Finite Group Databasebeta

← #219214

Subgroups — #219214

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23817C211yesmaximal, characteristic
2#21713C211yesmaximal, characteristic, complemented
3#21633C211yesmaximal, characteristic
4#9873C211yesmaximal, characteristic, complemented
5#9640C211yesmaximal, characteristic
6#9548C211yesmaximal, characteristic, complemented
7#9546C211yesmaximal, characteristic
8#9544C211yesmaximal, characteristic, complemented
9#9542C211yesmaximal, characteristic
10#8850C211yesmaximal, characteristic, complemented
11#8850C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic
13#8843C211yesmaximal, characteristic
14#8812C211yesmaximal, characteristic, complemented
15#8134C211yesmaximal, characteristic, complemented
16C2×#446—21no—
17C2×#446C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×#124C2211yescharacteristic
21#2101C411yescharacteristic
22#2101C411yescharacteristic
23#2072C411yescharacteristic
24#2071C411yescharacteristic
25#2042C411yescharacteristic
26#2042C411yescharacteristic
27#2024C411yescharacteristic
28#2023C411yescharacteristic
29#1985—21no—
30#1984C2211yescharacteristic
31#1954C2211yescharacteristic
32#1940—21no—
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1924—21no—
36#1924—21no—
37#1842C2211yescharacteristic
38#1831C2211yescharacteristic
39#1771—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#103C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1752—21no—
47#1752—21no—
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
50C4×#97C2211yescharacteristic
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
73#446—41no—
74#446—41no—
75#446—44no—
76C8×D4—41no—
77C8×D4—41no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#103C2×C411yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#102C2×C411yescharacteristic
92C2×#102C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97D411yescharacteristic
101C2×#97D411yescharacteristic
102C2×#97D411yescharacteristic
103C2×#97D411yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#391C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#389—21no—
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#388—21no—
119#384C2×C411yescharacteristic
120#382C2×C411yescharacteristic
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C43C2311yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
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#336—21no—
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#335—41no—
150#335—41no—
151#334—41no—
152#334—41no—
153Q8⋊C8—41no—
154Q8⋊C8—41no—
155D4⋊C8—41no—
156D4⋊C8—41no—
157C22×Q8—21no—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4⋊Q8—22no—
179C4⋊Q8—41no—
180#124—22no—
181#124—41no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232#103—22no—
233#103—41no—
234#103—41no—
235#103—41no—
236#102—22no—
237#102—22no—
238#102—22no—
239#102—41no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242C22⋊C8—42no—
243#97—22no—
244#97—22no—
245#97—41no—
246#97—41no—
247#97—42no—
248#97—42no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—81no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#13112yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
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—21no—
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—41no—
346C2×C8—84no—
347C2×C8AGL(1,ℤ/8)12yes—
348C2×C8#13712yes—
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—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4#13112yes—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—81no—
373C22⋊C4—81no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42H(5,𝔽2)11yescharacteristic
391C42H(5,𝔽2)11yescharacteristic
392C422−1+411yescharacteristic
393C422−1+411yescharacteristic
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×C4×D411yescharacteristic
402Q8—41no—
403Q8—41no—
404Q8—41no—
405Q8—41no—
406Q8—84no—
407D4—22no—
408D4—22no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
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—41no—
445C2×C4—41no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4#52011yescharacteristic
451C2×C4#52011yescharacteristic
452C2×C4#53511yescharacteristic
453C2×C4#53611yescharacteristic
454C2×C4#54211yescharacteristic
455C2×C4#54211yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4C2×#13711yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57511yescharacteristic
464C8—22no—
465C8—22no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#232611yescharacteristic
486C22#293311yescharacteristic
487C22#293311yescharacteristic
488C22C4×AGL(1,ℤ/8)11yescharacteristic
489C22C4×#13711yescharacteristic
490C22#295711yescharacteristic
491C22#295711yescharacteristic
492C4—21no—
493C4—21no—
494C4—41no—
495C4—42no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4#299211yescharacteristic
506C4#299311yescharacteristic
507C4#299411yescharacteristic
508C4#299511yescharacteristic
509C4#321311yescharacteristic
510C4#321611yescharacteristic
511C4#323911yescharacteristic
512C4#324011yescharacteristic
513C2—82no—
514C2#1800611yescharacteristic
515C2#1800711yescharacteristic
516C2#1800911yescharacteristic
517C2#1801211yescharacteristic
518C2#2221711yescharacteristic
519C2#2222311yescharacteristic
520C2#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 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21713C2semidirect
4. #9873C2semidirect
6. #9548C2semidirect
8. #9544C2semidirect
10. #8850C2semidirect
11. #8850C2semidirect
14. #8812C2semidirect
15. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#238172
#217132
#216332
#98732
#96402
#95482
#95462
#95442
#95422
#88502
#88432
#88122
#81342