Finite Group Databasebeta

← C2×#22588448 #31997307

Subgroups — C2×#22588448 #31997307

Order 18176.

1554 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556043C212yesmaximal
2#22588448C218yesmaximal, direct product factor
3C2×#17985765C214yesmaximal, complemented
4C2×#518×C71C211yesmaximal, characteristic
5C22×C22⋊Dic71C2214yes—
6C2×#14556043—24no—
7C2×#14556043—24no—
8C2×#14556043C2218yes—
9#17985765C22116yescomplemented
10C23×D4×C71C2211yescharacteristic
11#518×C71—24no—
12#518×C71C414yescomplemented
13#518×C71C2214yes—
14C2×C4×D4×C71—24no—
15C2×#117×C71C412yescomplemented
16C22×C22⋊C4×C71C412yescomplemented
17C22×C22⋊C4×C71C2212yes—
18C23×Dic71—24no—
19C2×C22⋊Dic71—24no—
20C2×C22⋊Dic71—24no—
21C2×C22⋊Dic71—24no—
22C2×C22⋊Dic71—24no—
23C2×C22⋊Dic71—24no—
24C2×C22⋊Dic71C2318yes—
25#14556043—216no—
26C25×C71C2×C411yescharacteristic
27C25×C71C2311yescharacteristic
28C22×D4×C71—24no—
29C22×D4×C71—24no—
30C22×D4×C71C2×C412yes—
31C22×D4×C71D414yes—
32C22×D4×C71D414yes—
33C22×D4×C71C2312yes—
34C23×C4×C71—22no—
35C23×C4×C71—22no—
36C23×C4×C71C2×C411yescharacteristic
37C4×D4×C71—216no—
38C4×D4×C71—216no—
39#117×C71—24no—
40#117×C71C2×C418yescomplemented
41C2×C4⋊C4×C71—22no—
42C2×C22⋊C4×C71—22no—
43C2×C22⋊C4×C71—24no—
44C2×C22⋊C4×C71—22no—
45C2×C22⋊C4×C71—22no—
46C2×C22⋊C4×C71—22no—
47C2×C22⋊C4×C71C2×C414yes—
48C2×C22⋊C4×C71C2×C414yes—
49C2×C22⋊C4×C71C2×C418yescomplemented
50C2×C22⋊C4×C71C2314yes—
51C2×C42×C71—22no—
52C22×Dic71—28no—
53C22×Dic71—44no—
54C22×Dic71—44no—
55C22×Dic71—88no—
56C22⋊Dic71—28no—
57C22⋊Dic71—28no—
58C22⋊Dic71—44no—
59C22⋊Dic71—44no—
60C24×C71—22no—
61C24×C71—22no—
62C24×C71—82no—
63C24×C71—82no—
64C24×C71C22⋊C412yes—
65C24×C71C22⋊C412yes—
66C24×C71C22×C412yes—
67C24×C71C22×C411yescharacteristic
68C24×C71C22×C411yescharacteristic
69C24×C71C2×D412yes—
70C24×C71C2×D412yes—
71C24×C71#4114yes—
72C24×C71#4114yes—
73C24×C71C2411yescharacteristic
74C2×D4×C71—28no—
75C2×D4×C71—28no—
76C2×D4×C71—28no—
77C2×D4×C71—28no—
78C2×D4×C71—28no—
79C2×D4×C71—28no—
80C2×D4×C71C22⋊C418yes—
81C2×D4×C71C2×D418yes—
82C22×C4×C71—22no—
83C22×C4×C71—22no—
84C22×C4×C71—24no—
85C22×C4×C71—24no—
86C22×C4×C71—28no—
87C22×C4×C71—24no—
88C22×C4×C71—84no—
89C22×C4×C71—84no—
90C22×C4×C71C22⋊C412yes—
91C22×C4×C71C22⋊C412yes—
92C22×C4×C71C22×C412yes—
93C4⋊C4×C71—28no—
94C22⋊C4×C71—24no—
95C22⋊C4×C71—24no—
96C22⋊C4×C71—28no—
97C22⋊C4×C71—24no—
98C22⋊C4×C71C22×C418yes—
99C42×C71—28no—
100C23×C71—22no—
101C23×C71—22no—
102C23×C71—24no—
103C23×C71—24no—
104C23×C71—44no—
105C23×C71—42no—
106C23×C71—48no—
107C23×C71—44no—
108C23×C71—84no—
109C23×C71—81no—
110C23×C71—82no—
111C23×C71—84no—
112C23×C71—81no—
113C23×C71—82no—
114C23×C71—82no—
115C23×C71—84no—
116C23×C71C2×C22⋊C411yescharacteristic
117C23×C71C2×C22⋊C414yes—
118C23×C71C2×C22⋊C411yescharacteristic
