Finite Group Databasebeta

← C2×#13936 #514141

Subgroups — C2×#13936 #514141

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2#13936C2116yesmaximal, direct product factor
3C2×#1443C214yesmaximal
4C2×C4⋊#95C212yesmaximal
5C2×C4⋊#95C212yesmaximal
6C2×C4⋊#95C211yesmaximal, characteristic
7C2×#1434C212yesmaximal
8C2×#1434C212yesmaximal
9C2×C4×#95C211yesmaximal, characteristic
10C23×C4⋊C4C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2212yes—
14C23×C42C2211yescharacteristic
15C23×C42C2212yes—
16C2×C4×C4⋊C4C4116yescomplemented
17C2×C4×C4⋊C4C2218yes—
18C22×#95C2212yes—
19C22×#95C2214yes—
20C22×#95C2211yescharacteristic
21C22×#95C2212yes—
22C22×#95C2212yes—
23C22×#95C2214yes—
24C22×#95C2214yes—
25C22×#95C2214yes—
26C22×#95C2212yes—
27C22×#95C2211yescharacteristic
28#1443C22132yes—
29C4⋊#95C2218yes—
30C4⋊#95C22116yes—
31C4⋊#95C22116yes—
32#1434C22116yes—
33#1434C22116yes—
34C4×#95C2218yes—
35C24×C4C2314yes—
36C24×C4C2312yes—
37C24×C4C2312yes—
38C24×C4C2311yescharacteristic
39C24×C4C2312yes—
40C24×C4C2312yes—
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×C4⋊C4—24no—
44C22×C4⋊C4—28no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4—28no—
47C22×C4⋊C4—28no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4C2×C418yes—
50C22×C4⋊C4C2×C418yes—
51C22×C4⋊C4C2318yes—
52C22×C4⋊C4C2314yes—
53C22×C4⋊C4C2314yes—
54C22×C42—28no—
55C22×C42—28no—
56C22×C42—28no—
57C22×C42—28no—
58C22×C42—48no—
59C22×C42C2×C418yes—
60C22×C42C2×C4116yes—
61C22×C42C2318yes—
62C22×C42C2314yes—
63C22×C42C2318yes—
64C4×C4⋊C4C2×C4164yescomplemented
65C2×#95—28no—
66C2×#95—24no—
67C2×#95—216no—
68C2×#95—28no—
69C2×#95—48no—
70C2×#95—48no—
71C2×#95—48no—
72C2×#95—44no—
73C2×#95C2×C4116yes—
74C2×#95Q818yes—
75C2×#95C2314yes—
76C2×#95C2318yes—
77C2×#95C2318yes—
78C2×#95C23116yes—
79C2×#95C23116yes—
80C2×#95C23116yes—
81C2×#95C23116yes—
82C2×#95C2318yes—
83C2×#95C2318yes—
84C2×#95C2314yes—
85C25C2411yescharacteristic
86C23×C4—24no—
87C23×C4—24no—
88C23×C4—28no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—28no—
108C23×C4—28no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—28no—
113C23×C4—28no—
114C23×C4—28no—
115C23×C4—28no—
116C23×C4—28no—
117C23×C4—28no—
118C23×C4—28no—
119C23×C4C22×C418yes—
120C23×C4C22×C418yes—
121C23×C4C22×C414yes—
122C23×C4C22×C414yes—
123C23×C4C22×C414yes—
124C23×C4C2×Q818yes—
125C23×C4C2×Q814yes—
126C23×C4#4118yes—
127C23×C4#4114yes—
128C23×C4#4114yes—
129C23×C4#4118yes—
130C23×C4#4118yes—
131C23×C4#4114yes—
132C23×C4C2418yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2412yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2412yes—
139C23×C4C2412yes—
140C2×C4⋊C4—232no—
141C2×C4⋊C4—232no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—216no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—216no—
146C2×C4⋊C4—216no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—432no—
149C2×C4⋊C4C22×C4116yes—
150C2×C4⋊C4C22×C4116yes—
151C2×C42—232no—
152C2×C42—216no—
153C2×C42—216no—
154C2×C42—216no—
155C2×C42—216no—
156C2×C42—432no—
157C2×C42—416no—
158C2×C42—432no—
159C2×C42C22×C4116yes—
160C2×C42C22×C4132yes—
161#95—216no—
162#95—28no—
163#95—232no—
164#95—216no—
165#95—416no—
166#95—416no—
167#95—416no—
168#95—48no—
169#95C22×C4132yes—
170#95C2×Q8116yes—
171C24C23×C412yes—
172C24C22×Q812yes—
173C24C2×#4114yes—
174C24C2×#4114yes—
175C24C2×#4112yes—
176C24C2×#4112yes—
177C24C2×#4112yes—
178C24C2×#4114yes—
179C24H(5,𝔽2)14yes—
180C24H(5,𝔽2)12yes—
181C242−1+412yes—
182C24C2511yescharacteristic
183C22×C4—28no—
184C22×C4—28no—
185C22×C4—216no—
186C22×C4—216no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4—216no—
192C22×C4—28no—
193C22×C4—216no—
194C22×C4—216no—
195C22×C4—216no—
196C22×C4—216no—
197C22×C4—28no—
198C22×C4—216no—
199C22×C4—216no—
200C22×C4—216no—
