Finite Group Databasebeta

← C2×#22589482 #32024735

Subgroups — C2×#22589482 #32024735

Order 18176.

1518 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556128C211yesmaximal, characteristic
2C22×#14556053C212yesmaximal
3C2×#17985905C211yesmaximal, characteristic
4#22589482C218yesmaximal, direct product factor
5C2×#17985762C212yesmaximal
6C2×#336×C71C211yesmaximal, characteristic
7C23×Dic142—21no—
8C23×Dic142C2211yescharacteristic
9C2×#14556128—28no—
10C2×#14556128—22no—
11C2×#14556128C2214yes—
12C22×C4⋊Dic71—21no—
13C22×C4⋊Dic71C2211yescharacteristic
14C22×C4⋊Dic71C2212yes—
15C2×#14556053—28no—
16C2×#14556053C2218yes—
17#17985905—22no—
18#17985905C2214yes—
19C2×C71⋊#95—21no—
20C2×C71⋊#95—21no—
21#17985762C2218yes—
22C23×C8×C71C2212yes—
23C22×C42×C71C2211yescharacteristic
24#336×C71C2214yes—
25C22×Dic142—28no—
26C22×Dic142—24no—
27C22×Dic142—21no—
28C22×Dic142—24no—
29C22×Dic142—22no—
30C22×Dic142—24no—
31C22×Dic142—44no—
32C22×Dic142—44no—
33C22×Dic142Q814yescomplemented
34C22×Dic142C2312yes—
35C23×Dic71—22no—
36C23×Dic71—21no—
37C23×Dic71—21no—
38#14556128—216no—
39#14556128—216no—
40C2×C4⋊Dic71—28no—
41C2×C4⋊Dic71—24no—
42C2×C4⋊Dic71—24no—
43C2×C4⋊Dic71—22no—
44C2×C4⋊Dic71—21no—
45C2×C4⋊Dic71—24no—
46C2×C4⋊Dic71D418yes—
47C2×C4⋊Dic71C2312yes—
48C2×C4⋊Dic71C2314yes—
49#14556053—216no—
50C71⋊#95—24no—
51C71⋊#95—24no—
52C23×C4×C71D411yescharacteristic
53C23×C4×C71D411yescharacteristic
54C23×C4×C71C2311yescharacteristic
55C22×C8×C71—24no—
56C22×C8×C71—24no—
57C22×C8×C71—24no—
58C22×C8×C71C2314yes—
59C2×C42×C71—24no—
60C2×C42×C71D412yes—
61C2×C42×C71C2312yes—
62C2×Dic142—216no—
63C2×Dic142—24no—
64C2×Dic142—24no—
65C2×Dic142—216no—
66C2×Dic142—24no—
67C2×Dic142—22no—
68C2×Dic142—48no—
69C2×Dic142—432no—
70C2×Dic142—42no—
71C2×Dic142—48no—
72C2×Dic142—42no—
73C2×Dic142—48no—
74C2×Dic142C2×Q814yescomplemented
75C22×Dic71—28no—
76C22×Dic71—24no—
77C22×Dic71—24no—
78C22×Dic71—22no—
79C22×Dic71—22no—
80C22×Dic71—42no—
81C22×Dic71—42no—
82C22×Dic71—44no—
83C22×Dic71—44no—
84C22×Dic71—42no—
85C22×Dic71—42no—
86C22×Dic71—42no—
87C4⋊Dic71—216no—
88C4⋊Dic71—28no—
89C4⋊Dic71—24no—
90C4⋊Dic71—24no—
91C4⋊Dic71—42no—
92C4⋊Dic71C2×D418yes—
93C24×C71C2×D411yescharacteristic
94C22×C4×C71—22no—
95C22×C4×C71—22no—
96C22×C4×C71—22no—
97C22×C4×C71—22no—
98C22×C4×C71SD1614yes—
99C22×C4×C71Q1614yes—
100C22×C4×C71C2×D414yes—
101C22×C4×C71C2×D412yes—
102C22×C4×C71C2×D412yes—
103C22×C4×C71C2×D411yescharacteristic
104C22×C4×C71C2×Q812yes—
105C22×C4×C71#4112yes—
106C22×C4×C71#4114yes—
107C22×C4×C71C2411yescharacteristic
108C2×C8×C71—24no—
109C2×C8×C71—24no—
110C2×C8×C71—24no—
111C2×C8×C71—48no—
112C42×C71—24no—
113C42×C71—24no—
114C23×C71#12012yes—
115C23×C71C22⋊Q812yes—
116C23×C71#12412yes—
117C23×C71C2×SD1612yes—
118C23×C71C2×Q1612yes—
119C23×C71AGL(1,ℤ/8)12yes—
120C23×C71#13712yes—
121C23×C71C22×D411yescharacteristic
122C2×C4×C71—22no—
123C2×C4×C71—22no—
124C2×C4×C71—22no—
125C2×C4×C71—22no—
126C2×C4×C71—28no—
