Finite Group Databasebeta

← C2×#29653 #6434701

Subgroups — C2×#29653 #6434701

Order 512.

3086 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3C2×#2502C211yesmaximal, characteristic
4#29653C2116yesmaximal, direct product factor
5C2×#2060C214yesmaximal, complemented
6C2×#2028C214yesmaximal, complemented
7C2×#1826C212yesmaximal
8C2×#1821C211yesmaximal, characteristic
9C2×#544—24no—
10C2×#544—28no—
11C2×#544—22no—
12C2×#544—24no—
13C2×#544C2214yes—
14C2×#544C2218yes—
15C22×C4⋊Q8—22no—
16C22×#125—22no—
17C22×C22⋊Q8—22no—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#3095—28no—
22#3095C22116yes—
23C2×#484—22no—
24C2×#485—24no—
25C2×#483—22no—
26C2×#428C2214yes—
27C2×#419C2214yes—
28C2×#407C2212yes—
29#2502—24no—
30#2502C2218yes—
31C2×#398—22no—
32C2×#391C2212yes—
33C2×#385—22no—
34C2×#384C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2060C22132yescomplemented
37#2028C22132yescomplemented
38#1826C22116yes—
39#1821C2218yes—
40C2×#343C2214yes—
41C2×#339C2218yes—
42C23×Q8—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C22×M16C2314yes—
47#544—216no—
48#544—232no—
49#544—216no—
50#544—232no—
51#544—28no—
52#544—216no—
53#544C23116yes—
54C2×C4⋊Q8—216no—
55C2×C4⋊Q8—28no—
56C2×C4⋊Q8—28no—
57C2×C4⋊Q8—216no—
58C2×#125—216no—
59C2×#125—28no—
60C2×#125—28no—
61C2×#125—216no—
62C2×C22⋊Q8—216no—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—28no—
65C2×C22⋊Q8—216no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8C2314yes—
71C2×#117—22no—
72C2×#117—24no—
73C2×#117—22no—
74C2×#117—24no—
75C2×#117D414yes—
76C2×#117D418yes—
77C2×#117C2318yes—
78C2×#117C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4D414yes—
88C22×C4⋊C4D412yes—
89C22×C4⋊C4D412yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C42—22no—
97C22×C42C2312yes—
98#485—232no—
99#484—216no—
100#483—216no—
101#428—28no—
102#428C23116yes—
103C2×#107—24no—
104C2×#106—24no—
105#419—28no—
106#419C23116yes—
107C2×#103—24no—
108C2×#103—24no—
109#407—24no—
110#407C2318yes—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113#398—216no—
114#391—24no—
115#391C2318yes—
116#385—216no—
117#384—24no—
118#384C2318yes—
119C4×C22⋊C4—22no—
120C4×C22⋊C4C2314yes—
121C2×#95—21no—
122C2×#95—24no—
123#343C23116yes—
124#339C23132yes—
125C25C2×D411yescharacteristic
126C22×Q8—216no—
127C22×Q8—28no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—28no—
132C22×Q8—88no—
133C22×Q8C2412yes—
134C23×C4—28no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—84no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D414yes—
149C23×C4C2×D414yes—
150C23×C4C2×D418yes—
151C23×C4C2412yes—
152C23×C4C2411yescharacteristic
153C2×M16—24no—
154C2×M16—216no—
155C2×M16C2418yes—
156C22×C8—24no—
157C4⋊Q8—264no—
158C4⋊Q8—232no—
159C4⋊Q8—232no—
160#125—264no—
161#125—232no—
162#125—232no—
163C22⋊Q8—264no—
164C22⋊Q8—232no—
165C22⋊Q8—232no—
166C22⋊Q8—28no—
167C22⋊Q8—28no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170#117—28no—
171#117—28no—
172#117—28no—
173#117—28no—
174#117—24no—
175#117—28no—
176#117—24no—
177#117C2×D418yes—
178#117C2×D4116yes—
179#117C2418yes—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—232no—
183C2×C4⋊C4—216no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—88no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D418yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D418yes—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D418yes—
210C2×C4⋊C4C2412yes—
211C2×C4⋊C4C2414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—28no—
225C2×C42—24no—
226C2×C42—28no—
227C2×C42—22no—
228C2×C42C2414yes—
229#107—216no—
230#106—216no—
231#103—216no—
232#103—216no—
233C22⋊C8—28no—
234C22⋊C8—28no—
235#95—24no—
236#95—216no—
237C24—24no—
238C24—24no—
239C24—42no—
240C24#12011yescharacteristic
241C24#12012yes—
242C24#12011yescharacteristic
243C24C22×D412yes—
244C24C22×D411yescharacteristic
245C2×Q8—232no—
246C2×Q8—24no—
247C2×Q8—28no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—832no—
253C2×Q8—816no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—216no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—44no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—88no—
285C22×C4—84no—
286C22×C4—88no—
287C22×C4#12014yes—
288C22×C4#12012yes—
289C22×C4#12014yes—
290C22×C4#12012yes—
291C22×C4C22×D412yes—
292C22×C4C22×D414yes—
293C22×C4C22×D414yes—
294C22×C4C22×D418yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C42−1+418yes—
299C22×C4C2511yescharacteristic
300M16—216no—
301M16—48no—
302C2×C8—28no—
303C2×C8—48no—
304C4⋊C4—232no—
305C4⋊C4—264no—
306C4⋊C4—216no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—816no—
317C4⋊C4C22×D418yes—
318C4⋊C4C22×D418yes—
319C22⋊C4—216no—
320C22⋊C4—216no—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C42—216no—
326C42—28no—
327C42—28no—
328C42—28no—
329C42—44no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—416no—
344C23#45614yes—
345C23C2×#12012yes—
346C23C2×#12011yescharacteristic
347C23C2×#12012yes—
348C23C2×#12011yescharacteristic
349C23#53614yes—
350C23C23×D411yescharacteristic
351Q8—48no—
352Q8—832no—
353Q8—84no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—44no—
366C2×C4—416no—
367C2×C4—416no—
368C2×C4—416no—
369C2×C4—44no—
370C2×C4—416no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4—88no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4C2×#12014yes—
386C2×C4C2×#12012yes—
387C2×C4#53614yes—
388C2×C4#53618yes—
389C2×C4#53618yes—
390C2×C4C23×D412yes—
391C2×C4C2×2−1+414yes—
392C8—48no—
393C22—22no—
394C22—28no—
395C22—28no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—48no—
405C22—48no—
406C22—44no—
407C22#261412yes—
408C22#261412yes—
409C22C2×#45614yes—
410C22C2×#45612yes—
411C22#301912yes—
412C22C22×#12011yescharacteristic
413C22C2×#53612yes—
414C4—24no—
415C4—48no—
416C4—48no—
417C4—48no—
418C4—48no—
419C4—82no—
420C4—82no—
421C4#261414yes—
422C4C2×#53614yes—
423C2—22no—
424C2—22no—
425C2—24no—
426C2—42no—
427C2#2965314yesdirect product factor
428C2C2×#261411yescharacteristic
429C2C22×#45611yescharacteristic
430C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #29653C2direct (N × H)
5. C2×#2060C2semidirect
6. C2×#2028C2semidirect
36. #2060C22semidirect
37. #2028C22semidirect
427. C2#29653direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#30952
C2×#25022
#296532
C2×#20602
C2×#20282
C2×#18262
C2×#18212