Finite Group Databasebeta

← C2×#71872 #10948022

Subgroups — C2×#71872 #10948022

Order 768.

1198 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C22×#836C212yesmaximal, complemented
3#71872C218yesmaximal, direct product factor
4C2×#4548C212yesmaximal, complemented
5C2×#4542C212yesmaximal, complemented
6C2×#2133—31nomaximal, Sylow
7C22×C3×#41C411yescharacteristic
8C23×C3×Q8C2211yescharacteristic
9C3×#536—24no—
10C3×#536C414yes—
11C3×#536C2214yes—
12C2×C3×C4⋊Q8C411yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#124C411yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#123—22no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×#117C411yescharacteristic
19C22×#208—22no—
20C2×#836—24no—
21C2×#836—22no—
22C2×#836C2214yes—
23C2×#836C2218yescomplemented
24#4548C2218yescomplemented
25#4542C2218yescomplemented
26C2×#536D311yescharacteristic
27C22×#101—32no—
28#2133—38no—
29C2×#356—32no—
30C2×#355—32no—
31C2×C3×#41—22no—
32C2×C3×#41—42no—
33C2×C3×#41—42no—
34C2×C3×#41C2×C414yes—
35C22×C3×Q8—42no—
36C22×C3×Q8—42no—
37C22×C3×Q8C2×C411yescharacteristic
38C22×C3×Q8C2×C411yescharacteristic
39C22×C3×Q8C2×C412yes—
40C22×C3×Q8D412yes—
41C22×C3×Q8D414yes—
42C22×C3×Q8C2311yescharacteristic
43C22×C3×Q8C2312yes—
44C22×C3×D4—21no—
45C22×C3×D4C2×C411yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4C2×C411yescharacteristic
48C23×C3×C4D412yes—
49C3×C4⋊Q8—22no—
50C3×C4⋊Q8—22no—
51C3×C4⋊Q8C2×C414yes—
52C3×C4⋊Q8C2×C412yes—
53C3×C4⋊Q8C2312yes—
54C3×#124—22no—
55C3×#124—22no—
56C3×#124C2×C414yes—
57C3×#124C2×C412yes—
58C3×#124C2312yes—
59C3×#123—28no—
60C3×#123—28no—
61C3×C22⋊Q8—28no—
62C3×C22⋊Q8—28no—
63C3×#117—22no—
64C3×#117C2×C414yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42C2×C411yescharacteristic
72C2×C3×C42C2×C411yescharacteristic
73C2×#208—24no—
74C2×#208—28no—
75C2×#208—42no—
76C2×#208—44no—
77C22×C3⋊C8—42no—
78#836—24no—
79#836—28no—
80#836C2318yes—
81C22×#41Dic311yescharacteristic
82C23×Q8C2×D311yescharacteristic
83C22×M16—62no—
84#536—24no—
85#536Dic314yes—
86#536C2×D314yes—
87C2×C4⋊Q8Dic311yescharacteristic
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×#124Dic311yescharacteristic
90C2×#124C2×D311yescharacteristic
91C2×#123—22no—
92C2×C22⋊Q8—22no—
93C2×#117Dic311yescharacteristic
94C2×#101—34no—
95C2×#101—38no—
96C2×#101—62no—
97C2×#101—64no—
98#356—38no—
99#355—38no—
100C24×C3—21no—
101C24×C3C22⋊C411yescharacteristic
102C3×#41—48no—
103C3×#41—48no—
104C2×C3×Q8—21no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—44no—
108C2×C3×Q8—42no—
109C2×C3×Q8—44no—
110C2×C3×Q8—42no—
111C2×C3×Q8—44no—
112C2×C3×Q8—41no—
113C2×C3×Q8—44no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C2×C3×Q8—41no—
117C2×C3×Q8C22⋊C414yes—
118C2×C3×Q8C22⋊C412yes—
119C2×C3×Q8C22×C412yes—
120C2×C3×Q8C22×C412yes—
121C2×C3×Q8C22×C411yescharacteristic
122C2×C3×Q8C2×D414yes—
123C2×C3×Q8C2×D412yes—
124C2×C3×Q8C2411yescharacteristic
125C2×C3×D4—24no—
126C2×C3×D4—21no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4C22×C412yes—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C22⋊C412yes—
144C22×C3×C4C22⋊C411yescharacteristic
145C22×C3×C4C22⋊C412yes—
146C22×C3×C4C22⋊C414yes—
147C22×C3×C4C22×C411yescharacteristic
148C22×C3×C4C22×C412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D414yes—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—28no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C42—21no—
160C3×C42—21no—
161C3×C42C22×C412yes—
162C3×C42C22×C412yes—
163#208—28no—
164#208—44no—
165#208—48no—
166C2×C3⋊C8—44no—
167C2×C3⋊C8—48no—
168C2×#41—22no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41C2×Dic314yes—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C2×Dic311yescharacteristic
175C22×Q8C2×Dic311yescharacteristic
176C22×Q8C2×Dic312yes—
177C22×Q8#6712yes—
178C22×Q8#6714yes—
179C22×Q8C22×D311yescharacteristic
180C22×Q8C22×D312yes—
181C22×D4—21no—
182C22×D4C2×Dic311yescharacteristic
183C23×C4—21no—
184C23×C4C2×Dic311yescharacteristic
185C23×C4#6712yes—
186C2×M16—64no—
187C2×M16—68no—
188C2×M16—124no—
