Finite Group Databasebeta

← C22×#5729 #11701953

Subgroups — C22×#5729 #11701953

Order 768.

7630 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C22×#946C212yesmaximal
3C22×#944C212yesmaximal, complemented
4C2×#5729C2148yesmaximal, direct product factor
5C22×#896C212yesmaximal
6C22×D3×C4⋊C4C211yesmaximal, characteristic
7C22×#888C211yesmaximal, characteristic, complemented
8C22×#890C211yesmaximal, characteristic, complemented
9C22×#889C212yesmaximal, complemented
10C22×#892C211yesmaximal, characteristic
11C22×#872C211yesmaximal, characteristic
12C22×#870C211yesmaximal, characteristic, complemented
13C22×#544—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C23×C4×D3C2212yes—
16C23×C4×D3C2211yescharacteristic
17C23×Dic6C2212yes—
18C2×C3×C4⋊Q8C22124yescomplemented
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2212yes—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C2×#946C22148yes—
24C2×#944C22148yescomplemented
25C22×(C2×D3)⋊C4C2211yescharacteristic
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2212yes—
28C22×C4⋊Dic3C2211yescharacteristic
29C22×C4⋊Dic3C2211yescharacteristic
30C22×C4⋊Dic3C2212yes—
31C22×Dic3⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×Dic3⋊C4C2211yescharacteristic
34C22×Dic3⋊C4C2211yescharacteristic
35C22×Dic3⋊C4C2212yes—
36C22×Dic3⋊C4C2212yes—
37C22×Dic3⋊C4C2212yes—
38C22×C4×Dic3C2211yescharacteristic
39C22×C4×Dic3C2212yes—
40#5729C221256yesdirect product factor
41C2×#896C22148yes—
42C2×#892C22124yes—
43C2×D3×C4⋊C4C22124yes—
44C2×#890C22124yescomplemented
45C2×#889C22148yescomplemented
46C2×#888C22124yescomplemented
47C2×#872C22124yes—
48C2×#870C22124yescomplemented
49C2×#544—348no—
50C22×C4⋊Q8—32no—
51C22×C4⋊Q8—31no—
52C22×C4⋊Q8D311yescharacteristic, complemented
53C22×#125—31no—
54C22×#125—32no—
55C22×#125—31no—
56C22×C22⋊Q8—32no—
57C22×C22⋊Q8—32no—
58C22×#117—31no—
59C22×#117—31no—
60C23×C4⋊C4—31no—
61C24×D3C2311yescharacteristic
62C22×C3×Q8—48no—
63C22×C3×Q8C23124yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2312yes—
73C23×Dic3C2311yescharacteristic
74C22×C4×D3—48no—
75C22×C4×D3Q8116yescomplemented
76C22×C4×D3C23124yes—
77C22×C4×D3C23112yes—
78C22×Dic6—48no—
79C22×Dic6C23124yes—
80C3×C4⋊Q8C23164yescomplemented
81C2×C3×C4⋊C4C23112yes—
82C2×C3×C4⋊C4C23124yes—
83C2×C3×C4⋊C4C23112yes—
84C2×C3×C42C23112yes—
85#946C231128yes—
86#944C231128yescomplemented
87C2×(C2×D3)⋊C4C23112yes—
88C2×(C2×D3)⋊C4C23112yes—
89C2×(C2×D3)⋊C4C23124yes—
90C2×C4⋊Dic3C23112yes—
91C2×C4⋊Dic3C23112yes—
92C2×C4⋊Dic3C23124yes—
