Finite Group Databasebeta

← #106721

Subgroups — #106721

Order 512.

915 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#413C211yesmaximal, characteristic
4#8073C212yesmaximal, complemented
5#7895C212yesmaximal, complemented
6C2×C4×#41C2211yescharacteristic
7#2990—24no—
8#2990—24no—
9C2×#446—21no—
10C2×#435—21no—
11C2×#435—21no—
12C2×C8×D4—21no—
13C2×#432C411yescharacteristic, complemented
14C2×#432C2211yescharacteristic
15C2×#431C411yescharacteristic, complemented
16C2×#423C411yescharacteristic, complemented
17C22×C4⋊C8C411yescharacteristic, complemented
18C22×C4⋊C8C2211yescharacteristic
19C2×#413—21no—
20C2×C4×M16C2212yes—
21#2283C2212yes—
22#1820—21no—
23#1799C412yes—
24#1799C412yes—
25#1782—21no—
26C4×#101—21no—
27C4×#101C412yescomplemented
28C4×#100—21no—
29C4×#100C412yescomplemented
30C4×#99—21no—
31C4×#99—21no—
32#1742—21no—
33#1300—24no—
34#1279—24no—
35C22×#41D411yescharacteristic
36C22×M16—21no—
37C22×M16—22no—
38C23×C8—21no—
39C23×C8—21no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C4×#41—22no—
44C2×C4×Q8C2×C411yescharacteristic
45C2×C4×Q8D411yescharacteristic
46C2×C4×D4C2×C411yescharacteristic
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D411yescharacteristic
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×#117D411yescharacteristic
53C2×#117Q811yescharacteristic
54C2×#117Q811yescharacteristic
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2311yescharacteristic
58#446—28no—
59#435—28no—
60#435—28no—
61C8×D4—28no—
62#432—22no—
63#432—22no—
64#432—22no—
65#432D414yes—
66#431—22no—
67#431—22no—
68#423—22no—
69#423—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8C2×C411yescharacteristic
77C2×C4⋊C8C2×C411yescharacteristic
78C2×C4⋊C8C2×C411yescharacteristic
79C2×C4⋊C8C2×C411yescharacteristic
80C2×C4⋊C8D414yes—
81#413—28no—
82C2×#101—22no—
83C2×#100—22no—
84C2×#99—22no—
85C2×#99—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C4×M16C2×C414yes—
93C2×#97—21no—
94C2×#97—21no—
95C2×#97—22no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98#383—22no—
99#382—22no—
100#381—22no—
101#380C2×C414yes—
102C4×C4⋊C4—22no—
103C4×C22⋊C4—22no—
104#376—22no—
105#343—22no—
106#343—22no—
107#342—22no—
108#342—22no—
109C2×#41—24no—
110C2×#41—22no—
111C2×#41—22no—
112C2×#41—42no—
113C22×Q8C22⋊C411yescharacteristic
114C22×D4C22⋊C411yescharacteristic
115C22×D4C22⋊C411yescharacteristic
116C22×D4C22⋊C411yescharacteristic
117C23×C4C4211yescharacteristic
118C23×C4C4211yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C411yescharacteristic
121C23×C4C4⋊C411yescharacteristic
122C23×C4C4⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C2×M16—24no—
127C2×M16—28no—
128C2×M16—24no—
129C2×M16—44no—
130C2×M16—42no—
131C2×M16—82no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—42no—
139C22×C8—82no—
140C22×C8—82no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×D4—22no—
146C4×D4—22no—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C4⋊C411yescharacteristic
167C2×C4⋊C4C4⋊C411yescharacteristic
168C2×C4⋊C4C4⋊C411yescharacteristic
169C2×C4⋊C4C4⋊C411yescharacteristic
170C2×C22⋊C4—24no—
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C4⋊C411yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42C4211yescharacteristic
187C2×C42C4211yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C4⋊C411yescharacteristic
191C2×C42C4⋊C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4112yes—
199C2×C42#4112yes—
200C4⋊C8—24no—
201C4⋊C8—24no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—24no—
207C4⋊C8C22⋊C412yes—
208C4⋊C8C22⋊C412yes—
209C4⋊C8C22⋊C412yes—
210C4⋊C8C22⋊C412yes—
211#101—28no—
212#100—28no—
213#99—44no—
214#99—44no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218#97—24no—
219#97—24no—
220#97—24no—
221#97—42no—
222C4×C8—24no—
223C4×C8—24no—
224C4×C8—42no—
225#95—24no—
226#95—42no—
227#95—42no—
228#95—42no—
229C24#9511yescharacteristic
230C24#9511yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232#41—48no—
233#41—48no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C2×D4#10414yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4#9511yescharacteristic
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9511yescharacteristic
278C22×C4#9511yescharacteristic
279C22×C4#9511yescharacteristic
280C22×C4C2×C4211yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4#12012yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292M16—28no—
293M16—48no—
294M16—44no—
295M16—84no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—48no—
305C2×C8—44no—
306C2×C8—84no—
307C2×C8—84no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—48no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—82no—
336C42—82no—
337C42C2×C22⋊C411yescharacteristic
338C42C2×C22⋊C411yescharacteristic
339C42#11711yescharacteristic
340C42#11711yescharacteristic
341C42C4×D412yes—
342C42C4×D412yes—
343C42#12012yes—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42#12312yes—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#33912yes—
359C23C2×#9511yescharacteristic
360C23#38612yes—
361C23C2×#10412yes—
362C23#42112yes—
363Q8—44no—
364D4—28no—
365D4—44no—
366D4—44no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4#33912yes—
414C2×C4C4×C22⋊C411yescharacteristic
415C2×C4C4×C22⋊C411yescharacteristic
416C2×C4#37911yescharacteristic
417C2×C4#38011yescharacteristic
418C2×C4#38011yescharacteristic
419C2×C4#38111yescharacteristic
420C2×C4#38512yes—
421C2×C4#38512yes—
422C2×C4#38612yes—
423C8—48no—
424C8—42no—
425C8—42no—
426C8—42no—
427C8—42no—
428C22—21no—
429C22—22no—
430C22—21no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—81no—
443C22—81no—
444C22C22⋊#9511yescharacteristic
445C22C2×#33911yescharacteristic
446C22#173311yescharacteristic
447C22#187114yes—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—164no—
462C4#175712yes—
463C4#179511yescharacteristic
464C4#179611yescharacteristic
465C2—24no—
466C2—21no—
467C2—21no—
468C2—42no—
469C2—81no—
470C2#1073211yescharacteristic
471C2#1076411yescharacteristic
472C2#1076511yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC22×#41D4
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. #8073C2semidirect
5. #7895C2semidirect
13. C2×#432C4semidirect
15. C2×#431C4semidirect
16. C2×#423C4semidirect
17. C22×C4⋊C8C4semidirect
27. C4×#101C4semidirect
29. C4×#100C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#207232
C4×#4132
#80732
#78952