127C2×C4×C71—28no—
128C2×C4×C71#12012yes—
129C2×C4×C71C22⋊Q818yes—
130C2×C4×C71C22⋊Q812yes—
131C2×C4×C71#12418yes—
132C2×C4×C71#12412yes—
133C2×C4×C71C2×SD1614yes—
134C2×C4×C71C2×Q1614yes—
135C2×C4×C71C22×D412yes—
136C2×C4×C71C22×Q811yescharacteristic
137C2×C4×C71C2×#4111yescharacteristic
138C2×C4×C71C2×#4112yes—
139C2×Dic71—28no—
140C2×Dic71—24no—
141C2×Dic71—42no—
142C2×Dic71—42no—
143C2×Dic71—44no—
144C2×Dic71—44no—
145C2×Dic71—48no—
146C2×Dic71—416no—
147C2×Dic71—42no—
148C2×Dic71—42no—
149C2×Dic71—42no—
150C2×Dic71—84no—
151Dic142—28no—
152Dic142—432no—
153Dic142—44no—
154Dic142—48no—
155C8×C71—44no—
156C22×C71#39314yes—
157C22×C71#45014yes—
158C22×C71#45114yes—
159C22×C71#47514yes—
160C22×C71#47714yes—
161C22×C71#48714yes—
162C22×C71#48914yes—
163C22×C71C2×#12011yescharacteristic
164C22×C71C2×C22⋊Q811yescharacteristic
165C22×C71C2×#12411yescharacteristic
166C22×C71C22×SD1611yescharacteristic
167C22×C71C22×Q1611yescharacteristic
168C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
169C22×C71C2×#13711yescharacteristic
170C4×C71—24no—
171C4×C71—24no—
172C4×C71#39314yes—
173C4×C71C2×C22⋊Q812yes—
174C4×C71C2×#12412yes—
175Dic71—44no—
176Dic71—48no—
177Dic71—82no—
178C2×#2024—711nomaximal, Sylow
179C2×C71#202418yescomplemented
180C2×C71C2×#39311yescharacteristic
181C2×C71C2×#45011yescharacteristic
182C2×C71C2×#45111yescharacteristic
183C2×C71C2×#47511yescharacteristic
184C2×C71C2×#47711yescharacteristic
185C2×C71C2×#48711yescharacteristic
186C2×C71C2×#48911yescharacteristic
187C22×C4⋊Q8—711no—
188C22×#103—712no—
189C2×#391—711no—
190#2024—718no—
191C2×#340—712no—
192C2×#336D7111yescharacteristic
193C71C2×#202411yescharacteristic, complemented, Sylow
194C23×Q8—711no—
195C23×Q8—1421no—
196C23×C8C2×D7112yes—
197C2×C4⋊Q8—714no—
198C2×C4⋊Q8—1428no—
199C2×C4⋊Q8—1422no—
200C22×C4⋊C4—712no—
201C22×C4⋊C4—711no—
202C22×C4⋊C4—1421no—
203C22×C42C2×D7111yescharacteristic
204C2×#103—718no—
205C2×#103—1428no—
206#391—714no—
207#391—1422no—
208C2×#95—1421no—
209C2×#95—1421no—
210#340—718no—
211#336C2×D7114yes—
212C22×Q8—2844no—
213C22×Q8—2844no—
214C22×Q8—714no—
215C22×Q8—712no—
216C22×Q8—1428no—
217C22×Q8—1424no—
218C22×Q8—1424no—
219C22×Q8—1424no—
220C22×Q8—1421no—
221C22×Q8—1422no—
222C23×C4—1421no—
223C23×C4—1422no—
224C23×C4—1421no—
225C23×C4D28411yescharacteristic
226C23×C4D28411yescharacteristic
227C23×C4C22×D7111yescharacteristic
228C22×C8—24no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C22×D7114yes—
232C4⋊Q8—14216no—
233C4⋊Q8—14216no—
234C2×C4⋊C4—718no—
235C2×C4⋊C4—714no—
236C2×C4⋊C4—712no—
237C2×C4⋊C4—1421no—
238C2×C4⋊C4—1424no—
239C2×C4⋊C4—1428no—
240C2×C4⋊C4—1424no—
241C2×C4⋊C4—1422no—
242C2×C4⋊C4—1424no—
243C2×C42—24no—
244C2×C42D28412yes—
245C2×C42C22×D7112yes—
246#103—14216no—
247#95—1424no—
248#95—1424no—
249C24C2×D28411yescharacteristic
250C2×Q8—28432no—
251C2×Q8—2848no—
252C2×Q8—2842no—
253C2×Q8—2848no—
254C2×Q8—2842no—
255C2×Q8—2848no—
256C2×Q8—714no—
257C2×Q8—14216no—
