Finite Group Databasebeta

← #225322

Subgroups — #225322

Order 512.

575 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22633C211yesmaximal, characteristic
2#22625C211yesmaximal, characteristic
3#21361C211yesmaximal, characteristic
4#9599C211yesmaximal, characteristic, complemented
5#9594C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9533C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic
11#8706C211yesmaximal, characteristic
12#8600C211yesmaximal, characteristic
13#8289C211yesmaximal, characteristic, complemented
14C8×#106C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C2×#431C2211yescharacteristic
17#2283C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1949C411yescharacteristic
20#1949C411yescharacteristic
21#1949C411yescharacteristic
22#1949C411yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1924—21no—
28#1923—21no—
29#1916—21no—
30#1916—21no—
31#1853C411yescharacteristic
32#1853C411yescharacteristic
33#1852C411yescharacteristic
34#1852C411yescharacteristic
35#1832—21no—
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×C22⋊C8—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×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73C2×#107C2×C411yescharacteristic
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#107C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8C2×C411yescharacteristic
95C2×C22⋊C8C2×C411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×C22⋊C8D411yescharacteristic
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#383—21no—
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#381—21no—
112#380C2×C411yescharacteristic
113#380C2×C411yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123#376—21no—
124C2×#95—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×C4—21no—
151C23×C4—21no—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C22×C411yescharacteristic
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—81no—
196C2×C42—81no—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#107—22no—
202#107—22no—
203#107—22no—
204#107—41no—
205#106—22no—
206#106—22no—
207#106—22no—
208#106—41no—
209C4⋊C8—22no—
210C4⋊C8—22no—
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—24no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228C22⋊C8—42no—
229C22⋊C8—81no—
230#97—42no—
231C4×C8—21no—
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—42no—
242C4×C8—81no—
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—22no—
263C22×C4—24no—
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—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
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—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#11712yes—
335C2×C8#11712yes—
336C2×C8#13712yes—
337C4⋊C4—24no—
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—21no—
350C4⋊C4—21no—
351C4⋊C4—42no—
352C4⋊C4#13112yes—
353C4⋊C4#13112yes—
354C4⋊C4#13112yes—
355C22⋊C4—42no—
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—
371C42—81no—
372C42—81no—
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C422−1+411yescharacteristic
376C422−1+411yescharacteristic
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23#44312yes—
384C23C2×C4×D411yescharacteristic
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—42no—
400C2×C4—42no—
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—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4C2×#11711yescharacteristic
433C2×C4#51911yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4#55211yescharacteristic
438C2×C4C2×#13111yescharacteristic
439C2×C4C2×#13111yescharacteristic
440C2×C4C2×#13111yescharacteristic
441C2×C4#56811yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57511yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C8—22no—
447C8—22no—
448C8—22no—
449C8—42no—
450C8—42no—
451C8—41no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22#234411yescharacteristic
468C22#292911yescharacteristic
469C22#293311yescharacteristic
470C22C4×#13711yescharacteristic
471C22#294611yescharacteristic
472C22C2×#44311yescharacteristic
473C22#295711yescharacteristic
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4#292211yescharacteristic
488C4#292511yescharacteristic
489C4#298511yescharacteristic
490C4#298611yescharacteristic
491C4#334411yescharacteristic
492C4#334711yescharacteristic
493C4#335111yescharacteristic
494C4#335711yescharacteristic
495C2—42no—
496C2#1791111yescharacteristic
497C2#1791811yescharacteristic
498C2#1801011yescharacteristic
499C2#1801411yescharacteristic
500C2#2260411yescharacteristic
501C2#2260611yescharacteristic
502C2#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. #9599C2semidirect
5. #9594C2semidirect
6. #9538C2semidirect
7. #9536C2semidirect
8. #9535C2semidirect
9. #9533C2semidirect
13. #8289C2semidirect
14. C8×#106C2semidirect

Maximal subgroups

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

subgroupindex
#226332
#226252
#213612
#95992
#95942
#95382
#95362
#95352
#95332
#87112
#87062
#86002
#82892
C8×#1062
#81482