Finite Group Databasebeta

← C22×#5546 #11701774

Subgroups — C22×#5546 #11701774

Order 768.

12270 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#237C211yesmaximal, characteristic
2C22×C3×C4×D4C211yesmaximal, characteristic, complemented
3C22×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×C4×#67C212yesmaximal, complemented
5C23×Dic3⋊C4C212yesmaximal, complemented
6C2×#5546C2148yesmaximal, direct product factor
7C22×D3×C4⋊C4C211yesmaximal, characteristic, complemented
8C22×#887C211yesmaximal, characteristic, complemented
9C22×#67⋊C4C212yesmaximal, complemented
10C22×#877C212yesmaximal, complemented
11C22×#872C211yesmaximal, characteristic, complemented
12C22×C4×Dic6C211yesmaximal, characteristic, complemented
13C22×#520—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2212yes—
16C23×#67C2212yes—
17C24×Dic3C2212yes—
18C24×Dic3C2212yes—
19C22×#237C4132yescomplemented
20C22×#237C22124yes—
21C23×C4×D3C2212yes—
22C23×C4×D3C2211yescharacteristic
23C23×Dic6C2211yescharacteristic
24C2×C3×C4×D4C22124yescomplemented
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C22⋊C4C2212yes—
27C22×C3×C42C2211yescharacteristic
28C22×#217C2212yes—
29C2×D4×Dic3C22124yescomplemented
30C2×C4×#67C22148yescomplemented
31C22×(C2×D3)⋊C4C2211yescharacteristic
32C22×(C2×D3)⋊C4C2211yescharacteristic
33C22×C4⋊Dic3C2211yescharacteristic
34C22×Dic3⋊C4C2212yes—
35C22×Dic3⋊C4C2211yescharacteristic
36C22×Dic3⋊C4C2211yescharacteristic
37C22×Dic3⋊C4C2212yes—
38C22×Dic3⋊C4C2212yes—
39C22×Dic3⋊C4C22148yescomplemented
40C22×C4×Dic3C2212yes—
41C22×C4×Dic3C2211yescharacteristic
42C22×C4×Dic3C2212yes—
43#5546C221256yesdirect product factor
44C2×D3×C4⋊C4C22124yescomplemented
45C2×#887C22124yescomplemented
46C2×#67⋊C4C22148yescomplemented
47C2×#877C22148yescomplemented
48C2×#872C22124yescomplemented
49C2×C4×Dic6C22124yescomplemented
50C23×#41—31no—
51C2×#520—348no—
52C22×C4×Q8—31no—
53C22×C4×Q8—31no—
54C22×C4×D4—31no—
55C22×C4×D4—32no—
56C22×C4×D4—32no—
57C22×C4×D4D311yescharacteristic, complemented
58C22×#117—31no—
59C22×#117—32no—
60C23×C4⋊C4—32no—
61C23×C4⋊C4—31no—
62C25×C3C2312yes—
63C24×D3C2311yescharacteristic
64C22×C3×D4C2×C4116yescomplemented
65C22×C3×D4C23112yes—
66C23×C3×C4—48no—
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C23124yes—
71C23×C3×C4C2311yescharacteristic
72C22×#67C2×C4132yescomplemented
73C22×#67C23124yes—
74C23×Dic3—48no—
75C23×Dic3C2×C4132yescomplemented
76C23×Dic3C2312yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2311yescharacteristic
