Finite Group Databasebeta

← #219155

Subgroups — #219155

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23817C211yesmaximal, characteristic
2#22311C211yesmaximal, characteristic, complemented
3#22208C211yesmaximal, characteristic
4#9874C211yesmaximal, characteristic, complemented
5#9645C211yesmaximal, characteristic
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic
9#9545C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8845C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic
14#8812C211yesmaximal, characteristic, complemented
15#8135C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×#124C2211yescharacteristic
20#2101C411yescharacteristic
21#2101C411yescharacteristic
22#2072C411yescharacteristic
23#2071C411yescharacteristic
24#2042C411yescharacteristic
25#2042C411yescharacteristic
26#2024C411yescharacteristic
27#2023C411yescharacteristic
28#1985—21no—
29#1985—21no—
30#1984C2211yescharacteristic
31#1954C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1917—21no—
37#1917—21no—
38#1850—21no—
39#1850—21no—
40#1842C2211yescharacteristic
41#1831C2211yescharacteristic
42#1770C2211yescharacteristic
43#1770C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C8×Q8—41no—
74C8×Q8—41no—
75C8×D4—41no—
76C8×D4—41no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#103C2×C411yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#102C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#391—21no—
107#391C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389—21no—
110#389C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#388—21no—
113#388—21no—
114#384C2×C411yescharacteristic
115#382C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4C2311yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C43C2311yescharacteristic
122#340C2×C411yescharacteristic
123#340C2×C411yescharacteristic
124#340C2×C411yescharacteristic
125#340C2×C411yescharacteristic
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—41no—
149#335—41no—
150#334—41no—
151#334—41no—
152Q8⋊C8—41no—
153Q8⋊C8—41no—
154D4⋊C8—41no—
155D4⋊C8—41no—
156C22×Q8—21no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
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
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179#124—22no—
180#124—22no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—81no—
186C4×Q8—81no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231#103—22no—
232#103—22no—
233#103—41no—
234#103—41no—
235#102—22no—
236#102—22no—
237#102—41no—
238#102—41no—
239C22⋊C8—42no—
240C22⋊C8—81no—
241C22⋊C8—81no—
242#97—81no—
243#97—81no—
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—21no—
255C4×C8—21no—
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—
265C24—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—42no—
272C2×Q8—81no—
273C2×Q8—81no—
274C2×Q8#13112yes—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4#13112yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
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—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
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—41no—
342C2×C8—41no—
343C2×C8—84no—
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—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4—81no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—81no—
368C22⋊C4—81no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42H(5,𝔽2)11yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C422−1+411yescharacteristic
388C422−1+411yescharacteristic
389C23—22no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×C4×D411yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—41no—
400Q8—41no—
401Q8—84no—
402D4—22no—
403D4—22no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
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—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—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4#52011yescharacteristic
446C2×C4#52011yescharacteristic
447C2×C4#53511yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#54211yescharacteristic
450C2×C4#54211yescharacteristic
451C2×C4C2×#13111yescharacteristic
452C2×C4C2×#13111yescharacteristic
453C2×C4#56811yescharacteristic
454C2×C4#56811yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—81no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22#232611yescharacteristic
481C22#292911yescharacteristic
482C22#293011yescharacteristic
483C22#294611yescharacteristic
484C22#294711yescharacteristic
485C22#295711yescharacteristic
486C22#295711yescharacteristic
487C4—21no—
488C4—21no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—41no—
493C4—42no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4#299211yescharacteristic
501C4#299211yescharacteristic
502C4#299211yescharacteristic
503C4#299211yescharacteristic
504C4#322411yescharacteristic
505C4#322511yescharacteristic
506C4#324311yescharacteristic
507C4#324411yescharacteristic
508C2—82no—
509C2#1792011yescharacteristic
510C2#1792111yescharacteristic
511C2#1794611yescharacteristic
512C2#1794711yescharacteristic
513C2#2221311yescharacteristic
514C2#2223311yescharacteristic
515C2#2223611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22311C2semidirect
4. #9874C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
10. #8846C2semidirect
12. #8844C2semidirect
14. #8812C2semidirect
15. #8135C2semidirect

Maximal subgroups

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

subgroupindex
#238172
#223112
#222082
#98742
#96452
#95482
#95472
#95462
#95452
#88462
#88452
#88442
#88432
#88122
#81352