Finite Group Databasebeta

← #195597

Subgroups — #195597

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5C22×#333C211yesmaximal, characteristic
6#9066C214yesmaximal, complemented
7#9060C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2369C2214yes—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1989—22no—
32#1988C2212yes—
33#1987C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1892—22no—
37#1803—22no—
38#1735—21no—
39#1735—21no—
40C2×#333—24no—
41C2×#333C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8D412yes—
45C22×Q16—24no—
46C23×C8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—24no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#494—24no—
76#494C2×C418yes—
77#487—24no—
78#487C2×C418yes—
79#462—216no—
80#451—216no—
81#431—22no—
82#431C2×C414yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—24no—
93C2×#103—24no—
94C2×C22⋊C8—21no—
95C2×C4×C8C2312yes—
96#391—24no—
97#391—28no—
98#389—24no—
99#389—24no—
100#389—24no—
101#387—28no—
102#384D412yes—
103#382—21no—
104#380—24no—
105#380—21no—
106#380D412yes—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#376—21no—
118C43—21no—
119#340—24no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#333—28no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—84no—
130C22×Q8C22⋊C418yes—
131C22×Q8C2×D414yes—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×Q16—216no—
137C2×Q16—88no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—82no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8C22⋊C418yes—
160C4⋊Q8C22×C412yes—
161C4⋊Q8C22×C412yes—
162C4⋊Q8C22×C414yes—
163C4⋊Q8C2×D414yes—
164#125—24no—
165#125—42no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—88no—
175#117—22no—
176#117—24no—
177#117—22no—
178#117—22no—
179#117—42no—
180#117—42no—
181#117—42no—
182#117Q1614yescomplemented
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—28no—
213C2×C42—41no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229#103—216no—
230#103—84no—
231C22⋊C8—24no—
232C4×C8—22no—
233C4×C8C22×C414yes—
234#95—28no—
235#95—28no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24#12011yescharacteristic
241C2×Q8—24no—
242C2×Q8—28no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—88no—
253C2×Q8—88no—
254C2×Q8C2×C22⋊C418yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—84no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12312yes—
275C22×C4C2×Q1612yes—
276C22×C4#13512yes—
277C22×C4#13712yes—
278C22×C4#13712yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282Q16—816no—
283C2×C8—24no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—44no—
290C2×C8—84no—
291C2×C8—84no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—88no—
316C4⋊C4C2×Q1614yes—
317C22⋊C4—24no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C42—24no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—44no—
328C42—42no—
329C42—88no—
330C42C2×C22⋊C412yes—
331C42C4×D412yes—
332C42C4×D412yes—
333C42#12112yes—
334C42#12112yes—
335C42#12312yes—
336C42C23×C411yescharacteristic
337C42C22×D411yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C23—22no—
341C23—22no—
342C23#38612yes—
343C23#45112yes—
344C23#45212yes—
345C23C2×#12011yescharacteristic
346Q8—48no—
347Q8—416no—
348Q8—44no—
349Q8—44no—
350Q8—84no—
351Q8—82no—
352Q8—81no—
353Q8—81no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—88no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#38612yes—
379C2×C4#38612yes—
380C2×C4#38612yes—
381C2×C4C4×Q1614yes—
382C2×C4#44114yes—
383C2×C4#45112yes—
384C2×C4#45212yes—
385C2×C4#46214yes—
386C2×C4#46414yes—
387C2×C4C22×C22⋊C411yescharacteristic
388C2×C4C2×C4×D411yescharacteristic
389C2×C4C2×C4×D411yescharacteristic
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×#12111yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C22×Q1611yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4#57712yes—
398C2×C4#57812yes—
399C8—44no—
400C8—82no—
401C8—82no—
402C22—24no—
403C22—24no—
404C22—22no—
405C22—22no—
406C22#187814yes—
407C22C2×#38611yescharacteristic
408C22C2×#45111yescharacteristic
409C22C2×#45211yescharacteristic
410C22#301312yes—
411C22#301912yes—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—22no—
416C4—44no—
417C4—48no—
418C4—82no—
419C4—84no—
420C4—82no—
421C4—82no—
422C4#187814yes—
423C4C2×C4×Q1611yescharacteristic
424C4C2×#44111yescharacteristic
425C4#294712yes—
426C4C2×#46211yescharacteristic
427C4C2×#46411yescharacteristic
428C4#304311yescharacteristic
429C4#304411yescharacteristic
430C2—24no—
431C2#1887312yes—
432C2#1887712yes—
433C2C2×#187811yescharacteristic
434C2#2093011yescharacteristic
435C2#2095311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9066C2semidirect
7. #9060C2semidirect
13. #3209C4semidirect
182. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#342162
#218852
#213112
C22×#3332
#90662
#90602
#78932
#78912