Finite Group Databasebeta

← #111133

Subgroups — #111133

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1520C212yesmaximal
2#14281C211yesmaximal, characteristic
3#13778C211yesmaximal, characteristic
4C2×#1289C212yesmaximal
5C2×#1280C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C22×C22⋊C8C2212yes—
8C2×#380C2211yescharacteristic
9C2×#380C2212yes—
10C2×C4×C22⋊C4—21no—
11C22×#95C2211yescharacteristic
12#1520—24no—
13#1520C418yescomplemented
14C2×#328—24no—
15#1446C412yescomplemented
16#1445C412yescomplemented
17#1441—22no—
18#1436—24no—
19C22⋊#95—21no—
20C22⋊#95—21no—
21C22⋊#95—21no—
22#1289—24no—
23#1280—22no—
24C23×Q8C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4C2311yescharacteristic
28C23×C8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8C2×C414yes—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C22⋊C4—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4—21no—
42C22×C42—22no—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—22no—
46C2×C22⋊C8—22no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8C2×C414yes—
49#380—22no—
50#380—22no—
51#380—22no—
52#380—22no—
53#380C2×C414yes—
54#380D414yes—
55C4×C22⋊C4—42no—
56C4×C22⋊C4—42no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95D412yes—
69C2×#95D414yes—
70#328—216no—
71C25C2×D411yescharacteristic
72C22×Q8—21no—
73C22×Q8—22no—
74C22×Q8—22no—
75C22×Q8—81no—
76C22×Q8—81no—
77C22×Q8C4212yescomplemented
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—24no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—81no—
91C23×C4—81no—
92C23×C4—81no—
93C23×C4—81no—
94C23×C4—81no—
95C23×C4—81no—
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C412yes—
98C23×C4SD1612yes—
99C23×C4Q1612yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4#4112yes—
106C23×C4#4112yes—
107C23×C4#4112yes—
108C22×C8—24no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—82no—
112C22×C8—82no—
113C22⋊Q8—24no—
114C22⋊Q8—24no—
115C22⋊Q8—24no—
116C22⋊Q8—24no—
117C22⋊Q8—42no—
118C22⋊Q8—42no—
119C22⋊Q8—42no—
120C22⋊Q8—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—82no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—81no—
135C2×C4⋊C4—81no—
136C2×C4⋊C4C4212yescomplemented
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C42—44no—
152C2×C42—82no—
153C2×C42—82no—
154C22⋊C8—24no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#95—24no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—44no—
163#95—82no—
164#95—82no—
165#95—82no—
166#95—82no—
167#95—82no—
168#95—82no—
169#95C22⋊C418yes—
170C24—41no—
171C24—41no—
172C24—41no—
173C24—41no—
174C24C2×C22⋊C411yescharacteristic
175C24C4×D412yes—
176C24#12011yescharacteristic
177C24#12112yes—
178C24#12311yescharacteristic
179C24C2×SD1611yescharacteristic
180C24C2×Q1611yescharacteristic
181C24#13512yes—
182C24AGL(1,ℤ/8)11yescharacteristic
183C24#13711yescharacteristic
184C24#13712yes—
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—84no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—28no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4#10314yes—
258C22×C4C2×C4211yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4#11712yes—
261C22×C4#11711yescharacteristic
262C22×C4C4×D412yes—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4#12112yes—
268C22×C4#12112yes—
269C22×C4#12312yes—
270C22×C4#12411yescharacteristic
271C22×C4#12412yes—
272C22×C4#12612yes—
273C22×C4#12711yescharacteristic
274C2×C8—44no—
275C2×C8—82no—
276C2×C8—82no—
277C2×C8—84no—
278C2×C8—84no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C4⋊C4—84no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—88no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—41no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23C4×C22⋊C411yescharacteristic
325C23#38612yes—
326C23#38611yescharacteristic
327C23#38812yes—
328C23#39011yescharacteristic
329C23C2×#10311yescharacteristic
330C23#41611yescharacteristic
331C23#41711yescharacteristic
332C23#41911yescharacteristic
333C23C4×SD1612yes—
334C23C4×Q1612yes—
335C23#44012yes—
336C23#44112yes—
337C23#45011yescharacteristic
338C23#45111yescharacteristic
339C23#45212yes—
340C23#46112yes—
341C23#46212yes—
342C23#46312yes—
343C23#46412yes—
344C23#48311yescharacteristic
345C23#48411yescharacteristic
346C23#48512yes—
347Q8—84no—
348Q8—84no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4C4×C22⋊C412yes—
408C2×C4#38812yes—
409C2×C4#38812yes—
410C2×C4#39012yes—
411C8—84no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—22no—
416C22—21no—
417C22—22no—
418C22—21no—
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—22no—
435C22—22no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22#143711yescharacteristic
449C22C4×#10311yescharacteristic
450C22#176311yescharacteristic
451C22#187211yescharacteristic
452C22#187311yescharacteristic
453C22#187712yes—
454C22#187812yes—
455C22#191111yescharacteristic
456C22#191211yescharacteristic
457C22#191311yescharacteristic
458C22#191311yescharacteristic
459C22#192712yes—
460C22#192812yes—
461C22#198111yescharacteristic
462C22#198311yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—44no—
468C4—88no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—162no—
473C2—22no—
474C2—21no—
475C2—22no—
476C2—22no—
477C2—22no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2#919911yescharacteristic
482C2#920011yescharacteristic
483C2#920111yescharacteristic
484C2#920211yescharacteristic
485C2#1093211yescharacteristic
486C2#1093811yescharacteristic
487C2#1094911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×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
13. #1520C4semidirect
15. #1446C4semidirect
16. #1445C4semidirect
77. C22×Q8C42semidirect
136. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#15202
#142812
#137782
C2×#12892
C2×#12802