Finite Group Databasebeta

← #120850

Subgroups — #120850

Order 512.

714 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14355C211yesmaximal, characteristic
2#10854C211yesmaximal, characteristic
3C2×#1305C211yesmaximal, characteristic
4C2×#1297C211yesmaximal, characteristic
5#8019C211yesmaximal, characteristic
6#8015C211yesmaximal, characteristic
7C2×#1284C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#381C2211yescharacteristic
11C2×#380C2211yescharacteristic
12C2×#380C2211yescharacteristic
13C2×C4×C22⋊C4C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15C2×#342—22no—
16#1444—21no—
17#1442—21no—
18#1439—22no—
19C4⋊#95—21no—
20#1434—21no—
21#1305—22no—
22#1297C414yescomplemented
23#1294—21no—
24#1294—21no—
25#1294—21no—
26#1294—21no—
27#1284—22no—
28C24×C4C2311yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4Q812yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4D412yes—
39C22×C42—21no—
40C22×C42—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407—21no—
46#407—21no—
47#407C2×C412yes—
48#407D412yes—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53#381—21no—
54#381—21no—
55#381—21no—
56#381—21no—
57#380—21no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65C4×C22⋊C4—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4C2×C412yes—
71C4×C22⋊C4C2×C412yes—
72C4×C22⋊C4Q812yes—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—21no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95D411yescharacteristic
81C2×#95D411yescharacteristic
82C2×#95D411yescharacteristic
83C2×#95D411yescharacteristic
84#342—44no—
85C25C2×D411yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—41no—
93C23×C4—82no—
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4C2×Q811yescharacteristic
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4111yescharacteristic
107C23×C4#4111yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—82no—
113C2×M16—82no—
114C22×C8—41no—
115C22×C8—41no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4—82no—
124C2×C4⋊C4—82no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—82no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—41no—
149C2×C42—82no—
150C2×C42—82no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C22⋊C8—22no—
154C22⋊C8—22no—
155C22⋊C8—41no—
156C22⋊C8—41no—
157C22⋊C8—41no—
158C22⋊C8—41no—
159#95—21no—
160#95—21no—
161#95—22no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—44no—
167#95—41no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—82no—
175#95—82no—
176#95C4⋊C412yes—
177#95C4⋊C412yes—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—42no—
183C24—42no—
184C24C4×D411yescharacteristic
185C24#12011yescharacteristic
186C24#12111yescharacteristic
187C24C22⋊Q811yescharacteristic
188C24C22⋊Q811yescharacteristic
189C24#12311yescharacteristic
190C24#12411yescharacteristic
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—44no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4#11711yescharacteristic
243C22×C4#11711yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×Q811yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12511yescharacteristic
260C22×C4#12511yescharacteristic
261C22×C4#12511yescharacteristic
262C22×C4#12511yescharacteristic
263C22×C4#12511yescharacteristic
264C22×C4#12611yescharacteristic
265C22×C4#12611yescharacteristic
266C22×C4#12611yescharacteristic
267C22×C4#12611yescharacteristic
268C22×C4C4⋊Q811yescharacteristic
269M16—82no—
270M16—82no—
271M16—82no—
272M16—82no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C2×C8—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—88no—
282C22⋊C4—24no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—82no—
290C22⋊C4—82no—
291C22⋊C4—82no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—82no—
295C42—82no—
296C42—82no—
297C42—82no—
298C42—82no—
299C23—22no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23#38511yescharacteristic
330C23#38711yescharacteristic
331C23#38811yescharacteristic
332C23#39311yescharacteristic
333C23#39411yescharacteristic
334C23#39811yescharacteristic
335C23#40011yescharacteristic
336C23#45311yescharacteristic
337C23#45411yescharacteristic
338C23#45511yescharacteristic
339C23#45611yescharacteristic
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—164no—
404C2×C4#38211yescharacteristic
405C2×C4#38711yescharacteristic
406C2×C4#38811yescharacteristic
407C2×C4#38911yescharacteristic
408C2×C4#39711yescharacteristic
409C2×C4#39811yescharacteristic
410C2×C4#40011yescharacteristic
411C2×C4#40011yescharacteristic
412C8—82no—
413C8—82no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#144211yescharacteristic
447C22#185911yescharacteristic
448C22#186011yescharacteristic
449C22#202711yescharacteristic
450C22#202811yescharacteristic
451C22#203112yes—
452C22#203212yes—
453C22#203711yescharacteristic
454C22#203811yescharacteristic
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—44no—
464C4—82no—
465C4—84no—
466C4—161no—
467C4—162no—
468C4—161no—
469C4—162no—
470C2—21no—
471C2—21no—
472C2—21no—
473C2—21no—
474C2—41no—
475C2—41no—
476C2—41no—
477C2—41no—
478C2#1038212yes—
479C2#1038512yes—
480C2#1126211yescharacteristic
481C2#1127611yescharacteristic
482C2#1127911yescharacteristic
subgroupG/H
center, socleC23#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1297C4semidirect

Maximal subgroups

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

subgroupindex
#143552
#108542
C2×#13052
C2×#12972
#80192
#80152
C2×#12842