Finite Group Databasebeta

← #106154

Subgroups — #106154

Order 512.

849 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2965C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#7973C212yesmaximal, complemented
5#7956C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2965—24no—
8#2965—24no—
9C2×#446—21no—
10C2×#436—21no—
11C2×#435—21no—
12C2×#435—21no—
13#2920C2212yes—
14C2×#431C411yescharacteristic, complemented
15C2×#431C411yescharacteristic, complemented
16C2×#431C2211yescharacteristic
17C2×C4×M16C411yescharacteristic, complemented
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C411yescharacteristic, complemented
20#2283C2212yes—
21#1820C412yes—
22#1799—21no—
23#1799—21no—
24#1782C412yes—
25C4×#99—21no—
26C4×#99—21no—
27#1465—21no—
28#1465—21no—
29#1464C412yescomplemented
30#1464C412yescomplemented
31#1461—21no—
32#1460—21no—
33#1288—24no—
34#1286—24no—
35C22×#41C2×C411yescharacteristic
36C22×M16—21no—
37C22×M16—21no—
38C23×C8—21no—
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C2×C4×Q8D411yescharacteristic
44C2×C4×Q8Q811yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4D411yescharacteristic
50C2×C4×D4Q811yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2311yescharacteristic
57#446—28no—
58#436—28no—
59#435—28no—
60#435—28no—
61#432—22no—
62#432C2×C414yes—
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431D414yes—
71#423—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×#99—22no—
76C2×#99—22no—
77#407—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C4×M16—22no—
82C4×M16—22no—
83C4×M16—22no—
84C4×M16—22no—
85C4×M16D414yes—
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8C2×C411yescharacteristic
92C2×C4×C8C2×C411yescharacteristic
93C2×C4×C8C2×C411yescharacteristic
94C2×C4×C8C2×C411yescharacteristic
95#383—22no—
96#382—22no—
97#381C2×C414yes—
98#380—22no—
99C4×C4⋊C4—22no—
100C4×C22⋊C4—22no—
101#376—22no—
102#342—22no—
103#342—22no—
104#324—22no—
105#324—22no—
106#323—22no—
107#323—22no—
108C2×#41—22no—
109C2×#41—22no—
110C2×#41—42no—
111C2×#41—42no—
112C22×Q8C4⋊C411yescharacteristic
113C22×D4C22⋊C411yescharacteristic
114C22×D4C22⋊C411yescharacteristic
115C22×D4C4⋊C411yescharacteristic
116C23×C4C4211yescharacteristic
117C23×C4C4211yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C4⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—44no—
128C2×M16—44no—
129C2×M16—42no—
130C22×C8—21no—
131C22×C8—24no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—44no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×D4—22no—
141C4×D4—22no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152#117—22no—
153#117—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C4⋊C4C4⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22⋊C411yescharacteristic
167C2×C22⋊C4C22⋊C411yescharacteristic
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C4⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42C4211yescharacteristic
182C2×C42C4211yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C4⋊C411yescharacteristic
186C2×C42C4⋊C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4112yes—
194C2×C42#4112yes—
195C4⋊C8—24no—
196C4⋊C8—24no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—42no—
200#99—44no—
201#99—44no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—42no—
207#97—24no—
208#97—24no—
209#97—24no—
210C4×C8—24no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—42no—
216C4×C8C22⋊C412yes—
217C4×C8C22⋊C412yes—
218C4×C8C22⋊C412yes—
219C4×C8C22⋊C412yes—
220#95—24no—
221#95—24no—
222#95—42no—
223#95—42no—
224C24#9511yescharacteristic
225C24#9511yescharacteristic
226C24C2×C22⋊C411yescharacteristic
227#41—48no—
228#41—48no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—41no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4#9511yescharacteristic
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4C2×C4211yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12012yes—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12312yes—
284M16—48no—
285M16—48no—
286M16—84no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—48no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—44no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—84no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—84no—
325C42C2×C22⋊C411yescharacteristic
326C42C2×C22⋊C411yescharacteristic
327C42#11711yescharacteristic
328C42#11711yescharacteristic
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12012yes—
332C42C22⋊Q812yes—
333C42C22⋊Q812yes—
334C42#12312yes—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23#34412yes—
349C23C2×#9511yescharacteristic
350C23#38612yes—
351Q8—44no—
352D4—44no—
353D4—44no—
354D4—44no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4#34412yes—
398C2×C4C4×C22⋊C411yescharacteristic
399C2×C4C4×C22⋊C411yescharacteristic
400C2×C4#37911yescharacteristic
401C2×C4#38011yescharacteristic
402C2×C4#38011yescharacteristic
403C2×C4#38111yescharacteristic
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4#38612yes—
407C8—24no—
408C8—42no—
409C8—42no—
410C8—42no—
411C8—84no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C22⋊#9511yescharacteristic
430C22C2×#34411yescharacteristic
431C22C2×#34411yescharacteristic
432C4—21no—
433C4—22no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—164no—
446C4#181212yes—
447C4#182712yes—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—42no—
452C2—81no—
453C2#1072911yescharacteristic
454C2#1075811yescharacteristic
455C2#1075911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7973C2semidirect
5. #7956C2semidirect
14. C2×#431C4semidirect
15. C2×#431C4semidirect
17. C2×C4×M16C4semidirect
19. C22×C4×C8C4semidirect
29. #1464C4semidirect
30. #1464C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29652
#207232
#173662
#79732
#79562