Finite Group Databasebeta

← #192554

Subgroups — #192554

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#8881C214yesmaximal, complemented
7#8879C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic
9#7920C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11Q8×M16—22no—
12Q8×M16—22no—
13#2962—22no—
14#2962—22no—
15#2961C2214yes—
16#2960—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2905—21no—
21C2×#423C2211yescharacteristic
22#2873—21no—
23C2×C4×M16C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1940C2212yes—
26#1940C2212yes—
27#1851—21no—
28#1851—21no—
29C4×C4⋊C8—22no—
30#1574—22no—
31#1571C2212yes—
32#1570C2212yes—
33#1565—22no—
34#1508C414yes—
35#1506C412yes—
36#1506C412yes—
37#1503C2214yes—
38#1485—22no—
39#1481C2212yes—
40#1481C2212yes—
41#1479—22no—
42#1303C2214yes—
43#1281C2214yes—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C22×M16D411yescharacteristic
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533C2×C412yes—
52#533D412yes—
53#533Q812yes—
54#533C2312yes—
55C2×C4⋊Q8Q811yescharacteristic
56C2×C4⋊Q8Q811yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8D412yes—
62C2×#117—21no—
63C2×#117—21no—
64C22×C42C2311yescharacteristic
65#446—24no—
66#446—24no—
67#446—28no—
68#446—28no—
69C8×Q8—24no—
70C8×Q8—24no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432C2×C412yes—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431—21no—
80#431—21no—
81#431C2312yes—
82#431C2312yes—
83#424—24no—
84#424—24no—
85#423—24no—
86#423—21no—
87#423—21no—
88#423—21no—
89#423—21no—
90#423D412yes—
91#423C2312yes—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98#407—21no—
99#407C2×C412yes—
100C2×C22⋊C8—21no—
101#405—24no—
102#405—24no—
103C4×M16—24no—
104C4×M16—21no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16—21no—
108C4×M16—21no—
109C4×M16—21no—
110C2×#97C2312yes—
111C2×C4×C8—21no—
112C2×C4×C8D412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4C2×C412yes—
117#376C2×C411yescharacteristic
118#376C2×C411yescharacteristic
119C43—21no—
120#336—22no—
121#336—22no—
122#335—24no—
123#334—24no—
124#328C2×C418yes—
125#328C2×C418yes—
126Q8⋊C8—24no—
127Q8⋊C8—24no—
128C22×Q8—21no—
129C22×Q8C4⋊C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4#4111yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—24no—
141C2×M16—42no—
142C2×M16—44no—
143C2×M16—41no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8C4⋊C412yes—
154C4⋊Q8C4⋊C412yes—
155C4⋊Q8C2×Q812yes—
156C4⋊Q8C2×Q812yes—
157#125—22no—
158#125—22no—
159#125—22no—
160#125—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8C4⋊C414yes—
168C22⋊Q8C22×C414yes—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8C4⋊C414yes—
176C4×Q8C2×D414yes—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C4⋊C412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×Q811yescharacteristic
203C2×C42C2×Q811yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222C4⋊C8—41no—
223C4⋊C8C2×D414yes—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8C22×C414yes—
230#97—28no—
231#97—22no—
232#97—22no—
233#97—22no—
234#97—41no—
235C4×C8—28no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—41no—
241C4×C8—41no—
242#95—21no—
243#95—21no—
244#95C22×C412yes—
245#95C22×C412yes—
246C24C4×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—82no—
252C2×Q8C2×C4⋊C414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4C22⋊Q811yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4#12311yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4111yescharacteristic
279M16—24no—
280M16—24no—
281M16—44no—
282M16—42no—
283M16—42no—
284M16—44no—
285M16—44no—
286M16—82no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—24no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—44no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8#13714yes—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—44no—
313C4⋊C4C2×C4⋊C414yes—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—44no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—44no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42C2×C4⋊C411yescharacteristic
325C42C2×C4⋊C411yescharacteristic
326C42C4×D411yescharacteristic
327C42C4×D411yescharacteristic
328C42#12011yescharacteristic
329C42#12011yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42#12311yescharacteristic
335C42#12311yescharacteristic
336C42C22×D411yescharacteristic
337C42C22×D411yescharacteristic
338C42C22×Q811yescharacteristic
339C42C2×#4111yescharacteristic
340C23—22no—
341C23—41no—
342C23—41no—
343C23#38511yescharacteristic
344C23#38511yescharacteristic
345C23C2×C4×D411yescharacteristic
346Q8—44no—
347Q8—44no—
348Q8—44no—
349Q8—82no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4#38511yescharacteristic
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4C22×C4⋊C411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×#12011yescharacteristic
388C2×C4C2×C22⋊Q811yescharacteristic
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×#13712yes—
392C2×C4#57512yes—
393C8—24no—
394C8—44no—
395C8—42no—
396C8—44no—
397C8—82no—
398C8—81no—
399C8—81no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—82no—
409C22C2×#38511yescharacteristic
410C22C4×#13712yes—
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#186112yes—
424C4#186812yes—
425C4#300411yescharacteristic
426C4#300511yescharacteristic
427C4#307812yes—
428C2—22no—
429C2—41no—
430C2—41no—
431C2C2×#186111yescharacteristic
432C2#2087411yescharacteristic
433C2C2×#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
#601922
#213502
#178092
#175452
#174642
#88812
#88792
#80972
#79202