Finite Group Databasebeta

← #217585

Subgroups — #217585

Order 512.

539 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21974C211yesmaximal, characteristic
2#21924C211yesmaximal, characteristic
3#21431C211yesmaximal, characteristic
4#9606C211yesmaximal, characteristic, complemented
5#9604C211yesmaximal, characteristic, complemented
6#9603C211yesmaximal, characteristic, complemented
7#9602C211yesmaximal, characteristic, complemented
8#8833C211yesmaximal, characteristic
9#8832C211yesmaximal, characteristic
10#8831C211yesmaximal, characteristic
11#8831C211yesmaximal, characteristic
12#8811C211yesmaximal, characteristic, complemented
13#8808C211yesmaximal, characteristic, complemented
14#8787C211yesmaximal, characteristic, complemented
15#8786C211yesmaximal, characteristic, complemented
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1939—21no—
20#1939—21no—
21#1924—21no—
22#1924C411yescharacteristic
23#1924C411yescharacteristic
24#1924C411yescharacteristic
25#1924C411yescharacteristic
26#1923—21no—
27#1917—21no—
28#1917C411yescharacteristic
29#1917C411yescharacteristic
30#1917C411yescharacteristic
31#1917C411yescharacteristic
32#1916—21no—
33#1851—21no—
34#1851—21no—
35#1842C2211yescharacteristic
36#1842C2211yescharacteristic
37#1842C2211yescharacteristic
38#1831C2211yescharacteristic
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43#1767C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1744C2211yescharacteristic
50C4×#97C2211yescharacteristic
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×#117Q811yescharacteristic
69C2×#117Q811yescharacteristic
70C22×C42C2311yescharacteristic
71#431—41no—
72#431—41no—
73#423—41no—
74#423—41no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2×C411yescharacteristic
83C2×C4⋊C8C2×C411yescharacteristic
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8C2×C411yescharacteristic
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8C2×C411yescharacteristic
101C2×#97—21no—
102C2×#97C2×C411yescharacteristic
103C2×#97C2×C411yescharacteristic
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4C2×C411yescharacteristic
118C4×C22⋊C4D411yescharacteristic
119C4×C22⋊C4D411yescharacteristic
120#376Q811yescharacteristic
121#376Q811yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—21no—
125#336—21no—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336C2×C411yescharacteristic
135#336C2×C411yescharacteristic
136#336C2×C411yescharacteristic
137#336C2×C411yescharacteristic
138#336C2×C411yescharacteristic
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#335—41no—
147#335—41no—
148#335—41no—
149#335—41no—
150#334—41no—
151#334—41no—
152#334—41no—
153#334—41no—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×Q811yescharacteristic
157C2×M16—42no—
158C2×M16—81no—
159C2×M16—81no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176#117—42no—
177#117—42no—
178C2×C4⋊C4C4⋊C411yescharacteristic
179C2×C4⋊C4C4⋊C411yescharacteristic
180C2×C4⋊C4C4⋊C411yescharacteristic
181C2×C4⋊C4C4⋊C411yescharacteristic
182C2×C22⋊C4C4⋊C411yescharacteristic
183C2×C22⋊C4C4⋊C411yescharacteristic
184C2×C22⋊C4C4⋊C411yescharacteristic
185C2×C22⋊C4C4⋊C411yescharacteristic
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q811yescharacteristic
201C2×C42C2×Q811yescharacteristic
202C2×C42C2×Q811yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42C2×Q811yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
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—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—42no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232C4⋊C8—81no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—42no—
238C22⋊C8—81no—
239C22⋊C8—81no—
240#97—22no—
241#97—22no—
242#97—41no—
243#97—41no—
244#97—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—81no—
255#95C4⋊C411yescharacteristic
256#95C4⋊C411yescharacteristic
257#95C4⋊C411yescharacteristic
258#95C4⋊C411yescharacteristic
259#95C4⋊C411yescharacteristic
260#95C4⋊C411yescharacteristic
261#95C4⋊C411yescharacteristic
262#95C4⋊C411yescharacteristic
263C24C2×C4⋊C411yescharacteristic
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—22no—
279C22×C4—41no—
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C2×C4⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294M16—84no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
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—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
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—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#13112yes—
342C2×C8#13112yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
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—22no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42H(5,𝔽2)11yescharacteristic
378C42H(5,𝔽2)11yescharacteristic
379C422−1+411yescharacteristic
380C422−1+411yescharacteristic
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23C22×C4⋊C411yescharacteristic
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
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—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4#43012yes—
433C2×C4#43012yes—
434C2×C4#52011yescharacteristic
435C2×C4#52011yescharacteristic
436C2×C4#53511yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54411yescharacteristic
439C2×C4#54411yescharacteristic
440C2×C4C2×#13111yescharacteristic
441C2×C4C2×#13111yescharacteristic
442C2×C4#56811yescharacteristic
443C2×C4#56811yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57811yescharacteristic
448C8—22no—
449C8—22no—
450C8—41no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#231711yescharacteristic
472C22#290611yescharacteristic
473C22#290611yescharacteristic
474C22#291211yescharacteristic
475C22#291311yescharacteristic
476C22C2×#43011yescharacteristic
477C22C2×#43011yescharacteristic
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4#299211yescharacteristic
492C4#299311yescharacteristic
493C4#299411yescharacteristic
494C4#299511yescharacteristic
495C4#322411yescharacteristic
496C4#322511yescharacteristic
497C4#324111yescharacteristic
498C4#324211yescharacteristic
499C2—81no—
500C2#1783811yescharacteristic
501C2#1783911yescharacteristic
502C2#1783911yescharacteristic
503C2#1784111yescharacteristic
504C2#2198011yescharacteristic
505C2#2199211yescharacteristic
506C2#2199411yescharacteristic
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. #9606C2semidirect
5. #9604C2semidirect
6. #9603C2semidirect
7. #9602C2semidirect
12. #8811C2semidirect
13. #8808C2semidirect
14. #8787C2semidirect
15. #8786C2semidirect

Maximal subgroups

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

subgroupindex
#219742
#219242
#214312
#96062
#96042
#96032
#96022
#88332
#88322
#88312
#88112
#88082
#87872
#87862