79C23×Dic3C23124yes—
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C23×Dic3C23124yes—
83C2×#237C2×C41384yescomplemented
84C2×#237C23164yes—
85C22×C4×D3—48no—
86C22×C4×D3C2×C4116yescomplemented
87C22×C4×D3C23124yes—
88C22×C4×D3C23112yes—
89C22×Dic6C2×C4116yescomplemented
90C22×Dic6C23112yes—
91C3×C4×D4C23164yescomplemented
92C2×C3×C4⋊C4C23112yes—
93C2×C3×C22⋊C4C23124yes—
94C2×C3×C42C23112yes—
95C2×#217C23124yes—
96D4×Dic3C23164yescomplemented
97C4×#67C231128yescomplemented
98C2×(C2×D3)⋊C4C23112yes—
99C2×(C2×D3)⋊C4C23112yes—
100C2×C4⋊Dic3C23112yes—
101C2×Dic3⋊C4C23124yes—
102C2×Dic3⋊C4C23112yes—
103C2×Dic3⋊C4C23112yes—
104C2×Dic3⋊C4C23124yes—
105C2×Dic3⋊C4C23124yes—
106C2×Dic3⋊C4C231128yescomplemented
107C2×C4×Dic3C23124yes—
108C2×C4×Dic3C23112yes—
109C2×C4×Dic3C23124yes—
110D3×C4⋊C4C23164yescomplemented
111#887C23164yescomplemented
112#67⋊C4C231128yescomplemented
113#877C231128yescomplemented
114#872C23164yescomplemented
115C4×Dic6C23164yescomplemented
116C22×#41—332no—
117C22×#41—324no—
118C23×Q8—31no—
119C23×D4—32no—
120C23×D4C2×D311yescharacteristic
121C24×C4—32no—
122C24×C4—31no—
123C24×C4—32no—
124C24×C4—32no—
125C24×C4C2×D312yes—
126#520—3256no—
127C2×C4×Q8—324no—
128C2×C4×Q8—324no—
129C2×C4×D4—324no—
130C2×C4×D4—348no—
131C2×C4×D4—348no—
132C2×C4×D4C2×D3124yescomplemented
133C2×#117—348no—
134C2×#117—324no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—32no—
137C22×C4⋊C4—32no—
138C22×C4⋊C4—32no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—348no—
141C22×C4⋊C4—31no—
142C22×C4⋊C4—324no—
143C22×C4⋊C4C2×D311yescharacteristic
144C22×C22⋊C4—32no—
145C22×C22⋊C4—31no—
146C22×C22⋊C4—31no—
147C22×C22⋊C4C2×D312yes—
148C22×C42—32no—
149C22×C42—32no—
150C22×C42—31no—
151C22×C42C2×D311yescharacteristic
152C24×C3—28no—
153C24×C3—28no—
154C24×C3C22×C4116yes—
155C24×C3C24112yes—
156C24×C3C2411yescharacteristic
157C23×D3—24no—
158C23×D3—24no—
159C23×D3C22×C418yes—
160C23×D3C2416yes—
161C2×C3×D4C22×C4196yescomplemented
162C2×C3×D4C24116yes—
163C22×C3×C4—28no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—448no—
167C22×C3×C4C22×C418yes—
168C22×C3×C4C2416yes—
169C22×C3×C4C24112yes—
170C22×C3×C4C2416yes—
171C22×C3×C4C24132yes—
172C22×C3×C4C2416yes—
173C2×#67C22×C41192yescomplemented
174C2×#67C24132yes—
175C22×Dic3—28no—
176C22×Dic3—24no—
177C22×Dic3—24no—
178C22×Dic3—448no—
179C22×Dic3C22×C4116yes—
180C22×Dic3C22×C418yes—
181C22×Dic3C22×C41192yescomplemented
182C22×Dic3C2416yes—
183C22×Dic3C2416yes—
184C22×Dic3C24112yes—
185C22×Dic3C24132yes—
