Finite Group Databasebeta

← #198159

Subgroups — #198159

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#17404C211yesmaximal, characteristic
6#9240C218yesmaximal
7#8010C211yesmaximal, characteristic
8#7930C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3237—22no—
12#3237—22no—
13#3237—22no—
14#3237—22no—
15C2×#501C412yescomplemented
16C2×#491C412yescomplemented
17C2×#485C2214yes—
18C2×#477C412yescomplemented
19C2×#475C412yescomplemented
20#2987—22no—
21#2985—22no—
22#2984—22no—
23#2983—22no—
24#2962—22no—
25#2961—22no—
26#2924C2214yes—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1994—22no—
33#1986—22no—
34#1826C2214yes—
35#1508C2214yes—
36#1506—22no—
37#1503—22no—
38#1472C2214yes—
39#1470—22no—
40#1467—22no—
41#1293C2214yes—
42#1282C2214yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533D412yes—
56#533D412yes—
57#533D412yes—
58#533D412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×#125—21no—
64C2×#125C2×C412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2312yes—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117C2312yes—
76C22×C42C2311yescharacteristic
77#501—24no—
78#501—24no—
79#491—24no—
80#491—24no—
81#485—28no—
82#485—28no—
83#477—24no—
84#477—24no—
85#475—24no—
86#475—24no—
87#446—28no—
88#446—28no—
89#446—28no—
90#446—24no—
91C8×Q8—24no—
92#433—28no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432—21no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—24no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#432D412yes—
105#432D412yes—
106#431—21no—
107#431—24no—
108#431—21no—
109#431—21no—
110#431D412yes—
111C2×#107C2×C414yes—
112C2×#106C2×C414yes—
113#424—24no—
114#423—21no—
115#423—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2×C412yes—
118C2×#103C2×C414yes—
119C2×#103C2×C414yes—
120#407C2312yes—
121#407C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2312yes—
124#405—24no—
125C4×M16—21no—
126C4×M16—21no—
127C2×#97—21no—
128C2×#97C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4C2312yes—
133#376—21no—
134C43—21no—
135#339—28no—
136#328—28no—
137#324—28no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8C22×C412yes—
142C23×C4#4111yescharacteristic
143C23×C4#4111yescharacteristic
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—22no—
147C22×C8—22no—
148C22×C8C22×C414yes—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C22⋊C412yes—
162C4⋊Q8C22⋊C412yes—
163C4⋊Q8C22⋊C412yes—
164C4⋊Q8C22⋊C412yes—
165#125—22no—
166#125—22no—
167#125—41no—
168#125—41no—
169#125—41no—
170#125—41no—
171#125C22⋊C414yes—
172#125C22⋊C414yes—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205#117—22no—
206#117—22no—
207#117—44no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—41no—
212#117C2×D414yes—
213#117C2×D414yes—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247#107—28no—
248#106—28no—
249C4⋊C8—28no—
250C4⋊C8—28no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259#103—28no—
260#103—28no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8C2×D414yes—
268C22⋊C8C2×D414yes—
269#97—28no—
270#97—22no—
271#97—22no—
272#97—22no—
273#97—22no—
274#97—22no—
275#97—41no—
276C4×C8—28no—
277C4×C8—22no—
278C4×C8—41no—
279#95—22no—
280#95—41no—
281C24C2×#4111yescharacteristic
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—44no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—82no—
293C2×Q8—82no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320M16—44no—
321M16—44no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4C2×C22⋊C414yes—
352C4⋊C4C2×C22⋊C414yes—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C42—24no—
360C42—24no—
361C42—44no—
362C42—44no—
363C42—42no—
364C42—42no—
365C42—82no—
366C42C2×C22⋊C411yescharacteristic
367C42C2×C22⋊C411yescharacteristic
368C42C2×C22⋊C411yescharacteristic
369C42C2×C22⋊C411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42#12011yescharacteristic
375C42#12011yescharacteristic
376C42#12011yescharacteristic
377C42#12011yescharacteristic
378C42#12111yescharacteristic
379C42#12111yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C23—22no—
383C23—41no—
384C23—41no—
385C23C2×C4×D411yescharacteristic
386C23C2×#12111yescharacteristic
387C23C2×#12311yescharacteristic
388Q8—44no—
389Q8—44no—
390Q8—82no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395Q8—82no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4C22×C22⋊C411yescharacteristic
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×#12111yescharacteristic
437C8—44no—
438C8—44no—
439C8—44no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—82no—
450C22C2×#38611yescharacteristic
451C22#295612yes—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—44no—
463C4—44no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4#190212yes—
468C4#288412yes—
469C2—22no—
470C2—41no—
471C2—41no—
472C2#2091811yescharacteristic
473C2#2098611yescharacteristic
474C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. C2×#501C4semidirect
16. C2×#491C4semidirect
18. C2×#477C4semidirect
19. C2×#475C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
#602512
#218442
#175452
#174042
#92402
#80102
#79302