Finite Group Databasebeta

← C2×#69536 #10822587

Subgroups — C2×#69536 #10822587

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic
2C2×C3×#381C211yesmaximal, characteristic
3C2×#5211C212yesmaximal
4C2×#4527C212yesmaximal
5C2×#4526C211yesmaximal, characteristic
6#69536C218yesmaximal, direct product factor
7C2×#2030—31nomaximal, Sylow
8C24×C3×C4C2211yescharacteristic
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217C2212yes—
12C22×#217C2212yes—
13C2×#924—22no—
14#5483—22no—
15#5483—24no—
16#5483C2214yes—
17C2×#878—24no—
18C3×#381—22no—
19C3×#381C2214yes—
20C2×C3×#95—22no—
21#5211—24no—
22#5211C2218yes—
23C2×C3⋊#95—24no—
24#4527C2218yes—
25#4526C2214yes—
26C2×#543—31no—
27C2×#381—32no—
28C2×#381D311yescharacteristic
29#2030—38no—
30C2×#342—31no—
31C2×#342—32no—
32C25×C3C2311yescharacteristic
33C23×C3×C4—22no—
34C23×C3×C4—22no—
35C23×C3×C4—21no—
36C23×C3×C4—21no—
37C23×C3×C4—22no—
38C23×C3×C4—24no—
39C23×C3×C4—21no—
40C23×C3×C4D412yes—
41C23×C3×C4Q814yescomplemented
42C23×C3×C4C2312yes—
43C23×Dic3—22no—
44C23×Dic3D414yes—
45C22×Dic6—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4D412yes—
51C2×C3×C22⋊C4C2312yes—
52C2×C3×C22⋊C4C2312yes—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—24no—
58C2×#217—24no—
59C2×#217—22no—
60C2×#217D414yes—
61C2×#217C2314yes—
62C2×#217C2314yes—
63C2×C4⋊Dic3—22no—
64#924—28no—
65#924—28no—
66C2×Dic3⋊C4—24no—
67#878—216no—
68#878—216no—
69C3×#95—28no—
70C3⋊#95—216no—
71C24×C4C2×D311yescharacteristic
72#543—34no—
73#543—64no—
74#543—62no—
75C2×C22⋊Q8—64no—
76C2×C22⋊Q8—62no—
77C22×C22⋊C4—32no—
78C22×C22⋊C4—32no—
79C22×C22⋊C4C2×D311yescharacteristic
80C22×C22⋊C4C2×D311yescharacteristic
81#381—22no—
82#381—38no—
83#381—64no—
84#381C2×D314yes—
85C2×#95—22no—
86C2×#95—64no—
87#342—34no—
88#342—38no—
89C24×C3—44no—
90C24×C3C2×D412yes—
91C24×C3C2×D412yes—
92C24×C3C2×D411yescharacteristic
93C24×C3C2×D411yescharacteristic
94C24×C3C2×Q812yes—
95C24×C3#4112yes—
96C24×C3#4114yes—
97C24×C3C2411yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—28no—
100C22×C3×C4—28no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4—42no—
111C22×C3×C4—41no—
112C22×C3×C4—44no—
113C22×C3×C4—44no—
114C22×C3×C4—42no—
115C22×C3×C4—42no—
116C22×C3×C4—44no—
117C22×C3×C4—44no—
118C22×C3×C4—42no—
119C22×C3×C4C2×D414yes—
120C22×C3×C4C2×Q814yescomplemented
121C22×Dic3—24no—
122C22×Dic3—24no—
123C22×Dic3—24no—
124C22×Dic3—48no—
125C22×Dic3—48no—
126C22×Dic3—44no—
127C22×Dic3C2×D418yes—
128C2×Dic6—28no—
129C2×Dic6—84no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—42no—
134#217—24no—
135#217—24no—
136#217—24no—
137#217—24no—
138#217—24no—
139#217—24no—
140#217—44no—
141C4⋊Dic3—28no—
142Dic3⋊C4—216no—
143C25C22×D311yescharacteristic
144C22×Q8—62no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—34no—
153C23×C4—62no—
154C23×C4Dic614yescomplemented
155C23×C4#6712yes—
156C23×C4C22×D312yes—
157C22⋊Q8—616no—
158C22⋊Q8—616no—
159C22⋊Q8—68no—
160C22⋊Q8—68no—
161C2×C4⋊C4—64no—
162C2×C4⋊C4—62no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—34no—
168C2×C22⋊C4—34no—
169C2×C22⋊C4—34no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—64no—
175C2×C22⋊C4—64no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4#6712yes—
178C2×C22⋊C4C22×D312yes—
179C2×C22⋊C4C22×D312yes—
180#95—28no—
181#95—616no—
182C23×C3—28no—
183C23×C3—24no—
184C23×C3—44no—
185C23×C3—42no—
186C23×C3—44no—
187C23×C3—44no—
188C23×C3—48no—
189C23×C3—42no—
190C23×C3#12011yescharacteristic
191C23×C3#12012yes—
192C23×C3#12011yescharacteristic
193C23×C3C22⋊Q814yes—
194C23×C3C22⋊Q814yes—
195C23×C3C22⋊Q812yes—
196C23×C3C22⋊Q812yes—
197C23×C3#12414yes—
198C23×C3#12414yes—
199C23×C3#12412yes—
200C23×C3#12412yes—
201C23×C3C22×D412yes—
202C23×C3C22×D411yescharacteristic
203C23×C3C22×Q811yescharacteristic
204C23×C3C2×#4111yescharacteristic
