Finite Group Databasebeta

← C2×#74192 #10966048

Subgroups — C2×#74192 #10966048

Order 768.

2030 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#382C211yesmaximal, characteristic
2C2×#4986C211yesmaximal, characteristic
3C2×#4982C212yesmaximal
4C2×#4979C211yesmaximal, characteristic
5C2×#4932C212yesmaximal
6#74192C2116yesmaximal, direct product factor
7C2×#4646C212yesmaximal
8C2×#4647C212yesmaximal
9C2×#4655C212yesmaximal
10C2×#4654C212yesmaximal
11C2×#2612—31nomaximal, Sylow
12C23×Dic6C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2212yes—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×Dic3⋊C4C2212yes—
20C22×Dic3⋊C4C2212yes—
21C22×C4×Dic3C2212yes—
22C22×C4×Dic3C2212yes—
23C3×#382C2218yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34#4986C2218yes—
35#4982C22116yes—
36#4979C2218yes—
37#4932C22116yes—
38#4655C22116yes—
39#4654C22116yes—
40#4647C22116yes—
41#4646C22116yes—
42#2612—316no—
43C2×#400—31no—
44C2×#398—32no—
45C2×#395—31no—
46C2×#391—32no—
47C2×#389—32no—
48C2×#387—32no—
49C2×#387—32no—
50C2×#382D311yescharacteristic
51C2×#376—32no—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C22×Dic6—28no—
62C22×Dic6—42no—
63C22×Dic6C2314yes—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4C2314yes—
67C2×C3×C4⋊C4C2314yes—
68C2×C3×C42Q818yescomplemented
69C2×C3×C42C2314yes—
70C2×C4⋊Dic3—28no—
71C2×C4⋊Dic3—42no—
72C2×C4⋊Dic3C2318yes—
73C2×C4⋊Dic3C2314yes—
74C2×Dic3⋊C4—28no—
75C2×Dic3⋊C4—28no—
76C2×Dic3⋊C4—44no—
77C2×Dic3⋊C4C2318yes—
78C2×Dic3⋊C4C2318yes—
79C2×Dic3⋊C4C2318yes—
80C2×C4×Dic3—28no—
81C2×C4×Dic3—28no—
82C2×C4×Dic3C2318yes—
83C2×C4×Dic3C2318yes—
84C3×#95C2318yes—
85C3×#95C2318yes—
86C3⋊#95C2318yes—
87C3⋊#95C2318yes—
88C3⋊#95C2318yes—
89C3⋊#95C2314yes—
90C3⋊#95C2318yes—
91C3⋊#95C2314yes—
92C3⋊#95C2318yes—
93C3⋊#95C2318yes—
94C23×Q8—31no—
95C22×C4⋊C4—32no—
96C22×C4⋊C4—32no—
97C22×C4⋊C4—32no—
98C22×C4⋊C4—32no—
99C22×C4⋊C4—31no—
100C22×C4⋊C4C2×D311yescharacteristic
101C22×C4⋊C4C2×D311yescharacteristic
102C22×C42—32no—
103C22×C42—32no—
104C22×C42C2×D311yescharacteristic
105#400—38no—
106#398—316no—
107#395—38no—
108#391—316no—
109#389—316no—
110#387—316no—
111#387—316no—
112#382C2×D318yes—
113#376—316no—
114C2×#95—32no—
115C2×#95—31no—
116C2×#95—32no—
117C2×#95—32no—
118C2×#95—31no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95C2×D312yes—
123C2×#95C2×D312yes—
124C24×C3C2411yescharacteristic
125C22×C3×C4—24no—
126C22×C3×C4—24no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4C2×Q818yes—
137C22×C3×C4C2×Q814yes—
138C22×C3×C4C2412yes—
139C22×C3×C4C2412yes—
140C22×C3×C4C2414yes—
141C22×C3×C4C2412yes—
142C22×C3×C4C2414yes—
143C22×Dic3—24no—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3#4118yes—
154C22×Dic3#4118yes—
155C22×Dic3C2414yes—
156C22×Dic3C2414yes—
157C22×Dic3C2414yes—
158C22×Dic3C2414yes—
159C2×Dic6—216no—
160C2×Dic6—44no—
161C2×Dic6—416no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4—44no—
164C3×C42C2×Q8116yescomplemented
165C4⋊Dic3—216no—
166C4⋊Dic3—44no—
167Dic3⋊C4—216no—
168Dic3⋊C4—216no—
169Dic3⋊C4—48no—
170C4×Dic3—216no—
171C4×Dic3—216no—
172C22×Q8—34no—
173C22×Q8—68no—
174C22×Q8—122no—
175C23×C4—32no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4C22×D311yescharacteristic
180C23×C4C22×D312yes—
181C23×C4C22×D311yescharacteristic
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D312yes—
184C2×C4⋊C4—38no—
185C2×C4⋊C4—38no—
186C2×C4⋊C4—38no—
187C2×C4⋊C4—34no—
188C2×C4⋊C4—38no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—68no—
192C2×C4⋊C4—68no—
193C2×C4⋊C4—68no—
194C2×C4⋊C4—124no—
195C2×C4⋊C4—122no—
196C2×C4⋊C4C22×D314yes—
197C2×C4⋊C4C22×D314yes—
198C2×C42—38no—
199C2×C42—38no—
200C2×C42—68no—
201C2×C42—68no—
202C2×C42Dic618yescomplemented