119C23×C71C2×C22⋊C412yes—
120C23×C71#11712yes—
121C23×C71#11712yes—
122C23×C71#11714yes—
123C23×C71#12314yes—
124C23×C71#12314yes—
125C23×C71#12314yes—
126C23×C71#12314yes—
127C23×C71C23×C411yescharacteristic
128C23×C71C22×D412yes—
129C23×C71C2×#4114yes—
130D4×C71—216no—
131D4×C71—216no—
132C2×C4×C71—24no—
133C2×C4×C71—22no—
134C2×C4×C71—24no—
135C2×C4×C71—22no—
136C2×C4×C71—28no—
137C2×C4×C71—48no—
138C2×C4×C71—84no—
139C2×C4×C71—84no—
140C2×C4×C71—88no—
141C2×C4×C71C2×C22⋊C414yes—
142C2×Dic71—48no—
143C2×Dic71—88no—
144C2×Dic71—88no—
145C2×Dic71—88no—
146C22×C71—28no—
147C22×C71—22no—
148C22×C71—24no—
149C22×C71—24no—
150C22×C71—44no—
151C22×C71—44no—
152C22×C71—42no—
153C22×C71—42no—
154C22×C71—41no—
155C22×C71—42no—
156C22×C71—41no—
157C22×C71—44no—
158C22×C71—44no—
159C22×C71—44no—
160C22×C71—44no—
161C22×C71—44no—
162C22×C71—44no—
163C22×C71—84no—
164C22×C71—84no—
165C22×C71#38111yescharacteristic
166C22×C71#38112yes—
167C22×C71#38114yes—
168C22×C71#38111yescharacteristic
169C22×C71C22×C22⋊C411yescharacteristic
170C22×C71C2×#11712yes—
171C22×C71C2×#12314yes—
172C4×C71—24no—
173C4×C71—48no—
174Dic71—88no—
175C2×#1828—711nomaximal, Sylow
176C2×C71—24no—
177C2×C71—21no—
178C2×C71—21no—
179C2×C71—22no—
180C2×C71—22no—
181C2×C71—24no—
182C2×C71—44no—
183C2×C71#182812yescomplemented
184C2×C71C2×#38111yescharacteristic
185C2×#518D7111yescharacteristic
186C22×#99—712no—
187#1828—718no—
188C2×#342—714no—
189C71C2×#182811yescharacteristic, complemented, Sylow
190C23×D4C2×D7111yescharacteristic
191#518—24no—
192#518Dic7114yescomplemented
193#518C2×D7114yes—
194C2×C4×D4—24no—
195C2×#117Dic7112yescomplemented
196C22×C22⋊C4—714no—
197C22×C22⋊C4Dic7112yescomplemented
198C22×C22⋊C4C2×D7112yes—
199C2×#99—718no—
200C2×#99—1424no—
201C2×#99—1424no—
202#342—7116no—
203C25C2×Dic7111yescharacteristic
204C25C22×D7111yescharacteristic
205C22×D4—24no—
206C22×D4—24no—
207C22×D4C2×Dic7112yes—
208C22×D4C22⋊D7114yes—
209C22×D4C22⋊D7114yes—
210C22×D4C22×D7112yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—1424no—
214C23×C4C2×Dic7111yescharacteristic
215C4×D4—216no—
216C4×D4—216no—
217#117—24no—
218#117C2×Dic7118yescomplemented
219C2×C4⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—718no—
226C2×C22⋊C4—1424no—
227C2×C22⋊C4—1424no—
228C2×C22⋊C4—1424no—
229C2×C22⋊C4—1424no—
230C2×C22⋊C4—1424no—
231C2×C22⋊C4C2×Dic7114yes—
232C2×C22⋊C4C2×Dic7114yes—
233C2×C22⋊C4C2×Dic7118yescomplemented
234C2×C22⋊C4C22×D7114yes—
235C2×C42—22no—
236#99—14216no—
237C24—22no—
238C24—22no—
239C24—82no—
240C24—82no—
241C24C22⋊Dic7112yes—
242C24C22⋊Dic7112yes—
243C24C22×Dic7112yes—
244C24C22×Dic7111yescharacteristic
245C24C22×Dic7111yescharacteristic
246C24C2×C22⋊D7112yes—
247C24C2×C22⋊D7112yes—
248C24#1176586514yes—
249C24#1176586514yes—
250C24C23×D7111yescharacteristic