186C22×Dic3C24112yes—
187C22×Dic3C2416yes—
188C22×Dic3C24132yes—
189#237C22×C41512yescomplemented
190C2×C4×D3—448no—
191C2×C4×D3C22×C4196yescomplemented
192C2×C4×D3C24132yes—
193C2×C4×D3C24116yes—
194C2×Dic6C22×C4196yescomplemented
195C2×Dic6C24116yes—
196C3×C4⋊C4C24116yes—
197C3×C22⋊C4C24132yes—
198C3×C42C24116yes—
199#217C24132yes—
200(C2×D3)⋊C4C24116yes—
201(C2×D3)⋊C4C24116yes—
202C4⋊Dic3C24116yes—
203Dic3⋊C4C24132yes—
204Dic3⋊C4C24116yes—
205Dic3⋊C4C24116yes—
206Dic3⋊C4C24132yes—
207Dic3⋊C4C24132yes—
208C4×Dic3C24132yes—
209C4×Dic3C24116yes—
210C4×Dic3C24132yes—
211C25—31no—
212C25C22×D312yes—
213C2×#41—3384no—
214C2×#41—364no—
215C22×Q8—316no—
216C22×Q8—312no—
217C22×D4—332no—
218C22×D4—324no—
219C22×D4C4×D3116yescomplemented
220C22×D4C22×D3112yes—
221C23×C4—332no—
222C23×C4—324no—
223C23×C4—31no—
224C23×C4—32no—
225C23×C4—316no—
226C23×C4—312no—
227C23×C4—31no—
228C23×C4—32no—
229C23×C4—31no—
230C23×C4—324no—
231C23×C4—324no—
232C23×C4—48no—
233C23×C4—128no—
234C23×C4—128no—
235C23×C4C22×D311yescharacteristic
236C23×C4C22×D311yescharacteristic
237C23×C4C22×D312yes—
238C23×C4C22×D3124yes—
239C23×C4C22×D311yescharacteristic
240C4×Q8—364no—
241C4×Q8—364no—
242C4×D4—364no—
243C4×D4—3128no—
244C4×D4—3128no—
245C4×D4C22×D3164yescomplemented
246#117—3128no—
247#117—364no—
248C2×C4⋊C4—324no—
249C2×C4⋊C4—312no—
250C2×C4⋊C4—324no—
251C2×C4⋊C4—324no—
252C2×C4⋊C4—312no—
253C2×C4⋊C4—3128no—
254C2×C4⋊C4—312no—
255C2×C4⋊C4—364no—
256C2×C4⋊C4C22×D3112yes—
257C2×C22⋊C4—324no—
258C2×C22⋊C4—312no—
259C2×C22⋊C4—312no—
260C2×C22⋊C4C22×D3124yes—
261C2×C42—324no—
262C2×C42—324no—
263C2×C42—312no—
264C2×C42C22×D3112yes—
265C23×C3—264no—
266C23×C3—224no—
267C23×C3—224no—
268C23×C3C23×C4148yes—
269C23×C3C23×C414yes—
270C23×C3H(5,𝔽2)14yes—
271C23×C32−1+414yes—
272C23×C3C2518yes—
273C23×C3C2513yes—
274C22×D3—232no—
275C22×D3—212no—
276C22×D3—212no—
277C22×D3C23×C4124yes—
278C22×D3C2514yes—
279C3×D4C23×C4164yescomplemented
280C2×C3×C4—224no—
281C2×C3×C4—212no—
282C2×C3×C4—212no—
283C2×C3×C4—432no—
284C2×C3×C4C23×C4124yes—
285C2×C3×C4C2514yes—
286C2×C3×C4C2518yes—
287C2×C3×C4C2514yes—
288C2×C3×C4C2514yes—
289#67C23×C41128yescomplemented
290C2×Dic3—224no—
291C2×Dic3—212no—
292C2×Dic3—212no—
293C2×Dic3—432no—
294C2×Dic3C23×C4148yes—
295C2×Dic3C23×C4124yes—
296C2×Dic3C23×C41128yescomplemented
297C2×Dic3C2514yes—
298C2×Dic3C2514yes—
299C2×Dic3C2518yes—
300C2×Dic3C2518yes—
301C2×Dic3C2514yes—
302C4×D3—432no—
303C4×D3C23×C4164yescomplemented
304Dic6C23×C4164yescomplemented
