Finite Group Databasebeta

← #194943

Subgroups — #194943

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2642C211yesmaximal, characteristic
2#34378C211yesmaximal, characteristic
3#21841C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5C2×#1470C211yesmaximal, characteristic
6#9023C214yesmaximal
7#9022C214yesmaximal
8#8022C211yesmaximal, characteristic
9#8013C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2642—24no—
15#2642C418yescomplemented
16C2×#400—22no—
17C2×#400C2214yes—
18#2382—22no—
19#2370—22no—
20#2345—21no—
21#2344—22no—
22#2344—21no—
23#2336—22no—
24#2330—22no—
25#2316C2212yes—
26#2313C2212yes—
27C2×#391—21no—
28C2×#391—21no—
29#2285C2211yescharacteristic
30#2283C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1993—22no—
33#1990—22no—
34#1988—22no—
35#1987—22no—
36#1824C2212yes—
37#1824C2214yes—
38#1824C2212yes—
39#1470—24no—
40C2×#324C2214yes—
41#1294C2212yes—
42#1294C2214yes—
43#1294C2212yes—
44C23×Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8C2×C414yes—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yes—
60C2×#125D414yes—
61C2×#117—22no—
62C2×#117—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432—22no—
73#431—22no—
74#431—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×#97—21no—
80C2×C4×C8—21no—
81#400—216no—
82#400—28no—
83#400C2×C4116yes—
84#391—28no—
85#391—28no—
86#389—22no—
87#389—24no—
88#389—22no—
89#389—24no—
90#389—24no—
91#387—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#384D412yes—
98#383—21no—
99#382—21no—
100#382D412yes—
101#381—22no—
102#381—21no—
103#380—22no—
104#380—21no—
105#380C2311yescharacteristic
106#380C2312yes—
107#380C2311yescharacteristic
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116#376—21no—
117#376—22no—
118#376C2×C414yes—
119C2×#95—21no—
120C2×#95—24no—
121C2×#95—24no—
122C2×#95C2312yes—
123#340—28no—
124#340—24no—
125#340—24no—
126#324—28no—
127C22×Q8—24no—
128C22×Q8—48no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8—42no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149#125—24no—
150#125—24no—
151#125—24no—
152#125—24no—
153#125—24no—
154#125—24no—
155#125—24no—
156#125—24no—
157#117—82no—
158#117—84no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C22⋊C414yes—
187C2×C4⋊C4C22⋊C418yes—
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—44no—
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217#97—42no—
218C4×C8—42no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—216no—
225#95—216no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95C22×C414yes—
235C24C22×D411yescharacteristic
236C2×Q8—416no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—84no—
240C2×Q8—88no—
241C2×Q8—84no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C22⋊C414yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C414yes—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—82no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—48no—
312C42—44no—
313C42—82no—
314C42—84no—
315C42—84no—
316C42C4×D412yes—
317C42C4×D412yes—
318C42C4×D412yes—
319C42C4×D412yes—
320C42#12112yes—
321C42#12112yes—
322C42#12112yes—
323C42#12112yes—
324C23—22no—
325C23—22no—
326C23C22×C22⋊C411yescharacteristic
327C23C2×#12011yescharacteristic
328C23C2×#12311yescharacteristic
329C23#57712yes—
330C23#57812yes—
331Q8—88no—
332Q8—88no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—28no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4#38612yes—
376C2×C4#38612yes—
377C2×C4#38612yes—
378C2×C4#38612yes—
379C2×C4#45414yes—
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12111yescharacteristic
384C8—82no—
385C8—82no—
386C22—24no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22C2×#38611yescharacteristic
393C22#289812yes—
394C22#300911yescharacteristic
395C22#301111yescharacteristic
396C22C2×#45412yes—
397C22#302812yes—
398C22#309712yes—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—44no—
408C4—84no—
409C4—82no—
410C4—84no—
411C4—82no—
412C4—84no—
413C4#188814yes—
414C2—22no—
415C2—22no—
416C2#1844112yes—
417C2#1844312yes—
418C2#2091311yescharacteristic
419C2C2×#188811yescharacteristic
420C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2642C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26422
#343782
#218412
#218342
C2×#14702
#90232
#90222
#80222
#80132