Finite Group Databasebeta

← #214049

Subgroups — #214049

Order 512.

753 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37423C211yesmaximal, characteristic
2#17409C211yesmaximal, characteristic
3#17407C211yesmaximal, characteristic
4#9456C212yesmaximal, complemented
5#9453C212yesmaximal, complemented
6#9450C212yesmaximal, complemented
7#9449C212yesmaximal, complemented
8#8745C211yesmaximal, characteristic, complemented
9#8744C211yesmaximal, characteristic, complemented
10#8525C211yesmaximal, characteristic, complemented
11#8525C211yesmaximal, characteristic, complemented
12C2×#544C2211yescharacteristic
13C2×#542C2211yescharacteristic
14#2983C2212yes—
15#2973C2212yes—
16#2862C411yescharacteristic
17#2861C411yescharacteristic
18#2716—22no—
19#2700—22no—
20#2690—22no—
21#2687—22no—
22#2660C411yescharacteristic
23#2640C411yescharacteristic
24#2568C2211yescharacteristic
25#2566C2211yescharacteristic
26#2565C2211yescharacteristic
27#2563C2211yescharacteristic
28#2504C411yescharacteristic
29#2503C411yescharacteristic
30#2500C411yescharacteristic
31#2499C411yescharacteristic
32#2298C2211yescharacteristic
33#1556C2212yes—
34#1556C2212yes—
35#1534C2212yes—
36#1534C2212yes—
37#1473—22no—
38#1472—22no—
39#1469—22no—
40#1468—22no—
41#1467C2212yes—
42#1466C2212yes—
43#1464C2211yescharacteristic
44#1464C2211yescharacteristic
45#1464C2211yescharacteristic
46#1464C2211yescharacteristic
47#1285C2212yes—
48#1285C2212yes—
49#1285C2212yes—
50#1285C2212yes—
51#544—42no—
52#544—42no—
53#542—42no—
54#542—42no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#126D412yes—
60C2×#125—21no—
61C2×#125C2×C411yescharacteristic
62C2×#125C2311yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124D412yes—
65C2×#123D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8D412yes—
70C2×#121C2×C411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76#446—42no—
77#446—42no—
78#435—42no—
79#435—42no—
80#432—22no—
81#432C2312yes—
82#432C2312yes—
83#432C2312yes—
84#432C2312yes—
85#431—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88#400—22no—
89#400—22no—
90#398—22no—
91#398—22no—
92#397—22no—
93#397—22no—
94#397—22no—
95#396—22no—
96#395—22no—
97#394—22no—
98#393—22no—
99#393—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2×C411yescharacteristic
111#388—22no—
112#388—22no—
113#388—22no—
114#388—21no—
115#388—21no—
116#387—22no—
117#387—21no—
118#387—21no—
119#387—22no—
120#386—22no—
121#385—22no—
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#382C2×C411yescharacteristic
130#381—22no—
131#380—22no—
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135C4×C22⋊C4C2×C411yescharacteristic
136#376C2×C411yescharacteristic
137C43C2×C411yescharacteristic
138#332—42no—
139#332—42no—
140#330—42no—
141#330—42no—
142#324—21no—
143#324—21no—
144#324—21no—
145#324—21no—
146#324—24no—
147#323—21no—
148#323—21no—
149#323—21no—
150#323—21no—
151#323—24no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8C22⋊C412yes—
155C22×Q8C22×C411yescharacteristic
156C22×D4C22⋊C412yes—
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C2×M16—42no—
162C22×C8—42no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—44no—
170#126—42no—
171#126—42no—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176#125—44no—
177#124—42no—
178#124—42no—
179#124—41no—
180#124—41no—
181#123—42no—
182#123—42no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189#121—41no—
190#121—41no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—22no—
194C4×D4—22no—
195#117—22no—
196#117—22no—
197#117—41no—
198#117—41no—
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—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D412yes—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2×D411yescharacteristic
236C2×C22⋊C4C2×D412yes—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2411yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—22no—
260#97—22no—
261#97—22no—
262#97—22no—
263#97—42no—
264C4×C8—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—21no—
280#95—21no—
281C24C2×C22⋊C412yes—
282C24C2×C22⋊C411yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—82no—
291C2×Q8—82no—
292C2×Q8—82no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—82no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325M16—162no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—162no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—81no—
372C22⋊C4—81no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C42—21no—
376C42—21no—
377C42—22no—
378C42—22no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—42no—
384C42—81no—
385C42—81no—
386C42H(5,𝔽2)12yes—
387C42H(5,𝔽2)12yes—
388C23—21no—
389C23—22no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—42no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23C22×C22⋊C411yescharacteristic
399Q8—81no—
400Q8—81no—
401Q8—162no—
402Q8—162no—
403Q8—162no—
404D4—81no—
405D4—81no—
406D4—162no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
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—82no—
448C2×C4—82no—
449C2×C4—162no—
450C2×C4#51811yescharacteristic
451C2×C4#51811yescharacteristic
452C2×C4#53412yes—
453C2×C4#53412yes—
454C8—82no—
455C8—82no—
456C8—82no—
457C8—82no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22—41no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—82no—
469C22—82no—
470C22#230811yescharacteristic
471C22#288911yescharacteristic
472C22#288911yescharacteristic
473C4—22no—
474C4—22no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C2—21no—
492C2—21no—
493C2—82no—
494C2—81no—
495C2#2175611yescharacteristic
496C2#2178411yescharacteristic
497C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9456C2semidirect
5. #9453C2semidirect
6. #9450C2semidirect
7. #9449C2semidirect
8. #8745C2semidirect
9. #8744C2semidirect
10. #8525C2semidirect
11. #8525C2semidirect

Maximal subgroups

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

subgroupindex
#374232
#174092
#174072
#94562
#94532
#94502
#94492
#87452
#87442
#85252