189C2×M16—122no—
190C22×C8—122no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8C2×Dic314yes—
194C4⋊Q8C2×Dic312yes—
195C4⋊Q8C22×D312yes—
196#124—22no—
197#124—22no—
198#124C2×Dic314yes—
199#124C2×Dic312yes—
200#124C22×D312yes—
201#123—28no—
202#123—28no—
203C22⋊Q8—28no—
204C22⋊Q8—28no—
205#117—22no—
206#117C2×Dic314yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42C2×Dic311yescharacteristic
214C2×C42C2×Dic311yescharacteristic
215#101—38no—
216#101—68no—
217#101—64no—
218C23×C3—22no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3#9912yes—
224C23×C3#9912yes—
225C23×C3C2×C22⋊C411yescharacteristic
226C23×C3C2×C22⋊C412yes—
227C3×Q8—44no—
228C3×Q8—42no—
229C3×Q8—42no—
230C3×Q8—41no—
231C3×Q8—44no—
232C3×Q8—42no—
233C3×Q8—41no—
234C3×Q8—42no—
235C3×Q8—42no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—48no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—48no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4C2×C22⋊C414yes—
263C2×C3×C4C2×C22⋊C412yes—
264C2×C3×C4C2×C22⋊C412yes—
265C2×C3×C4C23×C411yescharacteristic
266C2×C3×C4C22×D412yes—
267C3⋊C8—48no—
268C24—21no—
269C24#21711yescharacteristic
270#41—48no—
271#41—48no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8#21714yes—
286C2×Q8#21712yes—
287C2×Q8C22×Dic312yes—
288C2×Q8C22×Dic312yes—
289C2×Q8C22×Dic311yescharacteristic
290C2×Q8C2×#6714yes—
291C2×Q8C2×#6712yes—
292C2×Q8C23×D311yescharacteristic
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4C22×Dic312yes—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4#21712yes—
312C22×C4#21711yescharacteristic
313C22×C4#21712yes—
314C22×C4#21714yes—
315C22×C4C22×Dic311yescharacteristic
316C22×C4C22×Dic312yes—
317C22×C4C2×#6712yes—
318C22×C4C2×#6714yes—
319M16—68no—
320M16—124no—
321M16—128no—
322C2×C8—128no—
323C2×C8—124no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—21no—
333C42—21no—
334C42C22×Dic312yes—
335C42C22×Dic312yes—
336C22×C3—22no—
337C22×C3—24no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—44no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3C2×#9911yescharacteristic
345C22×C3C2×#9914yes—
346C22×C3C2×#9911yescharacteristic
347C22×C3C22×C22⋊C411yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—24no—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—42no—
357C23—22no—
358C23—42no—
359C23—44no—
360C23—41no—
361C23—41no—
362C23#83412yes—
363C23#83412yes—
364C23C2×#21711yescharacteristic
365C23C2×#21712yes—
366Q8—44no—
367Q8—42no—
368Q8—42no—
369Q8—41no—
370Q8—44no—
371Q8—42no—
372Q8—41no—
373Q8—42no—
374Q8—42no—
375D4—42no—
376D4—42no—
377D4—48no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4C2×#21711yescharacteristic
401C2×C4C2×#21714yes—
402C2×C4C2×#21712yes—
403C2×C4C2×#21712yes—
404C2×C4C23×Dic311yescharacteristic
405C2×C4C22×#6712yes—
406C8—128no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—24no—
410C2×C3—82no—
411C2×C3#213312yescomplemented
412C2×C3C22×#9911yescharacteristic
413C22—22no—
414C22—24no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—42no—
419C22—82no—
420C22—82no—
421C22C2×#83411yescharacteristic
422C22C2×#83414yes—
423C22C2×#83411yescharacteristic
424C22C22×#21711yescharacteristic
425C4—22no—
426C4—22no—
427C4—21no—
428C4—24no—
429C4—22no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—42no—
434C3C2×#213311yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—24no—
438C2—82no—
439C2#7187212yesdirect product factor
440C2C22×#83411yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC2×C3×C4C23×C4
FrattiniC2×Q8C23×D3
FittingC2×C3×#536C2
socleC22×C3C2×#99
prime radicalC22×C3×#41C4
cosocleC2×C3×Q8C24
hypercenterC2×#536D3
hypocenterC3C2×#2133
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
2. C22×#836C2semidirect
3. #71872C2direct (N × H)
4. C2×#4548C2semidirect
5. C2×#4542C2semidirect
23. C2×#836C22semidirect
24. #4548C22semidirect
25. #4542C22semidirect
411. C2×C3#2133semidirect
434. C3C2×#2133semidirect
439. C2#71872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C22×#8362
#718722
C2×#45482
C2×#45422
C2×#21333