Finite Group Databasebeta

← #121545

Subgroups — #121545

Order 512.

648 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21961C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5#10849C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2285C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1948—22no—
16#1947—21no—
17#1947—21no—
18#1947—21no—
19#1947—21no—
20#1853—22no—
21#1844—21no—
22#1836—22no—
23#1834—21no—
24#1834—21no—
25#1834—21no—
26#1808—22no—
27#1736—22no—
28C2×#340—22no—
29C2×#339—22no—
30#1294—22no—
31#1283C414yescomplemented
32C22×M16D412yes—
33C23×C8D412yes—
34C2×#117—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#431—21no—
43#431—21no—
44#431C2×C412yes—
45#431Q812yes—
46C2×#107—22no—
47C2×#107—22no—
48C2×#106—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2×C412yes—
55C2×C4⋊C8Q812yes—
56C2×C22⋊C8D412yes—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16C2×C412yes—
60C4×M16Q812yes—
61C2×#97—21no—
62C2×C4×C8—21no—
63C2×C4×C8—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384D411yescharacteristic
68#384D411yescharacteristic
69#384D411yescharacteristic
70#384D412yes—
71#384D411yescharacteristic
72#382—21no—
73#382—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4Q812yes—
86C2×#95—21no—
87#340—22no—
88#340—22no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—44no—
93#339—44no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16C4⋊C414yes—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8C4⋊C414yes—
112#117—42no—
113#117—42no—
114#117—42no—
115#117—42no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—82no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C42—22no—
148C2×C42—42no—
149C2×C42—42no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42C22×C411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165#107—82no—
166#107—82no—
167#106—84no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8C4⋊C414yes—
176#97—22no—
177#97—22no—
178C4×C8—22no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182#95—22no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C24#12711yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4C2×C4⋊C411yescharacteristic
212C22×C4C2×C4⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4C4⋊Q811yescharacteristic
227C22×C4C4⋊Q811yescharacteristic
228C22×C4C4⋊Q811yescharacteristic
229C22×C4#13512yes—
230C22×C4#13512yes—
231C22×C4AGL(1,ℤ/8)12yes—
232C22×C4#13712yes—
233M16—24no—
234M16—24no—
235M16—42no—
236M16—42no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—44no—
247C2×C8—82no—
248C4⋊C4—24no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—82no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—82no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C42—44no—
280C42—82no—
281C42—82no—
282C42—82no—
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42C22⋊Q811yescharacteristic
290C42C22⋊Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42#12511yescharacteristic
294C42#12511yescharacteristic
295C42#12511yescharacteristic
296C42C4⋊Q811yescharacteristic
297C42C4⋊Q811yescharacteristic
298C42C4⋊Q811yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23#38411yescharacteristic
304C23#39211yescharacteristic
305C23#39911yescharacteristic
306C23#49511yescharacteristic
307C23#49511yescharacteristic
308C23#49611yescharacteristic
309C23#49711yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—82no—
350C2×C4—162no—
351C2×C4—162no—
352C2×C4#38511yescharacteristic
353C2×C4#38511yescharacteristic
354C2×C4#38511yescharacteristic
355C2×C4#38911yescharacteristic
356C2×C4#38911yescharacteristic
357C2×C4#38911yescharacteristic
358C2×C4#39511yescharacteristic
359C2×C4#39511yescharacteristic
360C2×C4#39511yescharacteristic
361C2×C4#39811yescharacteristic
362C2×C4#39811yescharacteristic
363C2×C4#39811yescharacteristic
364C2×C4#42712yes—
365C2×C4#42812yes—
366C2×C4#44912yes—
367C2×C4#45212yes—
368C2×C4#45311yescharacteristic
369C2×C4#45411yescharacteristic
370C2×C4#45511yescharacteristic
371C2×C4#45611yescharacteristic
372C2×C4#46312yes—
373C2×C4#46412yes—
374C2×C4#47812yes—
375C2×C4#47912yes—
376C2×C4#48212yes—
377C2×C4#48512yes—
378C2×C4#49912yes—
379C2×C4#50112yes—
380C8—44no—
381C8—44no—
382C8—44no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22#144411yescharacteristic
396C22#184611yescharacteristic
397C22#184711yescharacteristic
398C22#200611yescharacteristic
399C22#200711yescharacteristic
400C22#200811yescharacteristic
401C22#200911yescharacteristic
402C22#201211yescharacteristic
403C22#201511yescharacteristic
404C22#209211yescharacteristic
405C22#209711yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—84no—
413C4—162no—
414C4—161no—
415C4—161no—
416C4—161no—
417C4—161no—
418C4#185412yes—
419C4#185611yescharacteristic
420C4#185711yescharacteristic
421C4#194912yes—
422C4#205912yes—
423C4#206011yescharacteristic
424C4#206111yescharacteristic
425C4#208512yes—
426C2—21no—
427C2—21no—
428C2—21no—
429C2—21no—
430C2#1019511yescharacteristic
431C2#1019611yescharacteristic
432C2#1020011yescharacteristic
433C2#1020711yescharacteristic
434C2#1132511yescharacteristic
435C2#1133311yescharacteristic
436C2#1134211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
31. #1283C4semidirect

Maximal subgroups

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

subgroupindex
#219812
#219612
#207642
#108512
#108492
#80142
C2×#12832