93C2×Dic3⋊C4C23124yes—
94C2×Dic3⋊C4C23124yes—
95C2×Dic3⋊C4C23112yes—
96C2×Dic3⋊C4C23112yes—
97C2×Dic3⋊C4C23124yes—
98C2×Dic3⋊C4C23124yes—
99C2×Dic3⋊C4C23124yes—
100C2×C4×Dic3C23112yes—
101C2×C4×Dic3C23124yes—
102#896C231128yes—
103#892C23164yes—
104D3×C4⋊C4C23164yes—
105#890C23164yescomplemented
106#889C231128yescomplemented
107#888C23164yescomplemented
108#872C23164yes—
109#870C23164yescomplemented
110C23×Q8—32no—
111C23×Q8C2×D312yes—
112C24×C4—31no—
113C24×C4—32no—
114#544—3256no—
115C2×C4⋊Q8—348no—
116C2×C4⋊Q8—324no—
117C2×C4⋊Q8C2×D3124yescomplemented
118C2×#125—348no—
119C2×#125—324no—
120C2×#125—324no—
121C2×C22⋊Q8—348no—
122C2×C22⋊Q8—348no—
123C2×#117—324no—
124C2×#117—324no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—31no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—32no—
131C22×C4⋊C4—32no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4—32no—
134C22×C4⋊C4—32no—
135C22×C4⋊C4—324no—
136C22×C4⋊C4C2×D311yescharacteristic
137C22×C4⋊C4C2×D311yescharacteristic
138C22×C4⋊C4C2×D312yes—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—31no—
142C22×C42—32no—
143C22×C42—31no—
144C22×C42C2×D311yescharacteristic
145C24×C3C2411yescharacteristic
146C23×D3—24no—
147C23×D3—24no—
148C23×D3C2×Q818yes—
149C23×D3C2416yes—
150C2×C3×Q8—448no—
151C2×C3×Q8C24132yes—
152C22×C3×C4—28no—
153C22×C3×C4—28no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4C2×Q818yes—
157C22×C3×C4C24112yes—
158C22×C3×C4C2416yes—
159C22×C3×C4C2416yes—
160C22×C3×C4C24112yes—
161C22×C3×C4C2416yes—
162C22×Dic3—24no—
163C22×Dic3—28no—
164C22×Dic3—24no—
165C22×Dic3—28no—
166C22×Dic3C2×Q818yes—
167C22×Dic3C2416yes—
168C22×Dic3C24112yes—
169C22×Dic3C2416yes—
170C22×Dic3C24112yes—
171C22×Dic3C2416yes—
172C2×C4×D3—448no—
173C2×C4×D3C2×Q8196yescomplemented
174C2×C4×D3C24132yes—
175C2×C4×D3C24116yes—
176C2×Dic6—448no—
177C2×Dic6C24132yes—
178C3×C4⋊C4C24116yes—
179C3×C4⋊C4C24132yes—
180C3×C4⋊C4C24116yes—
181C3×C42C24116yes—
182(C2×D3)⋊C4C24116yes—
183(C2×D3)⋊C4C24116yes—
184(C2×D3)⋊C4C24132yes—
185C4⋊Dic3C24116yes—
186C4⋊Dic3C24116yes—
187C4⋊Dic3C24132yes—
188Dic3⋊C4C24132yes—
189Dic3⋊C4C24132yes—
190Dic3⋊C4C24116yes—
191Dic3⋊C4C24116yes—
192Dic3⋊C4C24132yes—
193Dic3⋊C4C24132yes—
194Dic3⋊C4C24132yes—
195C4×Dic3C24116yes—
196C4×Dic3C24132yes—
197C25—31no—
198C22×Q8—324no—
199C22×Q8—48no—
200C22×Q8—128no—
201C22×Q8C22×D3124yes—
202C23×C4—31no—
203C23×C4—316no—
204C23×C4—312no—
205C23×C4—31no—
206C23×C4—32no—
207C23×C4—31no—
208C23×C4—32no—
209C23×C4—324no—
210C23×C4—128no—
