Finite Group Databasebeta

← #482105

Subgroups — #482105

Order 512.

1033 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#35700C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11883C214yesmaximal, complemented
5#11879C214yesmaximal, complemented
6#11700C212yesmaximal, complemented
7#11697C212yesmaximal
8C2×D4×Q8—21no—
9C2×D4×Q8C411yescharacteristic
10C2×#519C2211yescharacteristic
11#2982—21no—
12#2905—21no—
13C22×#101—21no—
14#2621C412yes—
15#2621C2211yescharacteristic
16#2546C411yescharacteristic
17#2546C2212yes—
18#2536C411yescharacteristic
19#2536C2212yes—
20#2429—22no—
21#2429—21no—
22#2415—22no—
23#2415—21no—
24#2392C412yes—
25#2392C2211yescharacteristic
26#2351C411yescharacteristic
27Q8×C22⋊C4—21no—
28#1801—22no—
29#1801C2214yes—
30#1742C2212yes—
31#1461—21no—
32#1461C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1346C2218yes—
36#1346C2214yes—
37#1345C2214yes—
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C22×M16—22no—
41D4×Q8—44no—
42D4×Q8—44no—
43D4×Q8—44no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—21no—
54#519—44no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×#117C2×C412yes—
67C2×#117D412yes—
68C2×#117D411yescharacteristic
69C2×#117C2311yescharacteristic
70C8×Q8—44no—
71#431—24no—
72#431—22no—
73#424—44no—
74#423—24no—
75C2×#101—24no—
76C2×#101—44no—
77#398—22no—
78#398—21no—
79#398C2×C412yes—
80#398C2×C412yes—
81#398C2×C412yes—
82#398C2312yes—
83#398C2311yescharacteristic
84#397—22no—
85#397—21no—
86#397C2×C411yescharacteristic
87#397C2×C412yes—
88#396—22no—
89#396—21no—
90#394—22no—
91#394—21no—
92#394C2×C412yes—
93#394C2311yescharacteristic
94#392C2×C411yescharacteristic
95#391—21no—
96#391—22no—
97#391—22no—
98#391—21no—
99#389—22no—
100#389—21no—
101#389—22no—
102#389—22no—
103#389—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#382C2×C412yes—
108#382C2×C411yescharacteristic
109#380—21no—
110#380—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115#343—22no—
116#343C2314yes—
117#324—24no—
118#324C2314yes—
119#323—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8D814yes—
130C22×Q8SD1614yes—
131C22×Q8C2×D412yes—
132C22×D4—21no—
133C22×D4C22×C411yescharacteristic
134C23×C4—22no—
135C23×C4—21no—
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C22⋊Q8—48no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C22⋊Q8—44no—
152C22⋊Q8—48no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C22⋊Q8—44no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—44no—
159C4×Q8—44no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—44no—
163C4×D4—42no—
164C4×D4—44no—
165C4×D4—48no—
166C4×D4—42no—
167C4×D4—42no—
168#117—24no—
169#117—42no—
170#117—44no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22×C411yescharacteristic
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C42—22no—
206C2×C42—21no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C4⋊C8—48no—
215C4⋊C8—44no—
216C4⋊C8—42no—
217#101—48no—
218C22⋊C8—24no—
219C4×C8—44no—
220C4×C8—42no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95C22×C411yescharacteristic
231#95C22×C412yes—
232#95C22×C411yescharacteristic
233C24—21no—
234C24C2×C22⋊C411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—48no—
242C2×Q8—48no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—816no—
250C2×Q8#10214yes—
251C2×Q8#10214yes—
252C2×Q8C2×D814yes—
253C2×Q8C2×SD1614yes—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4#10218yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×D812yes—
283C22×C4C2×SD1612yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—44no—
288M16—44no—
289M16—84no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—84no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4#9914yes—
316C22⋊C4—24no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C42—24no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—22no—
334C42—42no—
335C42—42no—
336C42—44no—
337C42—42no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×#10212yes—
350C23C22×C22⋊C411yescharacteristic
351Q8—28no—
352Q8—28no—
353Q8—44no—
354Q8—44no—
355Q8—88no—
356Q8—84no—
357D4—42no—
358D4—48no—
359D4—42no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—88no—
396C2×C4—84no—
397C2×C4C2×#9914yes—
398C2×C4C2×#9912yes—
399C2×C4C2×#10214yes—
400C2×C4C2×#10214yes—
401C2×C4C2×#10212yes—
402C2×C4#42112yes—
403C2×C4C22×D811yescharacteristic
404C2×C4C22×SD1611yescharacteristic
405C8—84no—
406C8—84no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—42no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22C22×#9911yescharacteristic
422C22C22×#10211yescharacteristic
423C22C2×#42111yescharacteristic
424C4—24no—
425C4—21no—
426C4—21no—
427C4—24no—
428C4—24no—
429C4—42no—
430C4—41no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—82no—
437C4—82no—
438C4—81no—
439C4#151914yes—
440C2—21no—
441C2—21no—
442C2—42no—
443C2—82no—
444C2C2×#151911yescharacteristic
445C2#3334911yescharacteristic
446C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#392C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20639C2semidirect
3. #17352C2semidirect
4. #11883C2semidirect
5. #11879C2semidirect
6. #11700C2semidirect

Maximal subgroups

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

subgroupindex
#357002
#206392
#173522
#118832
#118792
#117002
#116972