205C23×C3C2×#4112yes—
206C2×C3×C4—24no—
207C2×C3×C4—28no—
208C2×C3×C4—24no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—42no—
212C2×C3×C4—44no—
213C2×C3×C4—48no—
214C2×C3×C4—44no—
215C2×C3×C4—44no—
216C2×C3×C4—44no—
217C2×C3×C4—44no—
218C2×C3×C4—42no—
219C2×C3×C4—82no—
220C2×C3×C4—84no—
221C2×Dic3—28no—
222C2×Dic3—48no—
223C2×Dic3—48no—
224C2×Dic3—48no—
225C2×Dic3—44no—
226C2×Dic3—84no—
227C2×Dic3—88no—
228Dic6—88no—
229C24—44no—
230C24C2×Dic612yes—
231C24#23512yes—
232C24D3×D412yes—
233C24D3×D412yes—
234C24#23714yes—
235C24C2×#6711yescharacteristic
236C24C2×#6711yescharacteristic
237C24C23×D311yescharacteristic
238C2×Q8—68no—
239C2×Q8—244no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—38no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—64no—
263C22×C4—64no—
264C22×C4—64no—
265C22×C4—128no—
266C22×C4—128no—
267C22×C4—124no—
268C22×C4C2×Dic614yescomplemented
269C22×C4C2×#6714yes—
270C4⋊C4—616no—
271C4⋊C4—68no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4—64no—
277C22⋊C4—64no—
278C22⋊C4—64no—
279C22⋊C4—64no—
280C22⋊C4—64no—
281C22⋊C4—64no—
282C22⋊C4—124no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—28no—
288C22×C3—24no—
289C22×C3—28no—
290C22×C3—24no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—44no—
294C22×C3—42no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—42no—
298C22×C3#39312yes—
299C22×C3#39314yes—
300C22×C3#39312yes—
301C22×C3#45814yes—
302C22×C3C2×#12011yescharacteristic
303C22×C3C2×C22⋊Q812yes—
304C22×C3C2×C22⋊Q811yescharacteristic
305C22×C3C2×#12412yes—
306C22×C3C2×#12411yescharacteristic
307C3×C4—44no—
308C3×C4—44no—
309C3×C4—82no—
310Dic3—48no—
311Dic3—84no—
312C23—24no—
313C23—28no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23—48no—
318C23—42no—
319C23—42no—
320C23#87814yes—
321C23#87814yes—
322C23#88514yes—
323C23#88514yes—
324C23#92412yes—
325C23#92412yes—
326C23#93612yes—
327C23#93612yes—
328C23#93712yes—
329C23#93711yescharacteristic
330C23#93711yescharacteristic
331C23C22×Dic611yescharacteristic
332C23C2×#23511yescharacteristic
333C23C2×D3×D412yes—
334C23C2×#23712yes—
335C23C22×#6711yescharacteristic
336Q8—248no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—68no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4—128no—
354C2×C4—128no—
355C2×C4—128no—
356C2×C4—124no—
357C2×C4—244no—
358C2×C4—248no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—22no—
362C2×C3—21no—
363C2×C3—24no—
364C2×C3—22no—
365C2×C3#203014yescomplemented
366C2×C3C2×#39311yescharacteristic
367C2×C3C2×#45812yes—
368C22—24no—
369C22—28no—
370C22—28no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—24no—
376C22—44no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22#494714yes—
384C22#494712yes—
385C22#494712yes—
386C22#505314yes—
387C22C2×#87812yes—
388C22C2×#88512yes—
389C22C2×#92411yescharacteristic
390C22C2×#93611yescharacteristic
391C22C2×#93711yescharacteristic
392C4—44no—
393C4—44no—
394C4—82no—
395C4—128no—
396C4—244no—
397C3C2×#203011yescharacteristic, complemented, Sylow
398C2—21no—
399C2—22no—
400C2—22no—
401C2—24no—
402C2—22no—
403C2—21no—
404C2#6953614yesdirect product factor
405C2C2×#494711yescharacteristic
406C2C2×#505312yes—
subgroupG/H
centerC23#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#381C2
socleC23×C3#120
prime radicalC25×C3C23
hypercenterC2×#381D3
hypocenterC3C2×#2030
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C25 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
6. #69536C2direct (N × H)
41. C23×C3×C4Q8semidirect
120. C22×C3×C4C2×Q8semidirect
154. C23×C4Dic6semidirect
268. C22×C4C2×Dic6semidirect
365. C2×C3#2030semidirect
397. C3C2×#2030semidirect
404. C2#69536direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54832
C2×C3×#3812
C2×#52112
C2×#45272
C2×#45262
#695362
C2×#20303