211C23×C4C22×D311yescharacteristic
212C23×C4C22×D311yescharacteristic
213C23×C4C22×D312yes—
214C23×C4C22×D312yes—
215C23×C4C22×D311yescharacteristic
216C4⋊Q8—3128no—
217C4⋊Q8—364no—
218C4⋊Q8C22×D3164yescomplemented
219#125—3128no—
220#125—364no—
221#125—364no—
222C22⋊Q8—3128no—
223C22⋊Q8—3128no—
224#117—364no—
225#117—364no—
226C2×C4⋊C4—324no—
227C2×C4⋊C4—312no—
228C2×C4⋊C4—312no—
229C2×C4⋊C4—312no—
230C2×C4⋊C4—312no—
231C2×C4⋊C4—324no—
232C2×C4⋊C4—324no—
233C2×C4⋊C4—324no—
234C2×C4⋊C4—324no—
235C2×C4⋊C4—324no—
236C2×C4⋊C4—364no—
237C2×C4⋊C4C22×D3112yes—
238C2×C4⋊C4C22×D3112yes—
239C2×C4⋊C4C22×D3124yes—
240C2×C22⋊C4—324no—
241C2×C22⋊C4—312no—
242C2×C22⋊C4—312no—
243C2×C42—324no—
244C2×C42—312no—
245C2×C42C22×D3112yes—
246C23×C3C22×Q814yes—
247C23×C3H(5,𝔽2)14yes—
248C23×C32−1+414yes—
249C23×C3C2513yes—
250C22×D3—232no—
251C22×D3—212no—
252C22×D3—212no—
253C22×D3C22×Q8124yes—
254C22×D3C2514yes—
255C3×Q8—432no—
256C2×C3×C4—224no—
257C2×C3×C4—224no—
258C2×C3×C4—212no—
259C2×C3×C4—212no—
260C2×C3×C4C22×Q8124yes—
261C2×C3×C4C2518yes—
262C2×C3×C4C2514yes—
263C2×C3×C4C2514yes—
264C2×C3×C4C2518yes—
265C2×C3×C4C2514yes—
266C2×Dic3—212no—
267C2×Dic3—224no—
268C2×Dic3—212no—
269C2×Dic3—224no—
270C2×Dic3C22×Q8124yes—
271C2×Dic3C2514yes—
272C2×Dic3C2518yes—
273C2×Dic3C2514yes—
274C2×Dic3C2518yes—
275C2×Dic3C2514yes—
276C4×D3—432no—
277C4×D3C22×Q8164yescomplemented
278Dic6—432no—
279C24—38no—
280C24—36no—
281C24—64no—
282C24—64no—
283C24C23×D311yescharacteristic
284C2×Q8—332no—
285C2×Q8—448no—
286C2×Q8—1248no—
287C2×Q8C23×D3132yes—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—38no—
293C22×C4—36no—
294C22×C4—396no—
295C22×C4—316no—
296C22×C4—36no—
297C22×C4—312no—
298C22×C4—36no—
299C22×C4—312no—
300C22×C4—332no—
301C22×C4—68no—
302C22×C4—64no—
303C22×C4—64no—
304C22×C4—68no—
305C22×C4—1248no—
306C22×C4D3×Q818yes—
307C22×C4C23×D3112yes—
308C22×C4C23×D316yes—
309C22×C4C23×D316yes—
310C22×C4C23×D3112yes—
311C22×C4C23×D316yes—
312C4⋊C4—332no—
313C4⋊C4—316no—
314C4⋊C4—316no—
315C4⋊C4—316no—
316C4⋊C4—316no—
317C4⋊C4—332no—
318C4⋊C4—332no—
319C4⋊C4—332no—
320C4⋊C4—332no—
321C4⋊C4—332no—
322C4⋊C4C23×D3116yes—
323C4⋊C4C23×D3116yes—
324C4⋊C4C23×D3132yes—
325C22⋊C4—332no—
326C22⋊C4—316no—
327C22⋊C4—316no—
328C42—332no—
329C42—316no—
330C42C23×D3116yes—
331C22×C3#544116yescomplemented
332C22×C3C23×Q816yes—
333C22×C3C2×H(5,𝔽2)16yes—
334C22×C3C2×2−1+416yes—
335C22×C3C2611yescharacteristic
336C2×D3—24no—
337C2×D3—24no—
338C2×D3—248no—
339C2×D3C23×Q818yes—
340C3×C4—28no—
