Finite Group Databasebeta

← #195348

Subgroups — #195348

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#14155C211yesmaximal, characteristic
6#9046C218yesmaximal
7#8014C211yesmaximal, characteristic
8#7908C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—24no—
12#3230C418yescomplemented
13C2×#500—21no—
14C2×#487—21no—
15C2×#484C2214yes—
16C2×#477—22no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2370—22no—
20#2369—21no—
21#2365—21no—
22#2355—21no—
23#2355—22no—
24Q8×C4⋊C4—21no—
25#2329C2214yes—
26#2317—21no—
27C4×C22⋊Q8—22no—
28C2×#391—22no—
29#2285C2211yescharacteristic
30#2274—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990—21no—
34#1990—21no—
35#1987—22no—
36#1824C2214yes—
37#1503—24no—
38C2×#328C2214yes—
39#1445C2214yes—
40C4⋊#95—21no—
41C4⋊#95—21no—
42#1294C2214yes—
43#1280C2214yes—
44C23×Q8—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533D414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2312yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C42—21no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#500—28no—
76#487—28no—
77#484—28no—
78#484C2×C4116yes—
79#477—216no—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2×C414yes—
84C2×#107—24no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×#103—24no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#391—22no—
93#391—24no—
94#391—48no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—24no—
99#387—24no—
100#387—24no—
101#387—24no—
102#385—24no—
103#384—22no—
104#384D411yescharacteristic
105#384D411yescharacteristic
106#382—22no—
107#382—21no—
108#380—21no—
109#380C2312yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4D412yes—
124C4×C22⋊C4—24no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95C2312yes—
128C2×#95C2312yes—
129#340—24no—
130#340—24no—
131#328—28no—
132C22×Q8—24no—
133C22×Q8—21no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—42no—
139C22×Q8C2×D412yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8C22⋊C414yes—
160C4⋊Q8C22⋊C414yes—
161#125—24no—
162#125—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8C22×C414yes—
168C22⋊Q8C2×D414yes—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—84no—
173C4×Q8C22⋊C418yes—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4Q1614yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C22×C412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42#4112yes—
232C2×C42#4112yes—
233#107—216no—
234C4⋊C8—24no—
235C4⋊C8—42no—
236#103—216no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8C22×C414yes—
241C4×C8—24no—
242C4×C8—42no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—48no—
254#95—48no—
255C24C22×D411yescharacteristic
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—416no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8—88no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8C2×C22⋊C414yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4C2×Q1614yes—
309C22×C4C2×Q1612yes—
310C22×C4#13512yes—
311C22×C4#13712yes—
312C22×C4#13712yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4—84no—
351C4⋊C4C2×C22⋊C414yes—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C42—28no—
357C42—44no—
358C42—44no—
359C42—82no—
360C42—84no—
361C42—84no—
362C42—84no—
363C42C2×C22⋊C412yes—
364C42C2×C22⋊C412yes—
365C42C4×D412yes—
366C42C4×D412yes—
367C42#12112yes—
368C42#12112yes—
369C42#12312yes—
370C42#12312yes—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23C2×C4×D411yescharacteristic
376C23C2×#12011yescharacteristic
377C23C2×#12111yescharacteristic
378C23C22×Q1611yescharacteristic
379C23C2×#13511yescharacteristic
380C23C2×#13711yescharacteristic
381C23C2×#13711yescharacteristic
382Q8—44no—
383Q8—48no—
384Q8—82no—
385Q8—88no—
386Q8—88no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38612yes—
434C2×C4#38612yes—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4C4×Q1614yes—
438C2×C4#44114yes—
439C2×C4#45112yes—
440C2×C4#45112yes—
441C2×C4#45212yes—
442C2×C4#45212yes—
443C2×C4#45312yes—
444C2×C4#45412yes—
445C2×C4#46214yes—
446C2×C4#46414yes—
447C2×C4C22×C22⋊C411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×#12111yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C8—44no—
452C8—82no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22C2×#38611yescharacteristic
466C22C2×C4×Q1611yescharacteristic
467C22C2×#44111yescharacteristic
468C22C2×#45111yescharacteristic
469C22C2×#45211yescharacteristic
470C22C2×#45311yescharacteristic
471C22C2×#45411yescharacteristic
472C22#302811yescharacteristic
473C22#302811yescharacteristic
474C22C2×#46211yescharacteristic
475C22C2×#46411yescharacteristic
476C4—22no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—44no—
482C4—44no—
483C4—44no—
484C4—84no—
485C4—84no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4#187814yes—
490C4#187814yes—
491C4#188314yes—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2#1839711yescharacteristic
497C2#1839911yescharacteristic
498C2#1841111yescharacteristic
499C2#1841311yescharacteristic
500C2C2×#187811yescharacteristic
501C2C2×#188311yescharacteristic
502C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#342132
#218342
C2×#15032
#141552
#90462
#80142
#79082