Finite Group Databasebeta

← #190221

Subgroups — #190221

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34121C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8763C214yesmaximal, complemented
5#8762C214yesmaximal, complemented
6#8021C212yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C2×#432C2212yes—
10#2357C412yescomplemented
11#2356C412yescomplemented
12#2355—21no—
13#2354—21no—
14#2353C2212yes—
15#2353C2212yes—
16#2352—21no—
17#2352—21no—
18#2298—21no—
19#2298—21no—
20#2297—21no—
21#2285C2211yescharacteristic
22#2283C411yescharacteristic, complemented
23#2283C411yescharacteristic, complemented
24#2283C2212yes—
25C4×#117—21no—
26C4×#117C411yescharacteristic, complemented
27C4×#117C411yescharacteristic, complemented
28C2×C4×C4⋊C4C2211yescharacteristic
29#1820—21no—
30#1799C2212yes—
31#1799C2212yes—
32#1782—21no—
33C4×#99—21no—
34C4×#99—21no—
35#1467—21no—
36#1466—21no—
37#1464—21no—
38#1464—21no—
39#1459C2212yes—
40#1459C2212yes—
41#1294C2218yes—
42#1289C2218yes—
43C22×#41D411yescharacteristic
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C4⋊C4—21no—
63C22×C42—21no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2311yescharacteristic
67#432—22no—
68#432—22no—
69#432—22no—
70#432—22no—
71C2×C4⋊C8—22no—
72C2×#99—22no—
73C2×#99—22no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#391—22no—
77#390—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#386—22no—
87#386—22no—
88#384—21no—
89#383—22no—
90#383C2×C412yes—
91#383C2×C412yes—
92#382—21no—
93#382—21no—
94#382—22no—
95#382C2×C412yes—
96#381—22no—
97#381C2×C412yes—
98#381C2×C412yes—
99#380—21no—
100#380—21no—
101#380—21no—
102#380C2314yes—
103#380C2312yes—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4C2×C412yes—
126#376—21no—
127#376—21no—
128#376—22no—
129#376C2×C412yes—
130#376C2×C412yes—
131C2×#95C2312yes—
132C43—21no—
133C43—21no—
134#342—22no—
135#342—22no—
136#324—22no—
137#324—22no—
138#323—22no—
139#323—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4C22⋊C411yescharacteristic
146C22×D4C2×D411yescharacteristic
147C22×D4C2×D411yescharacteristic
148C23×C4—22no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—44no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4C22⋊C411yescharacteristic
206C2×C22⋊C4C22⋊C411yescharacteristic
207C2×C22⋊C4C2×D411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—42no—
245C4⋊C8—42no—
246#99—44no—
247#99—44no—
248C22⋊C8—24no—
249C22⋊C8—24no—
250#97—42no—
251#97—42no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95C22×C412yes—
267#95C22×C412yes—
268#95C22×C412yes—
269#95C22×C412yes—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24C22×D411yescharacteristic
273#41—48no—
274#41—48no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4#12312yes—
330C22×C4#12312yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C2×C8—44no—
338C2×C8—84no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—84no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—42no—
379C42—42no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42—44no—
384C42—44no—
385C42—84no—
386C42—82no—
387C42—82no—
388C42#11711yescharacteristic
389C42#11711yescharacteristic
390C42#11711yescharacteristic
391C42#11711yescharacteristic
392C42#11712yes—
393C42#11712yes—
394C42#12312yes—
395C42#12312yes—
396C42#12312yes—
397C42#12312yes—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23C22×C22⋊C411yescharacteristic
411C23C2×#12312yes—
412C23#57512yes—
413C23#57512yes—
414Q8—44no—
415D4—44no—
416D4—44no—
417D4—44no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—88no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4#38111yescharacteristic
475C2×C4#38111yescharacteristic
476C2×C4#38111yescharacteristic
477C2×C4#38112yes—
478C2×C4#38112yes—
479C2×C4#38111yescharacteristic
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×#12312yes—
483C8—84no—
484C22—21no—
485C22—22no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22C2×#38111yescharacteristic
501C22#289611yescharacteristic
502C22#289611yescharacteristic
503C22#309314yes—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—21no—
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—44no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—164no—
523C4#182712yes—
524C2—21no—
525C2—21no—
526C2—24no—
527C2—81no—
528C2—81no—
529C2#2070111yescharacteristic
530C2C2×#182711yescharacteristic
531C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8763C2semidirect
5. #8762C2semidirect
6. #8021C2semidirect
7. #7984C2semidirect
10. #2357C4semidirect
11. #2356C4semidirect
22. #2283C4semidirect
23. #2283C4semidirect
26. C4×#117C4semidirect
27. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#341212
#207232
#173632
#87632
#87622
#80212
#79842