203C2×C42C22×D314yes—
204#95—38no—
205#95—38no—
206#95—38no—
207#95—38no—
208#95—34no—
209#95—34no—
210#95—38no—
211#95—38no—
212#95C22×D318yes—
213#95C22×D318yes—
214C23×C3C22×Q812yes—
215C23×C3C2×#4114yes—
216C23×C3C2×#4114yes—
217C23×C3H(5,𝔽2)12yes—
218C23×C32−1+412yes—
219C23×C3C2511yescharacteristic
220C2×C3×C4—216no—
221C2×C3×C4—28no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—44no—
234C2×C3×C4—44no—
235C2×C3×C4—48no—
236C2×C3×C4C22×Q818yes—
237C2×C3×C4C22×Q814yes—
238C2×Dic3—216no—
239C2×Dic3—216no—
240C2×Dic3—24no—
241C2×Dic3—24no—
242C2×Dic3—24no—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—48no—
251C2×Dic3—48no—
252C2×Dic3C2×#4118yes—
253C2×Dic3C2×#4118yes—
254Dic6—416no—
255C24C23×D311yescharacteristic
256C2×Q8—616no—
257C2×Q8—124no—
258C2×Q8—1216no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—38no—
271C22×C4—38no—
272C22×C4—34no—
273C22×C4—34no—
274C22×C4—34no—
275C22×C4—34no—
276C22×C4—64no—
277C22×C4—64no—
278C22×C4—64no—
279C22×C4—64no—
280C22×C4—64no—
281C22×C4—64no—
282C22×C4—64no—
283C22×C4—64no—
284C22×C4—64no—
285C22×C4—64no—
286C22×C4C2×Dic614yes—
287C22×C4C2×Dic618yes—
288C22×C4C23×D312yes—
289C22×C4C23×D312yes—
290C22×C4C23×D314yes—
291C22×C4C23×D312yes—
292C22×C4C23×D314yes—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—616no—
296C4⋊C4—616no—
297C4⋊C4—616no—
298C4⋊C4—128no—
299C4⋊C4—124no—
300C42—616no—
301C42—616no—
302C42C2×Dic6116yescomplemented
303C22×C3#53318yes—
304C22×C3#54414yes—
305C22×C3#55114yes—
306C22×C3#55218yes—
307C22×C3#55514yes—
308C22×C3C23×Q811yescharacteristic
309C22×C3C22×#4112yes—
310C22×C3C22×#4112yes—
311C22×C3C2×H(5,𝔽2)11yescharacteristic
312C22×C3C2×2−1+411yescharacteristic
313C3×C4—28no—
314C3×C4—24no—
315C3×C4—42no—
316C3×C4—42no—
317C3×C4—44no—
318Dic3—28no—
319Dic3—28no—
320Dic3—44no—
321Dic3—44no—
322C23C22×Dic612yes—
323C23#100412yes—
324C23#100712yes—
325C23D3×#4114yes—
326C23D3×#4114yes—
327C23C24×D311yescharacteristic
328Q8—1216no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—28no—
334C2×C4—216no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—38no—
343C2×C4—38no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—616no—
348C2×C4—616no—
349C2×C4—64no—
350C2×C4—64no—
351C2×C4—64no—
352C2×C4—64no—
353C2×C4—64no—
354C2×C4—64no—
355C2×C4—64no—
356C2×C4—64no—
357C2×C4—64no—
358C2×C4—64no—
359C2×C4—128no—
360C2×C4—128no—
361C2×C4C22×Dic614yes—
362C2×C4C22×Dic618yes—
363C2×C3#261218yescomplemented
364C2×C3C2×#53312yes—
365C2×C3C2×#54411yescharacteristic
366C2×C3C2×#55111yescharacteristic
367C2×C3C2×#55212yes—
368C2×C3C2×#55511yescharacteristic
369C22#550014yes—
370C22#551718yes—
371C22#567014yes—
372C22#567914yes—
373C22#569318yes—
374C22C23×Dic611yescharacteristic
375C22C2×#100411yescharacteristic
376C22C2×#100711yescharacteristic
377C22C2×D3×#4112yes—
378C22C2×D3×#4112yes—
379C4—24no—
380C4—28no—
381C4—44no—
382C4—42no—
383C4—42no—
384C4—68no—
385C4—68no—
386C4—124no—
387C4—124no—
388C3C2×#261211yescharacteristic, complemented, Sylow
389C2#7419218yesdirect product factor
390C2C2×#550011yescharacteristic
391C2C2×#551712yes—
392C2C2×#567011yescharacteristic
393C2C2×#567911yescharacteristic
394C2C2×#569312yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#382C2
socle, prime radicalC24×C3C24
hypercenterC2×#382D3
hypocenterC3C2×#2612
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
6. #74192C2direct (N × H)
68. C2×C3×C42Q8semidirect
164. C3×C42C2×Q8semidirect
202. C2×C42Dic6semidirect
302. C42C2×Dic6semidirect
363. C2×C3#2612semidirect
388. C3C2×#2612semidirect
389. C2#74192direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3822
C2×#49862
C2×#49822
C2×#49792
C2×#49322
#741922
C2×#46462
C2×#46472
C2×#46552
C2×#46542
C2×#26123