Finite Group Databasebeta

← C22×C4×#237 #11701759

Subgroups — C22×C4×#237 #11701759

Order 768.

12878 conjugacy classes of subgroups.

The subgroup lattice has 468 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×C4×Dic3C212yesmaximal, complemented
6C2×C4×#237C2148yesmaximal, direct product factor
7C22×#887C211yesmaximal, characteristic, complemented
8C22×#892C211yesmaximal, characteristic, complemented
9C22×#67⋊C4C212yesmaximal, complemented
10C22×#877C212yesmaximal, complemented
11C22×C42×D3C211yesmaximal, characteristic, complemented
12C22×C4×Dic6C211yesmaximal, characteristic, complemented
13C22×C4×#41—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C24×C3×C4C2212yes—
16C23×#67C2212yes—
17C24×Dic3C2212yes—
18C24×Dic3C2212yes—
19C22×#237C4132yesdirect product factor
20C22×#237C22124yes—
21C23×C4×D3C2211yescharacteristic
22C23×C4×D3C2211yescharacteristic
23C23×C4×D3C2211yescharacteristic
24C23×Dic6C2211yescharacteristic
25C2×C3×C4×D4C22124yescomplemented
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×C3×C22⋊C4C2212yes—
28C22×C3×C42C2211yescharacteristic
29C22×#217C2212yes—
30C2×D4×Dic3C22124yescomplemented
31C2×C4×#67C22148yescomplemented
32C22×(C2×D3)⋊C4C2212yes—
33C22×C4⋊Dic3C2211yescharacteristic
34C22×Dic3⋊C4C2212yes—
35C22×Dic3⋊C4C2212yes—
36C22×C4×Dic3C2212yes—
37C22×C4×Dic3C2211yescharacteristic
38C22×C4×Dic3C2211yescharacteristic
39C22×C4×Dic3C2211yescharacteristic
40C22×C4×Dic3C2212yes—
41C22×C4×Dic3C2212yes—
42C22×C4×Dic3C22148yescomplemented
43C4×#237C221256yesdirect product factor
44C2×#892C22124yescomplemented
45C2×#887C22124yescomplemented
46C2×#67⋊C4C22148yescomplemented
47C2×#877C22148yescomplemented
48C2×C42×D3C22124yescomplemented
49C2×C4×Dic6C22124yescomplemented
50C23×#41—31no—
51C2×C4×#41—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×C42—31no—
61C23×C42—32no—
62C25×C3C2312yes—
63C24×D3C2311yescharacteristic
64C22×C3×D4C2×C4116yescomplemented
65C22×C3×D4C23112yes—
66C23×C3×C4—216no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2312yes—
69C23×C3×C4C23124yes—
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67C2×C4132yescomplemented
73C22×#67C23124yes—
74C23×Dic3—216no—
75C23×Dic3C2×C4132yescomplemented
76C23×Dic3C2311yescharacteristic
77C23×Dic3C23124yes—
78C23×Dic3C2312yes—
79C23×Dic3C2312yes—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C23124yes—
82C23×Dic3C2311yescharacteristic
83C2×#237C2×C41384yesdirect product factor
84C2×#237C23164yes—
85C22×C4×D3—28no—
86C22×C4×D3—28no—
87C22×C4×D3C2×C4116yescomplemented
88C22×C4×D3C23112yes—
89C22×C4×D3C23112yes—
90C22×C4×D3C23112yes—
91C22×Dic6C2×C4116yescomplemented
92C22×Dic6C23112yes—
93C3×C4×D4C23164yescomplemented
94C2×C3×C4⋊C4C23112yes—
95C2×C3×C22⋊C4C23124yes—
96C2×C3×C42C23112yes—
97C2×#217C23124yes—
98D4×Dic3C23164yescomplemented
99C4×#67C231128yescomplemented
100C2×(C2×D3)⋊C4C23124yes—
101C2×C4⋊Dic3C23112yes—
102C2×Dic3⋊C4C23124yes—
103C2×Dic3⋊C4C23124yes—
104C2×C4×Dic3C23124yes—
105C2×C4×Dic3C23112yes—
106C2×C4×Dic3C23112yes—
107C2×C4×Dic3C23112yes—
108C2×C4×Dic3C23124yes—
109C2×C4×Dic3C23124yes—
110C2×C4×Dic3C231128yescomplemented
111#892C23164yescomplemented
112#887C23164yescomplemented
113#67⋊C4C231128yescomplemented
