Finite Group Databasebeta

← #202264

Subgroups — #202264

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21360C211yesmaximal, characteristic
2#21349C211yesmaximal, characteristic
3#20476C211yesmaximal, characteristic
4#9540C211yesmaximal, characteristic, complemented
5#9539C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#7902C211yesmaximal, characteristic
13#7894C211yesmaximal, characteristic
14#7878C211yesmaximal, characteristic
15#7849C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2283C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1949C411yescharacteristic
24#1949C411yescharacteristic
25#1949C411yescharacteristic
26#1949C411yescharacteristic
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1854C411yescharacteristic
32#1854C411yescharacteristic
33#1854C411yescharacteristic
34#1854C411yescharacteristic
35#1850—21no—
36#1850—21no—
37C4×#107C2211yescharacteristic
38C4×#107C2211yescharacteristic
39C4×#106C2211yescharacteristic
40C4×#106C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×C4⋊C8—21no—
46C4×C4⋊C8—21no—
47#1752—21no—
48C8×C22⋊C4—21no—
49#1746—21no—
50#1726—21no—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54#1279C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—22no—
73#432Q814yes—
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8C2×C411yescharacteristic
103C2×C22⋊C8C2×C411yescharacteristic
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#383—21no—
121#382C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123#381—21no—
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136#376—21no—
137C43—21no—
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4#4111yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—41no—
168C22×C8—41no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177#117—22no—
178#117—22no—
179#117—41no—
180#117—41no—
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×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×Q812yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—22no—
210#107—22no—
211#107—41no—
212#107—41no—
213#106—22no—
214#106—22no—
215#106—41no—
216#106—41no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8C2×Q812yes—
231C4⋊C8C2×Q812yes—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237#97—24no—
238#97—42no—
239#97—42no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
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—21no—
254C4×C8—42no—
255C4×C8—42no—
256C4×C8—42no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24C4×D411yescharacteristic
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—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4#11712yes—
289C22×C4C4×D411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×Q812yes—
293C22×C4C22⋊Q812yes—
294C22×C4#12312yes—
295C22×C4#12512yes—
296C22×C4#12612yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
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—22no—
309C2×C8—22no—
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—44no—
325C2×C8—44no—
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#13512yes—
343C2×C8#13512yes—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4#13112yes—
359C4⋊C4#13112yes—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—24no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42#11712yes—
377C42C4×Q812yes—
378C42C22⋊Q812yes—
379C42#12312yes—
380C42#12512yes—
381C42#12612yes—
382C42C22×Q811yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C42C2×#4111yescharacteristic
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#38712yes—
393C23C2×C4×D411yescharacteristic
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—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
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—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4#38712yes—
437C2×C4#38712yes—
438C2×C4#38712yes—
439C2×C4C2×#11711yescharacteristic
440C2×C4C2×C4×Q811yescharacteristic
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×#12311yescharacteristic
443C2×C4C2×#12511yescharacteristic
444C2×C4C2×#12611yescharacteristic
445C2×C4C2×#13111yescharacteristic
446C2×C4C2×#13111yescharacteristic
447C2×C4#56811yescharacteristic
448C2×C4#56811yescharacteristic
449C2×C4C2×#13511yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57511yescharacteristic
453C8—22no—
454C8—22no—
455C8—42no—
456C8—42no—
457C8—41no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22C2×#38711yescharacteristic
475C22#292911yescharacteristic
476C22#293011yescharacteristic
477C22C4×#13511yescharacteristic
478C22#294311yescharacteristic
479C22#295511yescharacteristic
480C22#295511yescharacteristic
481C4—22no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4#292511yescharacteristic
495C4#292511yescharacteristic
496C4#296211yescharacteristic
497C4#296211yescharacteristic
498C4#307611yescharacteristic
499C4#307611yescharacteristic
500C4#309111yescharacteristic
501C4#309111yescharacteristic
502C2—42no—
503C2#1790411yescharacteristic
504C2#1790911yescharacteristic
505C2#1793211yescharacteristic
506C2#1793711yescharacteristic
507C2#2102411yescharacteristic
508C2#2103211yescharacteristic
509C2#2104111yescharacteristic
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 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9540C2semidirect
5. #9539C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
10. #9534C2semidirect
11. #9533C2semidirect

Maximal subgroups

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

subgroupindex
#213602
#213492
#204762
#95402
#95392
#95382
#95372
#95362
#95352
#95342
#95332
#79022
#78942
#78782
#78492