Finite Group Databasebeta

← #191280

Subgroups — #191280

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21976C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic
3#20472C211yesmaximal, characteristic
4#8789C211yesmaximal, characteristic, complemented
5#8789C211yesmaximal, characteristic, complemented
6#8788C211yesmaximal, characteristic, complemented
7#8787C211yesmaximal, characteristic, complemented
8#8786C211yesmaximal, characteristic, complemented
9#8786C211yesmaximal, characteristic, complemented
10#8785C211yesmaximal, characteristic, complemented
11#8784C211yesmaximal, characteristic, complemented
12#7878C211yesmaximal, characteristic
13#7877C211yesmaximal, characteristic
14#7876C211yesmaximal, characteristic
15#7873C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1940—21no—
24#1940—21no—
25#1939—21no—
26#1939—21no—
27#1851—21no—
28#1851—21no—
29#1850—21no—
30#1850—21no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1831C2211yescharacteristic
39#1771C411yescharacteristic
40#1771C411yescharacteristic
41#1771C411yescharacteristic
42#1771C411yescharacteristic
43#1753C411yescharacteristic
44#1753C411yescharacteristic
45#1753C411yescharacteristic
46#1753C411yescharacteristic
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1277C2211yescharacteristic
52#1277C2211yescharacteristic
53#1277C2211yescharacteristic
54#1277C2211yescharacteristic
55#1277C2211yescharacteristic
56#1277C2211yescharacteristic
57#1277C2211yescharacteristic
58#1277C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16C2×C411yescharacteristic
69C22×M16C2×C411yescharacteristic
70C23×C8—21no—
71C2×#117Q811yescharacteristic
72C2×#117Q811yescharacteristic
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#423—22no—
81#423—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8C2×C411yescharacteristic
107C2×C22⋊C8C2×C411yescharacteristic
108C4×M16—22no—
109C4×M16—22no—
110C2×#97—21no—
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97C2×C411yescharacteristic
114C2×#97C2×C411yescharacteristic
115C2×#97C2×C411yescharacteristic
116C2×#97C2×C411yescharacteristic
117C2×C4×C8—21no—
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2×C411yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C22⋊C4C2×C411yescharacteristic
131C4×C22⋊C4C2×C411yescharacteristic
132C4×C22⋊C4D411yescharacteristic
133C4×C22⋊C4D411yescharacteristic
134#376Q811yescharacteristic
135#376Q811yescharacteristic
136C43C2311yescharacteristic
137C43C2311yescharacteristic
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#336C2×C411yescharacteristic
152#336C2×C411yescharacteristic
153#336C2×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×Q811yescharacteristic
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—41no—
162C2×M16—41no—
163C2×M16—42no—
164C2×M16—81no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—42no—
175C22×C8—81no—
176C22×C8—81no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185#117—42no—
186#117—42no—
187C2×C4⋊C4C4⋊C411yescharacteristic
188C2×C4⋊C4C4⋊C411yescharacteristic
189C2×C4⋊C4C4⋊C411yescharacteristic
190C2×C4⋊C4C4⋊C411yescharacteristic
191C2×C22⋊C4C4⋊C411yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C22⋊C4C4⋊C411yescharacteristic
194C2×C22⋊C4C4⋊C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42C2×Q811yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42C2×Q811yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236C4⋊C8—81no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—22no—
248#97—22no—
249#97—22no—
250#97—22no—
251#97—41no—
252#97—41no—
253#97—42no—
254C4×C8—22no—
255C4×C8—22no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8—42no—
267#95C4⋊C411yescharacteristic
268#95C4⋊C411yescharacteristic
269#95C4⋊C411yescharacteristic
270#95C4⋊C411yescharacteristic
271#95C4⋊C411yescharacteristic
272#95C4⋊C411yescharacteristic
273#95C4⋊C411yescharacteristic
274#95C4⋊C411yescharacteristic
275C24C2×C4⋊C411yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C2×C4⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4#11712yes—
298C22×C4#11712yes—
299C22×C4#12412yes—
300C22×C4#12412yes—
301C22×C4#12512yes—
302C22×C4#12512yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q811yescharacteristic
306M16—44no—
307M16—44no—
308M16—84no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—84no—
356C2×C8#13112yes—
357C2×C8#13112yes—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—22no—
383C42—22no—
384C42—41no—
385C42—41no—
386C42#11712yes—
387C42#11712yes—
388C42#12412yes—
389C42#12412yes—
390C42#12512yes—
391C42#12512yes—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#38212yes—
403C23C22×C4⋊C411yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4#38212yes—
445C2×C4#38212yes—
446C2×C4#38212yes—
447C2×C4#43012yes—
448C2×C4#43012yes—
449C2×C4C2×#11711yescharacteristic
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×#12411yescharacteristic
452C2×C4C2×#12411yescharacteristic
453C2×C4C2×#12511yescharacteristic
454C2×C4C2×#12511yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57811yescharacteristic
463C8—22no—
464C8—22no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22C2×#38211yescharacteristic
487C22#290611yescharacteristic
488C22#290611yescharacteristic
489C22#291211yescharacteristic
490C22#291311yescharacteristic
491C22C2×#43011yescharacteristic
492C22C2×#43011yescharacteristic
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4#292211yescharacteristic
507C4#292311yescharacteristic
508C4#292411yescharacteristic
509C4#292411yescharacteristic
510C4#313611yescharacteristic
511C4#313611yescharacteristic
512C4#313711yescharacteristic
513C4#313711yescharacteristic
514C2—42no—
515C2#1783811yescharacteristic
516C2#1783911yescharacteristic
517C2#1784011yescharacteristic
518C2#1784111yescharacteristic
519C2#2075411yescharacteristic
520C2#2076311yescharacteristic
521C2#2076911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8789C2semidirect
5. #8789C2semidirect
6. #8788C2semidirect
7. #8787C2semidirect
8. #8786C2semidirect
9. #8786C2semidirect
10. #8785C2semidirect
11. #8784C2semidirect

Maximal subgroups

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

subgroupindex
#219762
#219742
#204722
#87892
#87882
#87872
#87862
#87852
#87842
#78782
#78772
#78762
#78732