114#877C231128yescomplemented
115C42×D3C23164yescomplemented
116C4×Dic6C23164yescomplemented
117C22×#41—332no—
118C22×#41—324no—
119C23×Q8—31no—
120C23×D4—32no—
121C23×D4C2×D311yescharacteristic
122C24×C4—32no—
123C24×C4—31no—
124C24×C4—32no—
125C24×C4—31no—
126C24×C4—31no—
127C24×C4C2×D312yes—
128C4×#41—3256no—
129C2×C4×Q8—324no—
130C2×C4×Q8—324no—
131C2×C4×D4—324no—
132C2×C4×D4—348no—
133C2×C4×D4—348no—
134C2×C4×D4C2×D3124yescomplemented
135C2×#117—324no—
136C2×#117—348no—
137C22×C4⋊C4—32no—
138C22×C4⋊C4—32no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4C2×D311yescharacteristic
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4C2×D312yes—
144C22×C42—32no—
145C22×C42—31no—
146C22×C42—32no—
147C22×C42—31no—
148C22×C42—348no—
149C22×C42—31no—
150C22×C42—32no—
151C22×C42—324no—
152C22×C42C2×D311yescharacteristic
153C24×C3—28no—
154C24×C3—28no—
155C24×C3C22×C4116yes—
156C24×C3C24112yes—
157C24×C3C2411yescharacteristic
158C23×D3—24no—
159C23×D3—24no—
160C23×D3C22×C418yes—
161C23×D3C2416yes—
162C2×C3×D4C22×C4196yescomplemented
163C2×C3×D4C24116yes—
164C22×C3×C4—24no—
165C22×C3×C4—28no—
166C22×C3×C4—296no—
167C22×C3×C4C22×C418yes—
168C22×C3×C4#4118yescomplemented
169C22×C3×C4C24112yes—
170C22×C3×C4C2416yes—
171C22×C3×C4C24132yes—
172C22×C3×C4C2416yes—
173C22×C3×C4C2416yes—
174C2×#67C22×C41192yescomplemented
175C2×#67C24132yes—
176C22×Dic3—24no—
177C22×Dic3—28no—
178C22×Dic3—296no—
179C22×Dic3C22×C4116yes—
180C22×Dic3C22×C418yes—
181C22×Dic3C22×C41192yescomplemented
182C22×Dic3#4118yescomplemented
183C22×Dic3C24132yes—
184C22×Dic3C2416yes—
185C22×Dic3C24112yes—
186C22×Dic3C2416yes—
187C22×Dic3C24112yes—
188C22×Dic3C24132yes—
189C22×Dic3C2416yes—
190#237C22×C41512yesdirect product factor
191C2×C4×D3—248no—
192C2×C4×D3—248no—
193C2×C4×D3C22×C4196yescomplemented
194C2×C4×D3C24116yes—
195C2×C4×D3C24116yes—
196C2×C4×D3C24116yes—
197C2×Dic6C22×C4196yescomplemented
198C2×Dic6C24116yes—
199C3×C4⋊C4C24116yes—
200C3×C22⋊C4C24132yes—
201C3×C42C24116yes—
202#217C24132yes—
203(C2×D3)⋊C4C24132yes—
204C4⋊Dic3C24116yes—
205Dic3⋊C4C24132yes—
206Dic3⋊C4C24132yes—
207C4×Dic3C24132yes—
208C4×Dic3C24116yes—
209C4×Dic3C24116yes—
210C4×Dic3C24116yes—
211C4×Dic3C24132yes—
212C4×Dic3C24132yes—
213C25—31no—
214C25C22×D312yes—
215C2×#41—3384no—
216C2×#41—364no—
217C22×Q8—316no—
218C22×Q8—312no—
219C22×D4—332no—
220C22×D4—324no—
221C22×D4C4×D3116yescomplemented
222C22×D4C22×D3112yes—
223C23×C4—216no—
224C23×C4—332no—
225C23×C4—32no—
226C23×C4—324no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—316no—
230C23×C4—312no—
231C23×C4—31no—
232C23×C4—32no—
233C23×C4—324no—
234C23×C4—312no—
235C23×C4—312no—
236C23×C4—68no—
237C23×C4—616no—
238C23×C4—68no—
239C23×C4C22×D312yes—
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D3124yes—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C4×Q8—364no—
245C4×Q8—364no—
246C4×D4—364no—
247C4×D4—3128no—
248C4×D4—3128no—
249C4×D4C22×D3164yescomplemented
250#117—364no—
251#117—3128no—
252C2×C4⋊C4—324no—
253C2×C4⋊C4—324no—
254C2×C4⋊C4—312no—
255C2×C4⋊C4C22×D3112yes—
256C2×C22⋊C4—324no—
257C2×C22⋊C4—324no—
258C2×C22⋊C4C22×D3124yes—
259C2×C42—324no—
260C2×C42—312no—
261C2×C42—324no—
262C2×C42—312no—