305C24—28no—
306C24—28no—
307C24—38no—
308C24—36no—
309C24—64no—
310C24—64no—
311C24C2×C4×D3116yes—
312C24C23×D3112yes—
313C24C23×D311yescharacteristic
314#41—3512no—
315C2×Q8—396no—
316C2×Q8—316no—
317C2×D4—3192no—
318C2×D4—332no—
319C2×D4C2×C4×D3196yescomplemented
320C2×D4C23×D3116yes—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—38no—
325C22×C4—316no—
326C22×C4—3192no—
327C22×C4—312no—
328C22×C4—36no—
329C22×C4—332no—
330C22×C4—396no—
331C22×C4—316no—
332C22×C4—36no—
333C22×C4—312no—
334C22×C4—36no—
335C22×C4—332no—
336C22×C4—332no—
337C22×C4—448no—
338C22×C4—68no—
339C22×C4—64no—
340C22×C4—64no—
341C22×C4—1248no—
342C22×C4—1248no—
343C22×C4C2×C4×D318yes—
344C22×C4C23×D316yes—
345C22×C4C23×D3112yes—
346C22×C4C23×D316yes—
347C22×C4C23×D3132yes—
348C22×C4C23×D316yes—
349C4⋊C4—332no—
350C4⋊C4—316no—
351C4⋊C4—332no—
352C4⋊C4—332no—
353C4⋊C4—316no—
354C4⋊C4—316no—
355C4⋊C4C23×D3116yes—
356C22⋊C4—332no—
357C22⋊C4—316no—
358C22⋊C4—316no—
359C22⋊C4C23×D3132yes—
360C42—332no—
361C42—332no—
362C42—316no—
363C42C23×D3116yes—
364C22×C3—28no—
365C22×C3—28no—
366C22×C3—296no—
367C22×C3#520116yescomplemented
368C22×C3C24×C4116yes—
369C22×C3C24×C416yes—
370C22×C3C2×H(5,𝔽2)16yes—
371C22×C3C2×2−1+416yes—
372C22×C3C2611yescharacteristic
373C2×D3—24no—
374C2×D3—24no—
375C2×D3—248no—
376C2×D3C24×C418yes—
377C3×C4—28no—
378C3×C4—24no—
379C3×C4—24no—
380C3×C4C24×C418yes—
381Dic3—28no—
382Dic3—24no—
383Dic3—24no—
384Dic3C24×C4116yes—
385Dic3C24×C418yes—
386C23—224no—
387C23—224no—
388C23—264no—
389C23—324no—
390C23—34no—
391C23—612no—
392C23—612no—
393C23—632no—
394C23C22×C4×D3148yes—
395C23C22×C4×D314yes—
396C23#100414yes—
397C23#101014yes—
398C23C24×D318yes—
399C23C24×D313yes—
400Q8—364no—
401D4—3128no—
402D4C22×C4×D3164yescomplemented
403C2×C4—212no—
404C2×C4—224no—
405C2×C4—212no—
406C2×C4—324no—
407C2×C4—348no—
408C2×C4—3128no—
409C2×C4—38no—
410C2×C4—34no—
411C2×C4—364no—
412C2×C4—34no—
413C2×C4—38no—
414C2×C4—34no—
415C2×C4—432no—
416C2×C4—624no—
417C2×C4—612no—
418C2×C4—612no—
419C2×C4—1232no—
420C2×C4—1232no—
421C2×C4C22×C4×D3124yes—
422C2×C4C24×D314yes—
423C2×C4C24×D318yes—
424C2×C4C24×D314yes—
425C2×C4C24×D314yes—
426C2×C3—216no—
427C2×C3C2×#520112yescomplemented
428C2×C3C25×C411yescharacteristic
429C2×C3C22×H(5,𝔽2)11yescharacteristic
430C2×C3C22×2−1+411yescharacteristic
431D3—28no—
432C22—28no—
433C22—28no—
434C22—296no—
435C22—38no—
436C22—64no—
437C22—64no—
438C22—648no—
439C22#5546116yesdirect product factor
440C22C23×C4×D3116yes—