201C22×C4—216no—
202C22×C4—216no—
203C22×C4—216no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—28no—
224C22×C4—28no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—416no—
238C22×C4—48no—
239C22×C4—416no—
240C22×C4—48no—
241C22×C4—48no—
242C22×C4—416no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—416no—
249C22×C4—416no—
250C22×C4—416no—
251C22×C4—416no—
252C22×C4—416no—
253C22×C4—416no—
254C22×C4—416no—
255C22×C4—416no—
256C22×C4#117116yes—
257C22×C4#117116yes—
258C22×C4#117116yes—
259C22×C4C4×Q8116yes—
260C22×C4#126132yes—
261C22×C4#126132yes—
262C22×C4C23×C414yes—
263C22×C4C23×C418yes—
264C22×C4C23×C418yes—
265C22×C4C23×C414yes—
266C22×C4C23×C414yes—
267C22×C4C22×Q818yes—
268C22×C4C22×Q814yes—
269C22×C4C2×#4118yes—
270C22×C4C2×#4114yes—
271C22×C4C2×#4118yes—
272C22×C4C2×#4114yes—
273C22×C4C2×#4118yes—
274C22×C4C2×#4114yes—
275C4⋊C4—232no—
276C4⋊C4—232no—
277C4⋊C4—432no—
278C42—232no—
279C42—432no—
280C42—432no—
281C23C2×#11714yes—
282C23C2×#11714yes—
283C23C2×#11714yes—
284C23C2×C4×Q814yes—
285C23C4×#4114yes—
286C23#51814yes—
287C23#52014yes—
288C23C2×#12618yes—
289C23C2×#12618yes—
290C23#52918yes—
291C23#53318yes—
292C23#54018yes—
293C23#54218yes—
294C23#55218yes—
295C23#55318yes—
296C23#55318yes—
297C23#55418yes—
298C23#55414yes—
299C23#55518yes—
300C23#55614yes—
301C23#55714yes—
302C23#55914yes—
303C23#55918yes—
304C23C24×C411yescharacteristic
305C23C23×Q811yescharacteristic
306C23C22×#4112yes—
307C23C22×#4112yes—
308C23C22×#4111yescharacteristic
309C23C22×#4111yescharacteristic
310C23C22×#4111yescharacteristic
311C23C22×#4112yes—
312C23C2×H(5,𝔽2)12yes—
313C23C2×H(5,𝔽2)11yescharacteristic
314C23C2×2−1+411yescharacteristic
315C2×C4—232no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—28no—
331C2×C4—28no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—432no—
336C2×C4—416no—
337C2×C4—432no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—432no—
353C2×C4—432no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—832no—
362C2×C4#383164yes—
363C2×C4C2×#11718yes—
364C2×C4C2×#11718yes—
365C2×C4C2×#11718yes—
366C2×C4C2×C4×Q818yes—
367C2×C4C2×#126116yes—
368C2×C4C2×#126116yes—
369C22C2×#38318yes—
370C22#229518yes—
371C22#231918yes—
372C22#234518yes—
373C22#234618yes—
374C22#235518yes—
375C22#235718yes—
376C22#2494116yes—
377C22#2497116yes—
378C22#2511116yes—
379C22#2513116yes—
380C22C22×#11711yescharacteristic
381C22C22×#11711yescharacteristic
382C22C22×#11711yescharacteristic
383C22C22×C4×Q811yescharacteristic
384C22C2×C4×#4111yescharacteristic
385C22C2×#51811yescharacteristic
386C22C2×#52011yescharacteristic
387C22C22×#12612yes—
388C22C22×#12612yes—
389C22C2×#52912yes—
390C22C2×#53312yes—
391C22C2×#54012yes—
392C22C2×#54212yes—
393C22C2×#55212yes—
394C22C2×#55312yes—
395C22C2×#55312yes—
396C22C2×#55412yes—
397C22C2×#55411yescharacteristic
398C22C2×#55512yes—
399C22C2×#55611yescharacteristic
400C22C2×#55711yescharacteristic
401C22C2×#55911yescharacteristic
402C22C2×#55912yes—
403C4—28no—
404C4—48no—
405C4—44no—
406C4—48no—
407C4—48no—
408C4—48no—
409C4—88no—
410C4C2×#383116yes—
411C2#13936116yesdirect product factor
412C2C22×#38311yescharacteristic
413C2C2×#229511yescharacteristic
414C2C2×#231911yescharacteristic
415C2C2×#234511yescharacteristic
416C2C2×#234611yescharacteristic
417C2C2×#235511yescharacteristic
418C2C2×#235711yescharacteristic
419C2C2×#249712yes—
420C2C2×#251112yes—
421C2C2×#249412yes—
422C2C2×#251312yes—
subgroupG/H
center, socle, prime radicalC25C24
commutatorC23C24×C4
Frattini, cosocleC24C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13936C2direct (N × H)
16. C2×C4×C4⋊C4C4semidirect
64. C4×C4⋊C4C2×C4semidirect
411. C2#13936direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
#139362
C2×#14432
C2×C4⋊#952
C2×#14342
C2×C4×#952