4⋊C4#9912yes—
369C4⋊C4#9912yes—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—21no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—21no—
386C42—21no—
387C42—22no—
388C42—22no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—81no—
393C42—81no—
394C42C4×Q812yes—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23C22×C22⋊C411yescharacteristic
401Q8—41no—
402Q8—41no—
403Q8—42no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407Q8—162no—
408D4—42no—
409D4—42no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
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—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—84no—
448C2×C4—161no—
449C2×C4—161no—
450C2×C4—162no—
451C2×C4C2×#9912yes—
452C2×C4C2×#9912yes—
453C2×C4C2×#9911yescharacteristic
454C2×C4C2×#9911yescharacteristic
455C2×C4C2×C4×Q811yescharacteristic
456C2×C4#51911yescharacteristic
457C2×C4#54812yes—
458C2×C4D4×Q812yes—
459C8—44no—
460C8—82no—
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—44no—
466C22—42no—
467C22—82no—
468C22—81no—
469C22—81no—
470C22Q8×C22⋊C411yescharacteristic
471C22C22×#9911yescharacteristic
472C22#288411yescharacteristic
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C2—21no—
495C2—21no—
496C2—82no—
497C2Q8×#9911yescharacteristic
498C2#3336911yescharacteristic
499C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22517C2semidirect
3. #13303C2semidirect
4. #13303C2semidirect
5. #13301C2semidirect
143. #355Q8semidirect

Maximal subgroups

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

subgroupindex
#396132
#225172
#133032
#133012
#128032
#128012
#126322
C4×#3552