341C3×C4—28no—
342C3×C4—24no—
343C3×C4—24no—
344C3×C4C23×Q818yes—
345Dic3—24no—
346Dic3—28no—
347Dic3—24no—
348Dic3—28no—
349Dic3C23×Q818yes—
350C23—324no—
351C23—34no—
352C23—612no—
353C23—612no—
354C23—632no—
355C23#100414yes—
356C23C2×D3×Q814yes—
357C23#100714yes—
358C23C24×D313yes—
359Q8—432no—
360Q8—1232no—
361C2×C4—224no—
362C2×C4—212no—
363C2×C4—224no—
364C2×C4—212no—
365C2×C4—324no—
366C2×C4—34no—
367C2×C4—364no—
368C2×C4—34no—
369C2×C4—38no—
370C2×C4—34no—
371C2×C4—38no—
372C2×C4—624no—
373C2×C4—612no—
374C2×C4—612no—
375C2×C4—624no—
376C2×C4—1232no—
377C2×C4C2×D3×Q8124yes—
378C2×C4C24×D318yes—
379C2×C4C24×D314yes—
380C2×C4C24×D314yes—
381C2×C4C24×D318yes—
382C2×C4C24×D314yes—
383C2×C3C2×#544112yescomplemented
384C2×C3C24×Q811yescharacteristic
385C2×C3C22×H(5,𝔽2)11yescharacteristic
386C2×C3C22×2−1+411yescharacteristic
387D3—28no—
388C22—38no—
389C22—64no—
390C22—64no—
391C22—648no—
392C22#5729116yesdirect product factor
393C22C2×#100416yes—
394C22C22×D3×Q816yes—
395C22C2×#100716yes—
396C22C25×D311yescharacteristic
397C4—28no—
398C4—24no—
399C4—28no—
400C4—24no—
401C4—38no—
402C4—68no—
403C4—64no—
404C4—64no—
405C4—68no—
406C4C22×D3×Q818yes—
407C3C22×#54411yescharacteristic, complemented, Sylow
408C2—68no—
409C2C2×#5729112yesdirect product factor
410C2C22×#100411yescharacteristic
411C2C23×D3×Q811yescharacteristic
412C2C22×#100711yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×C4⋊Q8C2
socleC24×C3C24
prime radicalC24×D3C23
hypercenterC22×C4⋊Q8D3
hypocenterC3C22×#544
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
3. C22×#944C2semidirect
4. C2×#5729C2direct (N × H)
7. C22×#888C2semidirect
8. C22×#890C2semidirect
9. C22×#889C2semidirect
12. C22×#870C2semidirect
18. C2×C3×C4⋊Q8C22semidirect
24. C2×#944C22semidirect
40. #5729C22direct (N × H)
44. C2×#890C22semidirect
45. C2×#889C22semidirect
46. C2×#888C22semidirect
48. C2×#870C22semidirect
52. C22×C4⋊Q8D3semidirect
75. C22×C4×D3Q8semidirect
80. C3×C4⋊Q8C23semidirect
86. #944C23semidirect
105. #890C23semidirect
106. #889C23semidirect
107. #888C23semidirect
109. #870C23semidirect
117. C2×C4⋊Q8C2×D3semidirect
173. C2×C4×D3C2×Q8semidirect
218. C4⋊Q8C22×D3semidirect
277. C4×D3C22×Q8semidirect
331. C22×C3#544semidirect
383. C2×C3C2×#544semidirect
392. C22#5729direct (N × H)
407. C3C22×#544semidirect
409. C2C2×#5729direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C22×#9462
C22×#9442
C2×#57292
C22×#8962
C22×D3×C4⋊C42
C22×#8882
C22×#8902
C22×#8892
C22×#8922
C22×#8722
C22×#8702
C22×#5443