Finite Group Databasebeta

← C2×#9336 #199567

Subgroups — C2×#9336 #199567

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C211yesmaximal, characteristic
2C2×#2987C211yesmaximal, characteristic
3C2×#1554C211yesmaximal, characteristic
4C2×#1534C211yesmaximal, characteristic
5C2×#1470C211yesmaximal, characteristic
6#9336C218yesmaximal, direct product factor
7C2×#1308C211yesmaximal, characteristic
8C2×#1287C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×#125C2211yescharacteristic
11#3236—22no—
12#3236—22no—
13#3236C414yes—
14#3236C2214yes—
15C2×#501—21no—
16C2×#491—21no—
17C2×#484C412yes—
18C2×#483C412yes—
19C2×#479—22no—
20#2987—22no—
21#2987—22no—
22#2987—22no—
23#2987C2214yes—
24C2×#446—22no—
25C2×#446—22no—
26C2×#433—22no—
27C2×#432C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#1554—22no—
33#1554C2214yes—
34C2×#333—21no—
35C2×#332—21no—
36#1534—22no—
37#1534C2214yes—
38C2×#330—22no—
39#1470—22no—
40#1470C2214yes—
41C2×#324—22no—
42#1308C2214yes—
43#1287C2214yes—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50#533D412yes—
51#533C2312yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×#125—21no—
62C2×#125—21no—
63C2×#125—21no—
64C2×#125—21no—
65C2×#125C2×C412yes—
66C2×#125C2312yes—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C412yes—
69C2×C4×Q8—21no—
70C2×C4×Q8D412yes—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C42C2311yescharacteristic
77#501—24no—
78#501—24no—
79#491—24no—
80#491—24no—
81#484—24no—
82#484C2×C418yes—
83#483—24no—
84#483C2×C418yes—
85#479—28no—
86#479—28no—
87#446—28no—
88#446—28no—
89#446—28no—
90#446—28no—
91#433—28no—
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—21no—
101#432—21no—
102#432C2×C412yes—
103#432D412yes—
104#432D412yes—
105#432C2312yes—
106#432C2312yes—
107#432C2312yes—
108#432C2312yes—
109#431—21no—
110#431—21no—
111#431—21no—
112#431C2312yes—
113C2×#107—22no—
114C2×#106—22no—
115C2×C4⋊C8—24no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×#103—22no—
122C2×#103—22no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8C2×C412yes—
127C2×C22⋊C8D412yes—
128C2×#97—22no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97D412yes—
133C2×C4×C8—22no—
134C2×C4×C8—21no—
135#333—24no—
136#332—24no—
137#330—28no—
138#324—28no—
139C22×Q8—21no—
140C22×Q8C22⋊C412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4#4111yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—41no—
150C22×C8—41no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8C22⋊C412yes—
156C4⋊Q8C22⋊C412yes—
157C4⋊Q8C2×D412yes—
158C4⋊Q8C2×D412yes—
159#125—22no—
160#125—22no—
161#125—22no—
162#125—22no—
163#125—22no—
164#125—22no—
165#125—22no—
166#125—22no—
167#125—22no—
168#125—22no—
169#125—22no—
170#125—22no—
171#125—41no—
172#125—41no—
173#125—41no—
174#125—41no—
175#125—41no—
176#125—41no—
177#125—41no—
178#125—41no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8C22⋊C414yes—
186C22⋊Q8C22×C414yes—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8C22⋊C414yes—
194C4×Q8C2×D414yes—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—22no—
230C2×C42—22no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#107—28no—
244#106—28no—
245C4⋊C8—216no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256#103—28no—
257#103—28no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8C22×C414yes—
267C22⋊C8C2×D414yes—
268#97—28no—
269#97—22no—
270#97—22no—
271#97—22no—
272#97—22no—
273#97—22no—
274#97—22no—
275#97—41no—
276#97C2×D414yes—
277C4×C8—28no—
278C4×C8—22no—
279C4×C8—41no—
280C24C4×D411yescharacteristic
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—44no—
285C2×Q8—82no—
286C2×Q8C2×C22⋊C414yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4C23×C411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—82no—
327C2×C8—82no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4C2×C22⋊C414yes—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C42—24no—
359C42—24no—
360C42C2×C22⋊C411yescharacteristic
361C42C2×C22⋊C411yescharacteristic
362C42C4×D411yescharacteristic
363C42C4×D411yescharacteristic
364C42#12011yescharacteristic
365C42#12011yescharacteristic
366C42#12111yescharacteristic
367C42#12111yescharacteristic
368C42#12111yescharacteristic
369C42#12111yescharacteristic
370C42#12311yescharacteristic
371C42#12311yescharacteristic
372C42C22×D411yescharacteristic
373C42C22×D411yescharacteristic
374C42C22×D411yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—22no—
378C23—22no—
379C23#38611yescharacteristic
380C23#38611yescharacteristic
381C23C2×C4×D411yescharacteristic
382Q8—44no—
383Q8—82no—
384Q8—82no—
385Q8—82no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4#38611yescharacteristic
421C2×C4#38611yescharacteristic
422C2×C4C22×C22⋊C411yescharacteristic
423C2×C4C2×C4×D411yescharacteristic
424C2×C4C2×#12011yescharacteristic
425C2×C4C2×#12111yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C8—44no—
429C8—44no—
430C8—44no—
431C8—82no—
432C8—82no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—44no—
440C22#188112yes—
441C22#190512yes—
442C22#191512yes—
443C22C2×#38611yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—44no—
457C4—44no—
458C4—82no—
459C4—82no—
460C2—41no—
461C2—41no—
462C2#933614yesdirect product factor
463C2C2×#188111yescharacteristic
464C2C2×#190511yescharacteristic
465C2C2×#191511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9336C2direct (N × H)
462. C2#9336direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#29872
C2×#15542
C2×#15342
C2×#14702
#93362
C2×#13082
C2×#12872