441C22C23×C4×D316yes—
442C22C2×#100416yes—
443C22C2×#101016yes—
444C22C25×D311yescharacteristic
445C4—24no—
446C4—28no—
447C4—24no—
448C4—38no—
449C4—316no—
450C4—68no—
451C4—64no—
452C4—64no—
453C4C23×C4×D318yes—
454C3C22×#52011yescharacteristic, complemented, Sylow
455C2—216no—
456C2—68no—
457C2C2×#5546112yesdirect product factor
458C2C24×C4×D311yescharacteristic
459C2C22×#100411yescharacteristic
460C2C22×#101011yescharacteristic
subgroupG/H
centerC24C23×D3
commutatorC2×C3C25×C4
FrattiniC22C25×D3
FittingC22×C3×C4×D4C2
socleC24×C3C24
prime radicalC23×#237C2
cosocleC22×C3C26
hypercenterC22×C4×D4D3
hypocenterC3C22×#520
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4×D4C2semidirect
3. C22×D4×Dic3C2semidirect
4. C22×C4×#67C2semidirect
5. C23×Dic3⋊C4C2semidirect
6. C2×#5546C2direct (N × H)
7. C22×D3×C4⋊C4C2semidirect
8. C22×#887C2semidirect
9. C22×#67⋊C4C2semidirect
10. C22×#877C2semidirect
11. C22×#872C2semidirect
12. C22×C4×Dic6C2semidirect
19. C22×#237C4semidirect
24. C2×C3×C4×D4C22semidirect
29. C2×D4×Dic3C22semidirect
30. C2×C4×#67C22semidirect
39. C22×Dic3⋊C4C22semidirect
43. #5546C22direct (N × H)
44. C2×D3×C4⋊C4C22semidirect
45. C2×#887C22semidirect
46. C2×#67⋊C4C22semidirect
47. C2×#877C22semidirect
48. C2×#872C22semidirect
49. C2×C4×Dic6C22semidirect
57. C22×C4×D4D3semidirect
64. C22×C3×D4C2×C4semidirect
72. C22×#67C2×C4semidirect
75. C23×Dic3C2×C4semidirect
83. C2×#237C2×C4semidirect
86. C22×C4×D3C2×C4semidirect
89. C22×Dic6C2×C4semidirect
91. C3×C4×D4C23semidirect
96. D4×Dic3C23semidirect
97. C4×#67C23semidirect
106. C2×Dic3⋊C4C23semidirect
110. D3×C4⋊C4C23semidirect
111. #887C23semidirect
112. #67⋊C4C23semidirect
113. #877C23semidirect
114. #872C23semidirect
115. C4×Dic6C23semidirect
132. C2×C4×D4C2×D3semidirect
161. C2×C3×D4C22×C4semidirect
173. C2×#67C22×C4semidirect
181. C22×Dic3C22×C4semidirect
189. #237C22×C4semidirect
191. C2×C4×D3C22×C4semidirect
194. C2×Dic6C22×C4semidirect
219. C22×D4C4×D3semidirect
245. C4×D4C22×D3semidirect
279. C3×D4C23×C4semidirect
289. #67C23×C4semidirect
296. C2×Dic3C23×C4semidirect
303. C4×D3C23×C4semidirect
304. Dic6C23×C4semidirect
319. C2×D4C2×C4×D3semidirect
367. C22×C3#520semidirect
402. D4C22×C4×D3semidirect
427. C2×C3C2×#520semidirect
439. C22#5546direct (N × H)
454. C3C22×#520semidirect
457. C2C2×#5546direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#2372
C22×C3×C4×D42
C22×D4×Dic32
C22×C4×#672
C23×Dic3⋊C42
C2×#55462
C22×D3×C4⋊C42
C22×#8872
C22×#67⋊C42
C22×#8772
C22×#8722
C22×C4×Dic62
C22×#5203