258C2×Q8—1424no—
259C2×Q8—1424no—
260C2×Q8—14216no—
261C2×Q8—1422no—
262C2×Q8—1424no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—2842no—
268C22×C4—2842no—
269C22×C4—2844no—
270C22×C4—2844no—
271C22×C4—2842no—
272C22×C4—2842no—
273C22×C4—2842no—
274C22×C4—1422no—
275C22×C4—1422no—
276C22×C4—1424no—
277C22×C4—1428no—
278C22×C4—1424no—
279C22×C4#1176585414yes—
280C22×C4Dic28414yes—
281C22×C4D4×D7114yes—
282C22×C4C2×D28412yes—
283C22×C4C2×D28412yes—
284C22×C4C2×D28411yescharacteristic
285C22×C4Q8×D7112yes—
286C22×C4#1176586412yes—
287C22×C4#1176586514yes—
288C22×C4C23×D7111yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—48no—
293C4⋊C4—2842no—
294C4⋊C4—718no—
295C4⋊C4—14216no—
296C4⋊C4—1424no—
297C4⋊C4—1424no—
298C4⋊C4—1428no—
299C42—24no—
300C42—24no—
301C23#1455609812yes—
302C23#1455610812yes—
303C23#1455611612yes—
304C23C2×#1176585412yes—
305C23C2×Dic28412yes—
306C23#1455615612yes—
307C23#1455616112yes—
308C23C22×D28411yescharacteristic
309Q8—28432no—
310Q8—2848no—
311Q8—2844no—
312Q8—1428no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28416no—
320C2×C4—2848no—
321C2×C4—2842no—
322C2×C4—2842no—
323C2×C4—2844no—
324C2×C4—2844no—
325C2×C4—2842no—
326C2×C4—2842no—
327C2×C4—2842no—
328C2×C4—5684no—
329C2×C4—1428no—
330C2×C4—1424no—
331C2×C4#1455609812yes—
332C2×C4#1455610618yes—
333C2×C4#1455610812yes—
334C2×C4#1455611418yes—
335C2×C4#1455611612yes—
336C2×C4C2×#1176585414yes—
337C2×C4C2×Dic28414yes—
338C2×C4C2×D4×D7112yes—
339C2×C4C2×Q8×D7111yescharacteristic
340C2×C4C2×#1176586411yescharacteristic
341C2×C4C2×#1176586512yes—
342C8—44no—
343C22#1798591214yes—
344C22#1798615214yes—
345C22#1798615714yes—
346C22#1798628514yes—
347C22#1798629514yes—
348C22#1798635014yes—
349C22#1798635914yes—
350C22C2×#1455609811yescharacteristic
351C22C2×#1455610811yescharacteristic
352C22C2×#1455611611yescharacteristic
353C22C22×#1176585411yescharacteristic
354C22C22×Dic28411yescharacteristic
355C22C2×#1455615611yescharacteristic
356C22C2×#1455616111yescharacteristic
357C4—24no—
358C4—24no—
359C4—2844no—
360C4—2848no—
361C4—5682no—
362C4#1798591214yes—
363C4C2×#1455610612yes—
364C4C2×#1455611412yes—
365C2#2258948218yesdirect product factor
366C2C2×#1798591211yescharacteristic
367C2C2×#1798615211yescharacteristic
368C2C2×#1798615711yescharacteristic
369C2C2×#1798628511yescharacteristic
370C2C2×#1798629511yescharacteristic
371C2C2×#1798635011yescharacteristic
372C2C2×#1798635911yescharacteristic
subgroupG/H
centerC24C2×D284
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#336×C71C2
socle, prime radicalC24×C71C2×D4
hypercenterC2×#336D71
hypocenterC71C2×#2024
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
4. #22589482C2direct (N × H)
33. C22×Dic142Q8semidirect
74. C2×Dic142C2×Q8semidirect
179. C2×C71#2024semidirect
193. C71C2×#2024semidirect
365. C2#22589482direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561282
C22×#145560532
C2×#179859052
#225894822
C2×#179857622
C2×#336×C712
C2×#202471