Finite Group Databasebeta

← #190362

Subgroups — #190362

Order 512.

963 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34189C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#8779C214yesmaximal, complemented
5#8776C214yesmaximal, complemented
6#8052C212yesmaximal, complemented
7#7940C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9#2920C2212yes—
10C2×#410—21no—
11#2357—21no—
12#2357C2212yes—
13#2356—21no—
14#2356C412yescomplemented
15#2353—21no—
16#2353—21no—
17#2353C412yescomplemented
18#2353C2212yes—
19#2314—21no—
20#2314—21no—
21#2311—21no—
22#2285C411yescharacteristic, complemented
23#2283C411yescharacteristic, complemented
24#2283C411yescharacteristic, complemented
25#2283C2211yescharacteristic
26#2283C2211yescharacteristic
27C2×#382C411yescharacteristic, complemented
28#2274—21no—
29C4×#117C2212yes—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740—21no—
35#1467—21no—
36#1466—21no—
37#1465C2212yes—
38#1465C2212yes—
39#1458—21no—
40#1458—21no—
41#1297C2218yes—
42#1284C2218yes—
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—22no—
58C2×#117D411yescharacteristic
59C2×#117D411yescharacteristic
60C2×#117D411yescharacteristic
61C22×C4⋊C4—21no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#431—22no—
68#431—22no—
69#423—22no—
70#410—28no—
71C2×#99—22no—
72C2×#99—22no—
73#407C2314yes—
74C4×M16—22no—
75#391—22no—
76#390—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#386—22no—
86#386—22no—
87#384—21no—
88#383—21no—
89#383—21no—
90#383C2×C412yes—
91#383C2×C412yes—
92#382—21no—
93#382—21no—
94#382C2×C412yes—
95#382C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#382C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#381—21no—
100#381—21no—
101#381C2×C412yes—
102#381C2312yes—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380C2×C412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
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⋊C4C2314yes—
126C4×C22⋊C4C2312yes—
127#376—21no—
128#376—21no—
129#376—22no—
130#376C2×C412yes—
131#376C2×C412yes—
132C2×#95—21no—
133C2×#95C2×C412yes—
134C43—22no—
135#342—22no—
136#342—22no—
137#324—22no—
138#324—22no—
139#323—22no—
140#323—22no—
141C2×#41—24no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—42no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
147C22×D4C2×D411yescharacteristic
148C22×D4C2×D411yescharacteristic
149C23×C4—22no—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×M16—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—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—
175C4×D4—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—28no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊C411yescharacteristic
207C2×C22⋊C4C22⋊C411yescharacteristic
208C2×C22⋊C4C2×D411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C22⋊C4C2×D411yescharacteristic
211C2×C22⋊C4C2×D411yescharacteristic
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—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C4⋊C8—42no—
243C4⋊C8—42no—
244#99—28no—
245#99—28no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248#97—42no—
249C4×C8—42no—
250#95—21no—
251#95—21no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—24no—
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—22no—
277C2×Q8—41no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4#10414yes—
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—
337M16—84no—
338C2×C8—44no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—28no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—82no—
363C22⋊C4—82no—
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—44no—
379C42—48no—
380C42—82no—
381C42—82no—
382C42—82no—
383C42—82no—
384C42—84no—
385C42—84no—
386C42#11711yescharacteristic
387C42#11711yescharacteristic
388C42#11711yescharacteristic
389C42#11711yescharacteristic
390C42#11712yes—
391C42#11712yes—
392C42#12312yes—
393C42#12312yes—
394C42#12312yes—
395C42#12312yes—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23C2×#10412yes—
410C23C22×C22⋊C411yescharacteristic
411C23C2×#12312yes—
412Q8—44no—
413D4—44no—
414D4—44no—
415D4—44no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—84no—
467C2×C4—84no—
468C2×C4—164no—
469C2×C4#38111yescharacteristic
470C2×C4#38111yescharacteristic
471C2×C4#38112yes—
472C2×C4#38112yes—
473C2×C4#38111yescharacteristic
474C2×C4#38111yescharacteristic
475C2×C4C2×#10412yes—
476C2×C4C2×#10412yes—
477C2×C4#42112yes—
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×#12312yes—
481C8—84no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C2×#38111yescharacteristic
500C22C22×#10411yescharacteristic
501C22C2×#42111yescharacteristic
502C4—22no—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—21no—
509C4—21no—
510C4—41no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—88no—
520C4#182612yes—
521C4#182612yes—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—42no—
526C2—81no—
527C2C2×#182611yescharacteristic
528C2#2070811yescharacteristic
529C2#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. #8779C2semidirect
5. #8776C2semidirect
6. #8052C2semidirect
7. #7940C2semidirect
14. #2356C4semidirect
17. #2353C4semidirect
22. #2285C4semidirect
23. #2283C4semidirect
24. #2283C4semidirect
27. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#341892
C4×#4102
#173612
#87792
#87762
#80522
#79402