Finite Group Databasebeta

← #217504

Subgroups — #217504

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21978C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3C4×#433C211yesmaximal, characteristic
4#9603C211yesmaximal, characteristic, complemented
5#9603C211yesmaximal, characteristic, complemented
6#9602C211yesmaximal, characteristic, complemented
7#9601C211yesmaximal, characteristic, complemented
8#8834C211yesmaximal, characteristic
9#8834C211yesmaximal, characteristic
10#8833C211yesmaximal, characteristic
11#8832C211yesmaximal, characteristic
12#8808C211yesmaximal, characteristic, complemented
13#8807C211yesmaximal, characteristic, complemented
14#8786C211yesmaximal, characteristic, complemented
15#8784C211yesmaximal, characteristic, complemented
16C2×#433C411yescharacteristic
17C2×#433C411yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1940—21no—
22#1940—21no—
23#1923—21no—
24#1923—21no—
25#1916—21no—
26#1916—21no—
27#1851—21no—
28#1851—21no—
29#1842C2211yescharacteristic
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1768C411yescharacteristic
38#1768C411yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1750C411yescharacteristic
44#1750C411yescharacteristic
45C4×C22⋊C8C2211yescharacteristic
46C4×C22⋊C8C2211yescharacteristic
47#1746C411yescharacteristic
48#1746C411yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C22×C42C2311yescharacteristic
71#433—24no—
72#433—24no—
73#432—41no—
74#432—41no—
75#423—41no—
76#423—41no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8C2×C411yescharacteristic
101C2×C22⋊C8C2×C411yescharacteristic
102C2×C22⋊C8C2×C411yescharacteristic
103C2×#97—21no—
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×#97C2×C411yescharacteristic
107C2×#97C2×C411yescharacteristic
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2×C411yescharacteristic
111C2×C4×C8C2×C411yescharacteristic
112C2×C4×C8C2×C411yescharacteristic
113C2×C4×C8C2×C411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4Q811yescharacteristic
121C4×C4⋊C4Q811yescharacteristic
122C4×C22⋊C4D411yescharacteristic
123C4×C22⋊C4D411yescharacteristic
124C4×C22⋊C4Q811yescharacteristic
125C4×C22⋊C4Q811yescharacteristic
126#376C2×C411yescharacteristic
127#376C2×C411yescharacteristic
128C43C2311yescharacteristic
129C43C2311yescharacteristic
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#335—41no—
149#335—41no—
150#335—41no—
151#335—41no—
152#334—41no—
153#334—41no—
154#334—41no—
155#334—41no—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q811yescharacteristic
159C2×M16—42no—
160C2×M16—81no—
161C2×M16—81no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182C2×C4⋊C4C4⋊C411yescharacteristic
183C2×C4⋊C4C4⋊C411yescharacteristic
184C2×C4⋊C4C4⋊C411yescharacteristic
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C22⋊C4C4⋊C411yescharacteristic
187C2×C22⋊C4C4⋊C411yescharacteristic
188C2×C22⋊C4C4⋊C411yescharacteristic
189C2×C22⋊C4C4⋊C411yescharacteristic
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q811yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42C2×Q811yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42C2×Q811yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
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—42no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8—81no—
235C4⋊C8M1614yescomplemented
236C4⋊C8M1614yescomplemented
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—42no—
242C22⋊C8—81no—
243C22⋊C8—81no—
244#97—22no—
245#97—22no—
246#97—22no—
247#97—22no—
248#97—42no—
249C4×C8—22no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—81no—
262C4×C8—81no—
263#95C4⋊C411yescharacteristic
264#95C4⋊C411yescharacteristic
265#95C4⋊C411yescharacteristic
266#95C4⋊C411yescharacteristic
267#95C4⋊C411yescharacteristic
268#95C4⋊C411yescharacteristic
269#95C4⋊C411yescharacteristic
270#95C4⋊C411yescharacteristic
271C24C2×C4⋊C411yescharacteristic
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—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
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×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q811yescharacteristic
302M16—84no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
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—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
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×C8C2×M1612yes—
348C2×C8C2×M1612yes—
349C2×C8C2×M1612yes—
350C2×C8C2×M1612yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42C2×M1612yes—
382C42C2×M1612yes—
383C42#13112yes—
384C42#13112yes—
385C42H(5,𝔽2)11yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C422−1+411yescharacteristic
388C422−1+411yescharacteristic
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C22×C4⋊C411yescharacteristic
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
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—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#42312yes—
439C2×C4#42312yes—
440C2×C4#42412yes—
441C2×C4#42412yes—
442C2×C4#52011yescharacteristic
443C2×C4#52011yescharacteristic
444C2×C4#53511yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#54411yescharacteristic
447C2×C4#54411yescharacteristic
448C2×C4C22×M1611yescharacteristic
449C2×C4C22×M1611yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8—22no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—81no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#231711yescharacteristic
480C22C2×#42311yescharacteristic
481C22C2×#42411yescharacteristic
482C22#291211yescharacteristic
483C22#291311yescharacteristic
484C22#291611yescharacteristic
485C22#291611yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4#299011yescharacteristic
500C4#299011yescharacteristic
501C4#299111yescharacteristic
502C4#299111yescharacteristic
503C4#322411yescharacteristic
504C4#322511yescharacteristic
505C4#324111yescharacteristic
506C4#324211yescharacteristic
507C2—81no—
508C2#1779811yescharacteristic
509C2#1779911yescharacteristic
510C2#1781111yescharacteristic
511C2#1781211yescharacteristic
512C2#2197411yescharacteristic
513C2#2199211yescharacteristic
514C2#2199511yescharacteristic
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. #9603C2semidirect
5. #9603C2semidirect
6. #9602C2semidirect
7. #9601C2semidirect
12. #8808C2semidirect
13. #8807C2semidirect
14. #8786C2semidirect
15. #8784C2semidirect
235. C4⋊C8M16semidirect
236. C4⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#219782
#219222
C4×#4332
#96032
#96022
#96012
#88342
#88332
#88322
#88082
#88072
#87862
#87842