Finite Group Databasebeta

← #343756

Subgroups — #343756

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26040C211yesmaximal, characteristic
2#26026C211yesmaximal, characteristic, complemented
3#25059C211yesmaximal, characteristic
4#25016C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10622C211yesmaximal, characteristic
8#10618C211yesmaximal, characteristic, complemented
9#10573C211yesmaximal, characteristic, complemented
10#10434C211yesmaximal, characteristic
11#9557C211yesmaximal, characteristic, complemented
12#9556C211yesmaximal, characteristic
13#8895C211yesmaximal, characteristic, complemented
14#8890C211yesmaximal, characteristic
15#8820C211yesmaximal, characteristic, complemented
16C2×#484—21no—
17C2×#474—21no—
18#2551C2211yescharacteristic
19#2538C2211yescharacteristic
20#2355C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2092—21no—
23#2086—21no—
24#2085—21no—
25#2076—21no—
26#2068—21no—
27#2057—21no—
28#2036—21no—
29#2024—21no—
30#1987—21no—
31#1984C2211yescharacteristic
32#1954C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1926—21no—
40#1921—21no—
41#1920—21no—
42#1862—21no—
43#1852—21no—
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1845C2211yescharacteristic
47#1844C2211yescharacteristic
48#1834C2211yescharacteristic
49#1834C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53#1802C2211yescharacteristic
54C4×#103C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74#484—44no—
75#474—44no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—41no—
85C2×#106—21no—
86C2×#106—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103D411yescharacteristic
96C2×#103D411yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102D411yescharacteristic
100C2×#102D411yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#396—41no—
107#395—41no—
108#391—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#387—41no—
115#385—41no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—41no—
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146Q8⋊C8—22no—
147D4⋊C8—22no—
148#322—22no—
149C22×Q8—41no—
150C22×Q8C2×D411yescharacteristic
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—41no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—81no—
166C4⋊Q8—81no—
167C22⋊Q8—42no—
168C22⋊Q8—81no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—42no—
178C4×D4—81no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42C2411yescharacteristic
213#107—42no—
214#107—42no—
215#107—42no—
216#107—42no—
217#107—42no—
218#107—81no—
219#107—81no—
220#107—81no—
221#107—82no—
222#106—42no—
223#106—81no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—81no—
234#103—22no—
235#103—41no—
236#103—42no—
237#103—81no—
238#102—22no—
239#102—42no—
240#102—81no—
241#102Q1614yes—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8#4112yes—
250#95—21no—
251#95—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258C24—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—161no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4C2×Q1612yes—
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×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—161no—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
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×C8C2×Q1612yes—
323C2×C8AGL(1,ℤ/8)12yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4C2×Q1612yes—
361C4⋊C4AGL(1,ℤ/8)12yes—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—161no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—161no—
374C42C2×#4111yescharacteristic
375C422−1+411yescharacteristic
376C422−1+411yescharacteristic
377C422−1+411yescharacteristic
378C23—22no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23D4211yescharacteristic
386Q8—41no—
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—164no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
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—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4#53611yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#53911yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#56311yescharacteristic
441C2×C4C22×Q1611yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57811yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—42no—
452C8—41no—
453C8—41no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22#268011yescharacteristic
466C22#326411yescharacteristic
467C22#326911yescharacteristic
468C22#327411yescharacteristic
469C22D4×Q1611yescharacteristic
470C22#328811yescharacteristic
471C22#328911yescharacteristic
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#323011yescharacteristic
486C4#323511yescharacteristic
487C4#324311yescharacteristic
488C4#324311yescharacteristic
489C4#333811yescharacteristic
490C4#334311yescharacteristic
491C4#334711yescharacteristic
492C4#335311yescharacteristic
493C2—82no—
494C2#1981111yescharacteristic
495C2#1981611yescharacteristic
496C2#2016011yescharacteristic
497C2#2023411yescharacteristic
498C2#3108111yescharacteristic
499C2#3110011yescharacteristic
500C2#3111311yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #26026C2semidirect
5. #10653C2semidirect
6. #10650C2semidirect
8. #10618C2semidirect
9. #10573C2semidirect
11. #9557C2semidirect
13. #8895C2semidirect
15. #8820C2semidirect

Maximal subgroups

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

subgroupindex
#260402
#260262
#250592
#250162
#106532
#106502
#106222
#106182
#105732
#104342
#95572
#95562
#88952
#88902
#88202