Finite Group Databasebeta

← #117471

Subgroups — #117471

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21202C211yesmaximal, characteristic
3C2×#1558C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8033C211yesmaximal, characteristic
7#8022C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#419—22no—
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15C2×#376C2211yescharacteristic
16#1960—22no—
17#1956—21no—
18#1956—21no—
19#1949—22no—
20#1853—21no—
21#1852—21no—
22#1847—21no—
23#1836—21no—
24#1558—22no—
25#1558C414yescomplemented
26C2×#333—22no—
27#1446—22no—
28#1445—22no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31#1300—22no—
32#1295—22no—
33#1294—22no—
34C23×Q8D412yes—
35C22×M16—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8Q812yes—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yes—
61#432C2×C412yes—
62#432D412yes—
63C2×#107—22no—
64C2×#106—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8D412yes—
67#419—48no—
68C2×#103—24no—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97D412yes—
79C2×C4×C8—21no—
80#384D411yescharacteristic
81#384D411yescharacteristic
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#382—21no—
87#380—21no—
88#380—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91#376—21no—
92#376—21no—
93#376—21no—
94#376—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95D412yes—
101#340—22no—
102#340—22no—
103#333—28no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8C4⋊C414yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—24no—
120C2×M16—42no—
121C2×M16—41no—
122C2×M16—41no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—41no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142#117—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C22⋊C4—41no—
168C2×C42—24no—
169C2×C42—42no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185#107—82no—
186#106—82no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—41no—
190C4⋊C8C22⋊C414yes—
191#103—48no—
192#103—48no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—22no—
201#97—22no—
202#97—41no—
203C4×C8—41no—
204C4×C8—41no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—84no—
216#95—84no—
217C24#12011yescharacteristic
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—48no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—84no—
227C2×Q8—82no—
228C2×Q8—82no—
229C2×Q8—84no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4C22⋊Q811yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274C22×C4#12411yescharacteristic
275C22×C4#12711yescharacteristic
276M16—44no—
277M16—44no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—81no—
286C2×C8—81no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C22⋊C4—82no—
306C42—44no—
307C42—44no—
308C42—82no—
309C42C2×C22⋊C411yescharacteristic
310C42#11711yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12411yescharacteristic
322C42#12411yescharacteristic
323C42#12611yescharacteristic
324C42#12611yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#38511yescharacteristic
330C23#39211yescharacteristic
331C23#39411yescharacteristic
332C23#45312yes—
333C23#45512yes—
334Q8—48no—
335Q8—84no—
336Q8—84no—
337Q8—84no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4#38011yescharacteristic
387C2×C4#38511yescharacteristic
388C2×C4#38611yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#39011yescharacteristic
392C2×C4#39311yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39611yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4#39911yescharacteristic
398C2×C4#45512yes—
399C2×C4#45612yes—
400C8—44no—
401C8—44no—
402C8—82no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22#143911yescharacteristic
412C22#185612yes—
413C22#200211yescharacteristic
414C22#200411yescharacteristic
415C22#200411yescharacteristic
416C22#200511yescharacteristic
417C22#200911yescharacteristic
418C22#200911yescharacteristic
419C22#203712yes—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—84no—
431C4—84no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#188212yes—
437C4#202812yes—
438C2—22no—
439C2—22no—
440C2#1009011yescharacteristic
441C2#1009111yescharacteristic
442C2#1009811yescharacteristic
443C2#1009911yescharacteristic
444C2#1106411yescharacteristic
445C2#1113011yescharacteristic
446C2#1113511yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1558C4semidirect

Maximal subgroups

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

subgroupindex
#219862
#212022
C2×#15582
#142822
C2×#13002
#80332
#80222