Finite Group Databasebeta

← #192551

Subgroups — #192551

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic
3#17807C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8092C211yesmaximal, characteristic
9#7921C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962—21no—
16#2962—21no—
17#2961C2212yes—
18#2961C2212yes—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C4×#117C2211yescharacteristic
26#1940C2214yes—
27#1850—22no—
28#1842—21no—
29#1831—21no—
30#1573—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1565—22no—
34#1508C412yes—
35#1508C412yes—
36#1506C414yes—
37#1503C2214yes—
38#1481—22no—
39#1481C2214yes—
40C2×Q8⋊C8—22no—
41#1303C2212yes—
42#1303C2212yes—
43#1281C2214yes—
44C23×C8—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533D412yes—
51#533Q812yes—
52#533C2312yes—
53C2×C4⋊Q8Q811yescharacteristic
54C2×C4⋊Q8Q811yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8D412yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#446—24no—
64#446—24no—
65#446—22no—
66#446—22no—
67#446—24no—
68#446—24no—
69#446—24no—
70#446—24no—
71C8×Q8—22no—
72C8×Q8—22no—
73#433—24no—
74#433—24no—
75#432—24no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432—21no—
80#432—21no—
81#432—21no—
82#432C2×C412yes—
83#432D412yes—
84#431—24no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431D412yes—
91#431C2312yes—
92#431C2312yes—
93#424—22no—
94#424—22no—
95#423—21no—
96#423C2311yescharacteristic
97#423C2311yescharacteristic
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—24no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2312yes—
108#407—21no—
109#407C2×C412yes—
110C2×C22⋊C8—21no—
111C2×C22⋊C8D412yes—
112#405—22no—
113#405—22no—
114C4×M16—21no—
115C4×M16—21no—
116C2×#97C2312yes—
117C2×C4×C8—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2312yes—
120C4×C22⋊C4—21no—
121C4×C22⋊C4C2×C411yescharacteristic
122C4×C22⋊C4C2×C411yescharacteristic
123#376C2×C412yes—
124C43—21no—
125#336—24no—
126#335—24no—
127#334—24no—
128#328C2×C418yes—
129#328C2×C418yes—
130Q8⋊C8—28no—
131C22×Q8—21no—
132C22×Q8C4⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—82no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C4⋊C412yes—
151C4⋊Q8C4⋊C412yes—
152C4⋊Q8C2×Q812yes—
153C4⋊Q8C2×Q812yes—
154#125—22no—
155#125—22no—
156#125—22no—
157#125—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8C4⋊C414yes—
165C22⋊Q8C22×C414yes—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8C4⋊C414yes—
173C4×Q8C2×D414yes—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—44no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C4⋊C412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×Q811yescharacteristic
201C2×C42C2×Q811yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—84no—
225C4⋊C8C2×D412yes—
226C4⋊C8C2×D412yes—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8C22×C414yes—
235#97—24no—
236#97—24no—
237#97—22no—
238#97—22no—
239#97—22no—
240#97—22no—
241C4×C8—24no—
242C4×C8—24no—
243C4×C8—22no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—41no—
247#95—22no—
248#95C22×C412yes—
249#95C22×C412yes—
250C24C4×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—44no—
255C2×Q8—82no—
256C2×Q8C2×C4⋊C414yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4C2×C4⋊C411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4C22⋊Q811yescharacteristic
275C22×C4C22⋊Q811yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12311yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283M16—44no—
284M16—42no—
285M16—42no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—44no—
296C2×C8—82no—
297C2×C8—82no—
298C2×C8—82no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4C2×C4⋊C414yes—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C42—24no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—44no—
321C42—42no—
322C42—42no—
323C42C2×C4⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42#12011yescharacteristic
328C42#12011yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C42C22×D411yescharacteristic
336C42C22×D411yescharacteristic
337C42C22×Q811yescharacteristic
338C42C2×#4111yescharacteristic
339C23—22no—
340C23—22no—
341C23—41no—
342C23#38511yescharacteristic
343C23#38511yescharacteristic
344C23C2×C4×D411yescharacteristic
345Q8—44no—
346Q8—44no—
347Q8—82no—
348Q8—82no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4#38511yescharacteristic
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4C22×C4⋊C411yescharacteristic
385C2×C4C2×C4×D411yescharacteristic
386C2×C4C2×#12011yescharacteristic
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×#12311yescharacteristic
390C2×C4#57712yes—
391C2×C4#57812yes—
392C8—44no—
393C8—42no—
394C8—42no—
395C8—42no—
396C8—42no—
397C8—82no—
398C8—82no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—44no—
407C22#186112yes—
408C22C2×#38511yescharacteristic
409C22#294712yes—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—44no—
420C4—44no—
421C4—44no—
422C4—82no—
423C4#186812yes—
424C4#301011yescharacteristic
425C4#301111yescharacteristic
426C4#308212yes—
427C2—22no—
428C2—22no—
429C2—41no—
430C2C2×#186111yescharacteristic
431C2#2087711yescharacteristic
432C2C2×#186811yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8881C2semidirect
7. #8879C2semidirect

Maximal subgroups

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

subgroupindex
#602512
#219762
#178072
#175452
C2×#14812
#88812
#88792
#80922
#79212