Finite Group Databasebeta

← #225255

Subgroups — #225255

Order 512.

548 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22633C211yesmaximal, characteristic
2#22626C211yesmaximal, characteristic
3#20819C211yesmaximal, characteristic
4#9595C211yesmaximal, characteristic, complemented
5#9594C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic
11#8709C211yesmaximal, characteristic
12#8597C211yesmaximal, characteristic
13#8286C211yesmaximal, characteristic, complemented
14C8×#106C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×C4×M16C2211yescharacteristic
17#2285C2211yescharacteristic
18#2283C2211yescharacteristic
19#1939C2211yescharacteristic
20#1939C2211yescharacteristic
21#1939C2211yescharacteristic
22#1939C2211yescharacteristic
23#1924—21no—
24#1923—21no—
25#1917—21no—
26#1916—21no—
27#1847C411yescharacteristic
28#1847C411yescharacteristic
29#1836C411yescharacteristic
30#1836C411yescharacteristic
31#1832—21no—
32#1809C411yescharacteristic
33#1809C411yescharacteristic
34#1809C411yescharacteristic
35#1809C411yescharacteristic
36#1798—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1750—21no—
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1746—21no—
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×C82C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C2×#117—21no—
70C22×C42C2311yescharacteristic
71C2×#107C2×C411yescharacteristic
72C2×#107C2×C411yescharacteristic
73C2×#107C2×C411yescharacteristic
74C2×#107C2×C411yescharacteristic
75C2×#106C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8D411yescharacteristic
91C2×C22⋊C8D411yescharacteristic
92C4×M16—22no—
93C4×M16—22no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97C2×C411yescharacteristic
97C2×#97C2×C411yescharacteristic
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#383—21no—
109#382—21no—
110#382C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#381—21no—
113#380—21no—
114#380C2×C411yescharacteristic
115#380C2×C411yescharacteristic
116#380C2×C411yescharacteristic
117#380C2×C411yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C22⋊C4—21no—
124#376—21no—
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148#322—22no—
149C82—22no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C22×C411yescharacteristic
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173#117—81no—
174#117—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—22no—
204#107—22no—
205#107—22no—
206#107—41no—
207#106—22no—
208#106—22no—
209#106—22no—
210#106—41no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—81no—
220C4⋊C8—81no—
221C4⋊C8—81no—
222C4⋊C8—81no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—42no—
226C22⋊C8—42no—
227C22⋊C8—81no—
228#97—22no—
229#97—22no—
230#97—42no—
231#97—81no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—42no—
243C4×C8—81no—
244C4×C8#4112yes—
245C4×C8#4112yes—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258C24C4×D411yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4111yescharacteristic
289M16—44no—
290M16—44no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8#11712yes—
328C2×C8#11712yes—
329C2×C8#13112yes—
330C2×C8#13112yes—
331C2×C8#13112yes—
332C2×C8#13712yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
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—21no—
348C4⋊C4—21no—
349C4⋊C4—42no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C22⋊C4—42no—
354C22⋊C4—81no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C422−1+411yescharacteristic
374C422−1+411yescharacteristic
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23C2×C4×D411yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4C2×#11711yescharacteristic
429C2×C4#51911yescharacteristic
430C2×C4#54711yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4#55211yescharacteristic
433C2×C4#55211yescharacteristic
434C2×C4C2×#13111yescharacteristic
435C2×C4C2×#13111yescharacteristic
436C2×C4C2×#13111yescharacteristic
437C2×C4#56811yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C8—22no—
443C8—22no—
444C8—22no—
445C8—42no—
446C8—41no—
447C8—41no—
448C8—41no—
449C8—41no—
450C8—81no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8#43314yes—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22#234411yescharacteristic
464C22#293111yescharacteristic
465C22#293511yescharacteristic
466C22C4×#13711yescharacteristic
467C22#294611yescharacteristic
468C22#295511yescharacteristic
469C22#295611yescharacteristic
470C4—21no—
471C4—21no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4C2×#43311yescharacteristic
484C4#292411yescharacteristic
485C4#298511yescharacteristic
486C4#298611yescharacteristic
487C4#334411yescharacteristic
488C4#334711yescharacteristic
489C4#335111yescharacteristic
490C4#335711yescharacteristic
491C2—81no—
492C2#1796511yescharacteristic
493C2#1796811yescharacteristic
494C2#1807711yescharacteristic
495C2#1808211yescharacteristic
496C2#2260011yescharacteristic
497C2#2260811yescharacteristic
498C2#2262111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9595C2semidirect
5. #9594C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
13. #8286C2semidirect
14. C8×#106C2semidirect

Maximal subgroups

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

subgroupindex
#226332
#226262
#208192
#95952
#95942
#95382
#95372
#95362
#95352
#87112
#87092
#85972
#82862
C8×#1062
#81462