263C2×C42—324no—
264C2×C42—3128no—
265C2×C42—312no—
266C2×C42—364no—
267C2×C42C22×D3112yes—
268C23×C3—264no—
269C23×C3—224no—
270C23×C3—224no—
271C23×C3C23×C4148yes—
272C23×C3C23×C414yes—
273C23×C3C2×#4114yes—
274C23×C3C2×#4114yes—
275C23×C3C2518yes—
276C23×C3C2513yes—
277C22×D3—232no—
278C22×D3—212no—
279C22×D3—212no—
280C22×D3C23×C4124yes—
281C22×D3C2514yes—
282C3×D4C23×C4164yescomplemented
283C2×C3×C4—212no—
284C2×C3×C4—224no—
285C2×C3×C4—264no—
286C2×C3×C4C23×C4124yes—
287C2×C3×C4C2×#41124yescomplemented
288C2×C3×C4C2518yes—
289C2×C3×C4C2514yes—
290C2×C3×C4C2514yes—
291C2×C3×C4C2514yes—
292#67C23×C41128yescomplemented
293C2×Dic3—212no—
294C2×Dic3—224no—
295C2×Dic3—264no—
296C2×Dic3C23×C4148yes—
297C2×Dic3C23×C4124yes—
298C2×Dic3C23×C41128yescomplemented
299C2×Dic3C2×#41124yescomplemented
300C2×Dic3C2514yes—
301C2×Dic3C2518yes—
302C2×Dic3C2514yes—
303C2×Dic3C2518yes—
304C2×Dic3C2514yes—
305C4×D3—232no—
306C4×D3—232no—
307C4×D3C23×C4164yescomplemented
308Dic6C23×C4164yescomplemented
309C24—28no—
310C24—28no—
311C24—38no—
312C24—36no—
313C24—64no—
314C24—64no—
315C24C2×C4×D3116yes—
316C24C23×D3112yes—
317C24C23×D311yescharacteristic
318#41—3512no—
319C2×Q8—396no—
320C2×Q8—316no—
321C2×D4—3192no—
322C2×D4—332no—
323C2×D4C2×C4×D3196yescomplemented
324C2×D4C23×D3116yes—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—296no—
328C22×C4—38no—
329C22×C4—316no—
330C22×C4—3192no—
331C22×C4—38no—
332C22×C4—36no—
333C22×C4—36no—
334C22×C4—312no—
335C22×C4—332no—
336C22×C4—396no—
337C22×C4—316no—
338C22×C4—36no—
339C22×C4—312no—
340C22×C4—332no—
341C22×C4—316no—
342C22×C4—316no—
343C22×C4—648no—
344C22×C4—696no—
345C22×C4—64no—
346C22×C4—68no—
347C22×C4—648no—
348C22×C4C2×C4×D318yes—
349C22×C4#23718yesdirect product factor
350C22×C4C23×D3112yes—
351C22×C4C23×D3132yes—
352C22×C4C23×D316yes—
353C22×C4C23×D316yes—
354C22×C4C23×D316yes—
355C4⋊C4—332no—
356C4⋊C4—332no—
357C4⋊C4—316no—
358C4⋊C4C23×D3116yes—
359C22⋊C4—332no—
360C22⋊C4—332no—
361C22⋊C4C23×D3132yes—
362C42—316no—
363C42—332no—
364C42—316no—
365C42—332no—
366C42—316no—
367C42—332no—
368C42C23×D3116yes—
369C22×C3—28no—
370C22×C3—28no—
371C22×C3—296no—
372C22×C3C4×#41116yescomplemented
373C22×C3C24×C4116yes—
374C22×C3C24×C416yes—
375C22×C3C22×#4116yes—
376C22×C3C22×#4116yes—
377C22×C3C2611yescharacteristic
378C2×D3—24no—
379C2×D3—24no—
380C2×D3—248no—
381C2×D3C24×C418yes—
382C3×C4—24no—
383C3×C4—28no—
384C3×C4C24×C418yes—
385C3×C4C22×#4118yescomplemented
386Dic3—24no—
387Dic3—28no—
388Dic3C24×C4116yes—
389Dic3C24×C418yes—
390Dic3C22×#4118yescomplemented
391C23—224no—
392C23—224no—
393C23—264no—
394C23—324no—
395C23—34no—
396C23—612no—
397C23—612no—
398C23—632no—
399C23C22×C4×D3148yes—
400C23C22×C4×D314yes—
401C23C2×#23714yes—
402C23D3×#4114yes—
403C23C24×D318yes—
404C23C24×D313yes—
405Q8—364no—
406D4—3128no—
407D4C22×C4×D3164yescomplemented
408C2×C4—224no—
409C2×C4—212no—
410C2×C4—264no—
411C2×C4—324no—
412C2×C4—348no—
413C2×C4—3128no—
414C2×C4—324no—
415C2×C4—34no—
416C2×C4—34no—
417C2×C4—38no—
418C2×C4—364no—
419C2×C4—34no—
420C2×C4—38no—
421C2×C4—632no—
422C2×C4—664no—
423C2×C4—612no—
424C2×C4—624no—
425C2×C4—632no—
426C2×C4C22×C4×D3124yes—
427C2×C4C2×#237124yesdirect product factor
