Finite Group Databasebeta

← #214048

Subgroups — #214048

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37303C211yesmaximal, characteristic
2#17409C211yesmaximal, characteristic
3#17407C211yesmaximal, characteristic
4#9455C212yesmaximal, complemented
5#9454C212yesmaximal, 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#2868C411yescharacteristic
17#2861C411yescharacteristic
18#2721—22no—
19#2680—22no—
20#2677—22no—
21#2674—22no—
22#2660C411yescharacteristic
23#2640C411yescharacteristic
24#2567C2211yescharacteristic
25#2566C2211yescharacteristic
26#2565C2211yescharacteristic
27#2542C2211yescharacteristic
28#2502C411yescharacteristic
29#2501C411yescharacteristic
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—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×#126D412yes—
61C2×#125—21no—
62C2×#125C2×C411yescharacteristic
63C2×#125C2311yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×#124D412yes—
66C2×#123—22no—
67C2×#123D412yes—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C22⋊Q8D412yes—
73C2×#121C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79#446—42no—
80#446—42no—
81#435—42no—
82#435—42no—
83#432—22no—
84#432C2312yes—
85#432C2312yes—
86#432C2312yes—
87#432C2312yes—
88#431—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91#399—22no—
92#399—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#397—22no—
97#397—22no—
98#396—22no—
99#395—22no—
100#395—22no—
101#394—22no—
102#393—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#389—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—21no—
116#388—21no—
117#387—22no—
118#387—21no—
119#387—21no—
120#387—22no—
121#385—22no—
122#385—22no—
123#384—22no—
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#384C2×C411yescharacteristic
127#384C2×C411yescharacteristic
128#384C2×C411yescharacteristic
129#384C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#382C2×C411yescharacteristic
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—
170C4⋊Q8—84no—
171#126—42no—
172#126—42no—
173#125—22no—
174#125—22no—
175#125—41no—
176#125—41no—
177#125—44no—
178#124—42no—
179#124—42no—
180#124—41no—
181#124—41no—
182#123—42no—
183#123—42no—
184#123—84no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—84no—
192#121—41no—
193#121—41no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×D4—22no—
197C4×D4—22no—
198#117—22no—
199#117—22no—
200#117—41no—
201#117—41no—
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—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D412yes—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C2×D411yescharacteristic
242C2×C22⋊C4C2×D411yescharacteristic
243C2×C22⋊C4C2×D412yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2411yescharacteristic
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—42no—
265C22⋊C8—42no—
266#97—22no—
267#97—22no—
268#97—22no—
269#97—22no—
270#97—42no—
271C4×C8—42no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—21no—
283#95—21no—
284C24C2×C22⋊C412yes—
285C24C2×C22⋊C411yescharacteristic
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8—82no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328M16—162no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—162no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42H(5,𝔽2)12yes—
394C42H(5,𝔽2)12yes—
395C23—21no—
396C23—22no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—41no—
403C23—42no—
404C23—41no—
405C23C22×C22⋊C411yescharacteristic
406Q8—81no—
407Q8—81no—
408Q8—162no—
409Q8—162no—
410Q8—162no—
411D4—81no—
412D4—81no—
413D4—162no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
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—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—162no—
457C2×C4#51811yescharacteristic
458C2×C4#51811yescharacteristic
459C2×C4#53412yes—
460C2×C4#53412yes—
461C8—82no—
462C8—82no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—42no—
468C22—41no—
469C22—41no—
470C22—42no—
471C22—41no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—82no—
476C22—82no—
477C22#230811yescharacteristic
478C22#288911yescharacteristic
479C22#288911yescharacteristic
480C4—22no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C2—21no—
499C2—21no—
500C2—82no—
501C2—81no—
502C2#2175611yescharacteristic
503C2#2178211yescharacteristic
504C2#2178311yescharacteristic
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. #9455C2semidirect
5. #9454C2semidirect
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
#373032
#174092
#174072
#94552
#94542
#94502
#94492
#87452
#87442
#85252