Finite Group Databasebeta

← #223148

Subgroups — #223148

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36184C211yesmaximal, characteristic
2#21271C212yesmaximal, complemented
3#9891C212yesmaximal, complemented
4#9124C212yesmaximal, complemented
5#9123C212yesmaximal, complemented
6#8721C211yesmaximal, characteristic, complemented
7#8718C211yesmaximal, characteristic, complemented
8#8583C211yesmaximal, characteristic, complemented
9C4×#329C211yesmaximal, characteristic, complemented
10#8031C211yesmaximal, characteristic, complemented
11#8030C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#552C411yescharacteristic
14C2×#432C2212yes—
15C2×#432C2212yes—
16#2859C2211yescharacteristic
17#2850C2211yescharacteristic
18#2850C2211yescharacteristic
19#2566C411yescharacteristic
20#2565C411yescharacteristic
21#2553—22no—
22#2519—22no—
23#2502C411yescharacteristic
24#2501C411yescharacteristic
25#2500C411yescharacteristic
26#2499C411yescharacteristic
27#2425—22no—
28#2359—22no—
29C4×#124C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C42×D4C2211yescharacteristic
33#1955—22no—
34#1940—22no—
35#1923—22no—
36#1923—22no—
37#1767C2212yes—
38#1749C2212yes—
39#1749C2212yes—
40C4×#97C2212yes—
41C2×#329C2212yes—
42C2×#329C2212yes—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45#1295C2212yes—
46#1295C2211yescharacteristic
47#1295C2212yes—
48#1295C2211yescharacteristic
49#1295C2212yes—
50#1295C2212yes—
51#553—44no—
52#552—44no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#125—21no—
56C2×#125C2×C411yescharacteristic
57C2×#125C2×C411yescharacteristic
58C2×#124C2311yescharacteristic
59C2×#123—21no—
60C2×#123—22no—
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×#121—21no—
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D412yes—
72C2×#117—21no—
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—24no—
80#432—24no—
81#432—42no—
82#432—42no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×#97C2312yes—
94C2×#97C2312yes—
95C2×C4×C8—22no—
96#399—22no—
97#398—22no—
98#398—22no—
99#397—22no—
100#396—22no—
101#394—22no—
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#389—22no—
109#389—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#387—22no—
115#387—22no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382—22no—
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#382C2311yescharacteristic
129#382C2311yescharacteristic
130#380—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4D412yes—
134C4×C22⋊C4—22no—
135C4×C22⋊C4C2×C411yescharacteristic
136C4×C22⋊C4C2×C411yescharacteristic
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139C2×#95—22no—
140C43C2311yescharacteristic
141#336—22no—
142#336—22no—
143#336—22no—
144#329—24no—
145#329—42no—
146C22×Q8C22×C411yescharacteristic
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162#126—44no—
163#126—44no—
164#125—42no—
165#125—42no—
166#125—44no—
167#124—22no—
168#124—21no—
169#124—21no—
170#123—48no—
171#123—44no—
172#123—44no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#121—42no—
176#121—44no—
177#121—42no—
178C4×Q8—42no—
179C4×D4—24no—
180C4×D4—44no—
181C4×D4—42no—
182C4×D4—44no—
183#117—44no—
184#117—42no—
185#117—42no—
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—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4C22⋊C411yescharacteristic
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C411yescharacteristic
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—44no—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249#97—22no—
250#97—22no—
251#97—22no—
252#97—22no—
253#97—22no—
254#97—22no—
255#97—22no—
256#97—44no—
257#97#4114yes—
258C4×C8—44no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270C24—21no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—82no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D412yes—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—22no—
359C22⋊C4—24no—
360C22⋊C4—24no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—82no—
373C22⋊C4#10014yes—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—42no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42C2×#4112yes—
390C42H(5,𝔽2)11yescharacteristic
391C422−1+411yescharacteristic
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23C2×#10012yes—
407C23C22×C22⋊C411yescharacteristic
408Q8—81no—
409Q8—81no—
410D4—44no—
411D4—81no—
412D4—81no—
413D4—84no—
414D4—162no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4C2×#10012yes—
462C2×C4C2×#10012yes—
463C2×C4#42114yes—
464C2×C4C4×#4111yescharacteristic
465C2×C4#52011yescharacteristic
466C2×C4#54612yes—
467C2×C4#54712yes—
468C8—44no—
469C8—44no—
470C8—82no—
471C8—82no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—41no—
482C22—82no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22—81no—
487C22#233911yescharacteristic
488C22C22×#10011yescharacteristic
489C22#288911yescharacteristic
490C22C2×#42112yes—
491C22#290112yes—
492C4—24no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—41no—
500C4—42no—
501C4—42no—
502C4—41no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C2—42no—
511C2—81no—
512C2—161no—
513C2#1766311yescharacteristic
514C2#1768012yes—
515C2#1768511yescharacteristic
516C2#2247911yescharacteristic
517C2#2250712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#553C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21271C2semidirect
3. #9891C2semidirect
4. #9124C2semidirect
5. #9123C2semidirect
6. #8721C2semidirect
7. #8718C2semidirect
8. #8583C2semidirect
9. C4×#329C2semidirect
10. #8031C2semidirect
11. #8030C2semidirect

Maximal subgroups

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

subgroupindex
#361842
#212712
#98912
#91242
#91232
#87212
#87182
#85832
C4×#3292
#80312
#80302