428C2×C4C24×D318yes—
429C2×C4C24×D314yes—
430C2×C4C24×D314yes—
431C2×C4C24×D314yes—
432C2×C3—216no—
433C2×C3C2×C4×#41112yescomplemented
434C2×C3C25×C411yescharacteristic
435C2×C3C23×#4111yescharacteristic
436C2×C3C23×#4111yescharacteristic
437D3—28no—
438C22—28no—
439C22—28no—
440C22—296no—
441C22—38no—
442C22—64no—
443C22—64no—
444C22—648no—
445C22C4×#237116yesdirect product factor
446C22C23×C4×D3116yes—
447C22C23×C4×D316yes—
448C22C22×#23716yes—
449C22C2×D3×#4116yes—
450C22C25×D311yescharacteristic
451C4—28no—
452C4—24no—
453C4—38no—
454C4—316no—
455C4—38no—
456C4—64no—
457C4—68no—
458C4C23×C4×D318yes—
459C4C22×#23718yesdirect product factor
460C3C22×C4×#4111yescharacteristic, complemented, Sylow
461C2—216no—
462C2—68no—
463C2C2×C4×#237112yesdirect product factor
464C2C24×C4×D311yescharacteristic
465C2C23×#23711yescharacteristic
466C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC23×C4C22×D3
commutatorC2×C3C25×C4
FrattiniC22C25×D3
FittingC22×C3×C4×D4C2
socleC24×C3C24
prime radicalC23×#237C2
cosocleC22×C3C26
hypercenterC22×C4×D4D3
hypocenterC3C22×C4×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ 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×C4×Dic3C2semidirect
6. C2×C4×#237C2direct (N × H)
7. C22×#887C2semidirect
8. C22×#892C2semidirect
9. C22×#67⋊C4C2semidirect
10. C22×#877C2semidirect
11. C22×C42×D3C2semidirect
12. C22×C4×Dic6C2semidirect
19. C22×#237C4direct (N × H)
25. C2×C3×C4×D4C22semidirect
30. C2×D4×Dic3C22semidirect
31. C2×C4×#67C22semidirect
42. C22×C4×Dic3C22semidirect
43. C4×#237C22direct (N × H)
44. C2×#892C22semidirect
45. C2×#887C22semidirect
46. C2×#67⋊C4C22semidirect
47. C2×#877C22semidirect
48. C2×C42×D3C22semidirect
49. C2×C4×Dic6C22semidirect
57. C22×C4×D4D3semidirect
64. C22×C3×D4C2×C4semidirect
72. C22×#67C2×C4semidirect
75. C23×Dic3C2×C4semidirect
83. C2×#237C2×C4direct (N × H)
87. C22×C4×D3C2×C4semidirect
91. C22×Dic6C2×C4semidirect
93. C3×C4×D4C23semidirect
98. D4×Dic3C23semidirect
99. C4×#67C23semidirect
110. C2×C4×Dic3C23semidirect
111. #892C23semidirect
112. #887C23semidirect
113. #67⋊C4C23semidirect
114. #877C23semidirect
115. C42×D3C23semidirect
116. C4×Dic6C23semidirect
134. C2×C4×D4C2×D3semidirect
162. C2×C3×D4C22×C4semidirect
168. C22×C3×C4#41semidirect
174. C2×#67C22×C4semidirect
181. C22×Dic3C22×C4semidirect
182. C22×Dic3#41semidirect
190. #237C22×C4direct (N × H)
193. C2×C4×D3C22×C4semidirect
197. C2×Dic6C22×C4semidirect
221. C22×D4C4×D3semidirect
249. C4×D4C22×D3semidirect
282. C3×D4C23×C4semidirect
287. C2×C3×C4C2×#41semidirect
292. #67C23×C4semidirect
298. C2×Dic3C23×C4semidirect
299. C2×Dic3C2×#41semidirect
307. C4×D3C23×C4semidirect
308. Dic6C23×C4semidirect
323. C2×D4C2×C4×D3semidirect
349. C22×C4#237direct (N × H)
372. C22×C3C4×#41semidirect
385. C3×C4C22×#41semidirect
390. Dic3C22×#41semidirect
407. D4C22×C4×D3semidirect
427. C2×C4C2×#237direct (N × H)
433. C2×C3C2×C4×#41semidirect
445. C22C4×#237direct (N × H)
459. C4C22×#237direct (N × H)
460. C3C22×C4×#41semidirect
463. C2C2×C4×#237direct (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×C4×Dic32
C2×C4×#2372
C22×#8872
C22×#8922
C22×#67⋊C42
C22×#8772
C22×C42×D32
C22×C4×Dic62
C22×C4×#413