Finite Group Databasebeta

← #218161

Subgroups — #218161

Order 512.

549 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#23759C211yesmaximal, characteristic
2#22102C211yesmaximal, characteristic
3#21973C211yesmaximal, characteristic
4#9598C211yesmaximal, characteristic, complemented
5#9596C211yesmaximal, characteristic, complemented
6#9557C211yesmaximal, characteristic, complemented
7#9555C211yesmaximal, characteristic, complemented
8#9534C211yesmaximal, characteristic, complemented
9#9533C211yesmaximal, characteristic, complemented
10#8937C211yesmaximal, characteristic
11#8848C211yesmaximal, characteristic, complemented
12#8848C211yesmaximal, characteristic, complemented
13#8829C211yesmaximal, characteristic
14#8711C211yesmaximal, characteristic
15#8149C211yesmaximal, characteristic
16C2×#446C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2283C2211yescharacteristic
21#2101C411yescharacteristic
22#2097C411yescharacteristic
23#2085C411yescharacteristic
24#2077C411yescharacteristic
25#2075C411yescharacteristic
26#2074C411yescharacteristic
27#2036C411yescharacteristic
28#2035C411yescharacteristic
29#1984C2211yescharacteristic
30#1939—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1924—21no—
37#1923—21no—
38#1917—21no—
39#1850—21no—
40#1850—21no—
41#1843—21no—
42#1832—21no—
43#1797C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#106C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#446—41no—
72#446—41no—
73#431—41no—
74#431—41no—
75C2×#107C2×C411yescharacteristic
76C2×#107C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8C2×C411yescharacteristic
101C2×C22⋊C8C2×C411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#387—21no—
115#385—21no—
116#383—21no—
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119#381—21no—
120#380C2×C411yescharacteristic
121#380C2×C411yescharacteristic
122#380C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336C2×C411yescharacteristic
149#336C2×C411yescharacteristic
150#335—41no—
151#335—41no—
152Q8⋊C8—41no—
153Q8⋊C8—41no—
154#322—41no—
155#322—41no—
156C22×Q8C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—81no—
168C22×C8—81no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42C2411yescharacteristic
209#107—22no—
210#107—41no—
211#106—22no—
212#106—22no—
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—41no—
222C4⋊C8—41no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—81no—
226C4⋊C8—81no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230#103—22no—
231#103—41no—
232#103—41no—
233#103—41no—
234C22⋊C8—22no—
235C22⋊C8—41no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238C22⋊C8—42no—
239C22⋊C8—81no—
240#97—42no—
241C4×C8—21no—
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—81no—
254#95—21no—
255#95—21no—
256#95—21no—
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—
267C24C4×D411yescharacteristic
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C22×C4—21no—
275C22×C4—22no—
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—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
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—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
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—84no—
346C2×C8#13112yes—
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—21no—
359C4⋊C4—21no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4#13112yes—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42H(5,𝔽2)11yescharacteristic
384C42H(5,𝔽2)11yescharacteristic
385C422−1+411yescharacteristic
386C422−1+411yescharacteristic
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23#44412yes—
394C23C2×C4×D411yescharacteristic
395Q8—41no—
396Q8—41no—
397Q8—41no—
398Q8—41no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
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—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4#52011yescharacteristic
445C2×C4#52011yescharacteristic
446C2×C4#53411yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53911yescharacteristic
449C2×C4#53911yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4#56811yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57811yescharacteristic
457C2×C4#57811yescharacteristic
458C8—22no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#232211yescharacteristic
480C22#293311yescharacteristic
481C22#293511yescharacteristic
482C22#294711yescharacteristic
483C22#295111yescharacteristic
484C22C2×#44411yescharacteristic
485C22#295511yescharacteristic
486C4—21no—
487C4—21no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4#299311yescharacteristic
500C4#299311yescharacteristic
501C4#299411yescharacteristic
502C4#299511yescharacteristic
503C4#319811yescharacteristic
504C4#320211yescharacteristic
505C4#324211yescharacteristic
506C4#324311yescharacteristic
507C2—42no—
508C2#1802311yescharacteristic
509C2#1802711yescharacteristic
510C2#1808111yescharacteristic
511C2#1809211yescharacteristic
512C2#2207411yescharacteristic
513C2#2209411yescharacteristic
514C2#2209611yescharacteristic
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 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9598C2semidirect
5. #9596C2semidirect
6. #9557C2semidirect
7. #9555C2semidirect
8. #9534C2semidirect
9. #9533C2semidirect
11. #8848C2semidirect
12. #8848C2semidirect

Maximal subgroups

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

subgroupindex
#237592
#221022
#219732
#95982
#95962
#95572
#95552
#95342
#95332
#89372
#88482
#88292
#87112
#81492