Finite Group Databasebeta

← #192581

Subgroups — #192581

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#17496C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8098C211yesmaximal, characteristic
9#7922C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983—22no—
15#2962—22no—
16#2961C2214yes—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1940—22no—
25#1940C2214yes—
26#1797—22no—
27#1572—22no—
28#1571C2212yes—
29#1570C2212yes—
30#1568—22no—
31#1503C414yes—
32#1503C2212yes—
33#1503C2212yes—
34C2×#328C412yes—
35C2×#328C412yes—
36#1497—22no—
37#1491—22no—
38#1481C2214yes—
39#1303C2214yes—
40#1281C2212yes—
41#1281C2212yes—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533D412yes—
48#533Q812yes—
49#533C2312yes—
50C2×C4⋊Q8Q811yescharacteristic
51C2×C4⋊Q8Q811yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—21no—
56C2×C4×Q8D412yes—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#446—28no—
62#446—24no—
63#446—28no—
64#446—28no—
65C8×Q8—24no—
66#433—28no—
67#432—24no—
68#432—21no—
69#432—21no—
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432D412yes—
80#431—24no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431C2×C412yes—
86#431D412yes—
87#431C2312yes—
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#424—24no—
91#423—21no—
92#423C2312yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8C2312yes—
97#407—21no—
98#407—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8C2×C412yes—
101C2×C22⋊C8D412yes—
102#405—24no—
103C4×M16—21no—
104C4×M16—21no—
105C2×#97—21no—
106C2×#97C2312yes—
107C2×C4×C8—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2312yes—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114#336—24no—
115#335—24no—
116#334—24no—
117#328C2×C414yes—
118#328C2×C414yes—
119#328C2×C414yes—
120#328C2×C414yes—
121Q8⋊C8—28no—
122C22×Q8—21no—
123C22×Q8C4⋊C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4111yescharacteristic
129C2×M16—22no—
130C2×M16—41no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D414yes—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8C4⋊C412yes—
139C4⋊Q8C4⋊C412yes—
140C4⋊Q8C2×Q812yes—
141C4⋊Q8C2×Q812yes—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8C4⋊C414yes—
153C22⋊Q8C22×C414yes—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8C4⋊C414yes—
161C4×Q8C2×D414yes—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C4⋊C412yes—
175C2×C22⋊C4—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—82no—
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×Q811yescharacteristic
187C2×C42C2×Q811yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2411yescharacteristic
192C4⋊C8—28no—
193C4⋊C8—28no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—41no—
204C4⋊C8—84no—
205C4⋊C8C2×D414yes—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8C22×C412yes—
216C22⋊C8C22×C412yes—
217#97—28no—
218#97—22no—
219#97—22no—
220#97—22no—
221#97—41no—
222#97—41no—
223#97—41no—
224C4×C8—28no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—22no—
230#95—21no—
231#95—21no—
232#95C22×C412yes—
233#95C22×C412yes—
234C24C4×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8C2×C4⋊C414yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4C4×D411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12011yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4C22⋊Q811yescharacteristic
268C22×C4#12311yescharacteristic
269C22×C4#12311yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273M16—42no—
274M16—42no—
275M16—82no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—82no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4C2×C4⋊C414yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C42—24no—
304C42—24no—
305C42—44no—
306C42—82no—
307C42C2×C4⋊C411yescharacteristic
308C42C2×C4⋊C411yescharacteristic
309C42C4×D411yescharacteristic
310C42C4×D411yescharacteristic
311C42#12011yescharacteristic
312C42#12011yescharacteristic
313C42C22⋊Q811yescharacteristic
314C42C22⋊Q811yescharacteristic
315C42C22⋊Q811yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42#12311yescharacteristic
318C42#12311yescharacteristic
319C42C22×D411yescharacteristic
320C42C22×D411yescharacteristic
321C42C22×Q811yescharacteristic
322C42C2×#4111yescharacteristic
323C23—22no—
324C23—21no—
325C23—21no—
326C23—41no—
327C23#38511yescharacteristic
328C23#38511yescharacteristic
329C23C2×C4×D411yescharacteristic
330Q8—44no—
331Q8—44no—
332Q8—82no—
333Q8—82no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4#38511yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38511yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4C22×C4⋊C411yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C2×#12011yescharacteristic
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4C2×#12311yescharacteristic
376C2×C4#57712yes—
377C2×C4#57812yes—
378C8—42no—
379C8—42no—
380C8—44no—
381C8—44no—
382C8—82no—
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22#186112yes—
394C22C2×#38511yescharacteristic
395C22#294911yescharacteristic
396C22#295011yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—44no—
408C4—44no—
409C4—44no—
410C4—82no—
411C4#301312yes—
412C4#308212yes—
413C2—21no—
414C2—22no—
415C2—21no—
416C2—41no—
417C2C2×#186111yescharacteristic
418C2#2088011yescharacteristic
419C2#2089211yescharacteristic
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
#219552
#178162
C2×#15032
#174962
#88812
#88792
#80982
#79222