Finite Group Databasebeta

← #238274

Subgroups — #238274

Order 512.

800 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35339C211yesmaximal, characteristic
2#21332C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#10377C218yesmaximal
6#8026C211yesmaximal, characteristic
7#7981C211yesmaximal, characteristic
8#7924C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2864—21no—
13#2864—21no—
14#2647—22no—
15#2570—22no—
16#2570—22no—
17#2569—22no—
18#2569C2214yes—
19#2317—21no—
20#2316—21no—
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1996—22no—
26#1996C2214yes—
27#1740—21no—
28#1740—21no—
29#1520C2214yes—
30#1518—22no—
31#1518—22no—
32#1518—21no—
33#1518—21no—
34#1503—22no—
35C2×#328C2214yes—
36#1294C2214yes—
37#1289C2214yes—
38#1282C2214yes—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47C2×C4⋊Q8Q811yescharacteristic
48C2×C4⋊Q8Q811yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2312yes—
52C2×C4×Q8—21no—
53C2×C4×Q8Q812yes—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C42C2311yescharacteristic
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#400—24no—
76#400—24no—
77#398—24no—
78#398—24no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#385—24no—
90#384—22no—
91#384—21no—
92#384—21no—
93#382—22no—
94#382—21no—
95#382—21no—
96#382—21no—
97#382D412yes—
98#382D411yescharacteristic
99#382D411yescharacteristic
100#382D412yes—
101#380—21no—
102#380—24no—
103#380—21no—
104#380C2312yes—
105#380C2312yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4D412yes—
113C4×C22⋊C4C2312yes—
114#376—21no—
115C2×#95—21no—
116C2×#95C2312yes—
117C43—21no—
118#342—24no—
119#342—24no—
120#328—28no—
121#328—24no—
122#328—24no—
123C22×Q8—21no—
124C22×Q8C2×Q812yes—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140#125—41no—
141#125—41no—
142#125—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—41no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—42no—
168#117—42no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C2×Q812yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q811yescharacteristic
211C2×C42C2×Q811yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—41no—
243#95C2×D414yes—
244#95C2×D412yes—
245#95C2×D412yes—
246C24C22×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—82no—
252C2×Q8—82no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4#12411yescharacteristic
276C22×C4#12411yescharacteristic
277C22×C4#12411yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#13512yes—
280C22×C4#13512yes—
281C22×C4#13712yes—
282C22×C4#13712yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—82no—
292C2×C8—82no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—44no—
335C42—82no—
336C42—82no—
337C42—82no—
338C42C22⋊Q811yescharacteristic
339C42C22⋊Q811yescharacteristic
340C42C22⋊Q811yescharacteristic
341C42C22⋊Q811yescharacteristic
342C42C22⋊Q811yescharacteristic
343C42C22⋊Q811yescharacteristic
344C42C22⋊Q811yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42#12411yescharacteristic
347C42#12411yescharacteristic
348C42#12411yescharacteristic
349C42#12411yescharacteristic
350C42#12411yescharacteristic
351C42#12411yescharacteristic
352C42#12411yescharacteristic
353C42#12411yescharacteristic
354C23—41no—
355C23—41no—
356C23C2×#12011yescharacteristic
357C23C2×C22⋊Q811yescharacteristic
358C23C2×#12411yescharacteristic
359C23C2×#13511yescharacteristic
360C23C2×#13511yescharacteristic
361C23C2×#13711yescharacteristic
362C23C2×#13711yescharacteristic
363Q8—82no—
364Q8—82no—
365Q8—82no—
366Q8—82no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4#39311yescharacteristic
404C2×C4#39311yescharacteristic
405C2×C4#39311yescharacteristic
406C2×C4#39311yescharacteristic
407C2×C4#39311yescharacteristic
408C2×C4#39311yescharacteristic
409C2×C4#39311yescharacteristic
410C2×C4#39311yescharacteristic
411C2×C4#45212yes—
412C2×C4#45212yes—
413C2×C4#45212yes—
414C2×C4#45212yes—
415C2×C4#47912yes—
416C2×C4#47912yes—
417C2×C4#47912yes—
418C2×C4#47912yes—
419C2×C4#48812yes—
420C2×C4#48812yes—
421C2×C4#49112yes—
422C2×C4#49112yes—
423C2×C4C2×C22⋊Q811yescharacteristic
424C2×C4C2×C22⋊Q811yescharacteristic
425C2×C4C2×#12411yescharacteristic
426C2×C4C2×#12411yescharacteristic
427C8—82no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22C2×#39311yescharacteristic
440C22C2×#45211yescharacteristic
441C22C2×#45211yescharacteristic
442C22C2×#47911yescharacteristic
443C22C2×#47911yescharacteristic
444C22C2×#48811yescharacteristic
445C22C2×#49111yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4—84no—
459C4—82no—
460C4—82no—
461C4—162no—
462C4#202612yes—
463C4#202612yes—
464C4#202612yes—
465C4#202612yes—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—21no—
472C2C2×#202611yescharacteristic
473C2#2347211yescharacteristic
474C2#2347411yescharacteristic
subgroupG/H
center, socleC22C2×#393
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#353392
#213322
#175752
C2×#15182
#103772
#80262
#79812
#79242