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—28no—
254C2×D4—28no—
255C2×D4—28no—
256C2×D4—28no—
257C2×D4C22⋊Dic7118yes—
258C2×D4C2×C22⋊D7118yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—2844no—
268C22×C4—2844no—
269C22×C4—5688no—
270C22×C4—1428no—
271C22×C4C22⋊Dic7112yes—
272C22×C4C22⋊Dic7112yes—
273C22×C4C22×Dic7112yes—
274C4⋊C4—28no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—28no—
279C22⋊C4—2844no—
280C22⋊C4—2844no—
281C22⋊C4—1428no—
282C22⋊C4—1428no—
283C22⋊C4C22×Dic7118yes—
284C42—28no—
285C23—22no—
286C23—22no—
287C23—24no—
288C23—24no—
289C23—44no—
290C23—48no—
291C23—42no—
292C23—44no—
293C23—84no—
294C23—81no—
295C23—82no—
296C23—84no—
297C23—81no—
298C23—82no—
299C23—82no—
300C23—84no—
301C23C2×C22⋊Dic7111yescharacteristic
302C23C2×C22⋊Dic7114yes—
303C23C2×C22⋊Dic7111yescharacteristic
304C23C2×C22⋊Dic7112yes—
305C23#1455608712yes—
306C23#1455608712yes—
307C23#1455608714yes—
308C23#1455611114yes—
309C23#1455611114yes—
310C23#1455611114yes—
311C23#1455611114yes—
312C23C23×Dic7111yescharacteristic
313C23C22×C22⋊D7112yes—
314C23C2×#1176586514yes—
315D4—216no—
316D4—216no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—48no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—88no—
326C2×C4—2848no—
327C2×C4—5688no—
328C2×C4—5688no—
329C2×C4—5688no—
330C2×C4C2×C22⋊Dic7114yes—
331C22—22no—
332C22—28no—
333C22—24no—
334C22—24no—
335C22—42no—
336C22—42no—
337C22—44no—
338C22—44no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—84no—
349C22—84no—
350C22#1798586211yescharacteristic
351C22#1798586212yes—
352C22#1798586214yes—
353C22#1798586211yescharacteristic
354C22C22×C22⋊Dic7111yescharacteristic
355C22C2×#1455608712yes—
356C22C2×#1455611114yes—
357C4—24no—
358C4—48no—
359C4—5688no—
360C2—21no—
361C2—24no—
362C2—21no—
363C2—22no—
364C2—22no—
365C2—24no—
366C2—44no—
367C2#2258844812yesdirect product factor
368C2C2×#1798586211yescharacteristic
subgroupG/H
centerC22#17985862
commutatorC23×C71C23×C4
FrattiniC24C23×D71
FittingC2×#518×C71C2
socleC22×C71#381
prime radicalC23×D4×C71C22
cosocleC24×C71C24
hypercenterC2×#518D71
hypocenterC71C2×#1828
LCSG ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C25 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
2. #22588448C2direct (N × H)
3. C2×#17985765C2semidirect
9. #17985765C22semidirect
12. #518×C71C4semidirect
15. C2×#117×C71C4semidirect
16. C22×C22⋊C4×C71C4semidirect
40. #117×C71C2×C4semidirect
49. C2×C22⋊C4×C71C2×C4semidirect
183. C2×C71#1828semidirect
189. C71C2×#1828semidirect
192. #518Dic71semidirect
195. C2×#117Dic71semidirect
197. C22×C22⋊C4Dic71semidirect
218. #117C2×Dic71semidirect
233. C2×C22⋊C4C2×Dic71semidirect
367. C2#22588448direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145560432
#225884482
C2×#179857652
C2×#518×C712
C2×#182871