Finite Group Databasebeta

← C23×#19387 #19167013

Subgroups — C23×#19387 #19167013

Order 5888.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3209C211yesmaximal, characteristic, complemented
2C23×#3077C211yesmaximal, characteristic
3C2×C23×#474C211yesmaximal, characteristic
4C23×#2972C211yesmaximal, characteristic
5C23×#1683C214yesmaximal, complemented
6C23×#1669C214yesmaximal, complemented
7C23×#1571C211yesmaximal, characteristic, complemented
8C23×#1549C211yesmaximal, characteristic, complemented
9C23×#1542C211yesmaximal, characteristic, complemented
10C23×#533C2211yescharacteristic
11C2×C23×C4⋊Q8C2211yescharacteristic
12C2×C4×D4×C23C2211yescharacteristic
13C23×#494C2212yescomplemented
14C23×#487C2212yescomplemented
15C23×#478—22no—
16C23×#476C2212yes—
17C23×#474—22no—
18C23×#474C2212yes—
19C23×#474C2214yes—
20C23×#462—22no—
21C23×#451—22no—
22C8×D4×C23—22no—
23C8×D4×C23C2214yes—
24C23×#431C2211yescharacteristic
25C23×#431C2211yescharacteristic
26C23×#428—21no—
27C2×C23×#107—21no—
28C23×#423C2211yescharacteristic
29C2×C23×C4⋊C8C2211yescharacteristic
30C23×#418—21no—
31C2×C23×#102—21no—
32C2×C23×C22⋊C8—22no—
33C23×#335C2214yescomplemented
34C23×#333C2212yescomplemented
35C23×#332C2212yescomplemented
36C23×#332C2214yescomplemented
37C22×Q8×C23—21no—
38C22×D4×C23D411yescharacteristic
39C23×C4×C23—21no—
40C2×Q16×C23—24no—
41C2×M16×C23—21no—
42C22×C8×C23—21no—
43C22×C8×C23—24no—
44C23×C4⋊Q8—21no—
45C23×C4⋊Q8—21no—
46C23×C4⋊Q8—21no—
47C23×C4⋊Q8C2312yes—
48C23×C4⋊Q8C2311yescharacteristic
49C23×C4⋊Q8C2311yescharacteristic
50C23×#125—21no—
51C23×C22⋊Q8—21no—
52C23×C22⋊Q8—21no—
53C4×Q8×C23—21no—
54C4×Q8×C23—21no—
55C4×D4×C23—21no—
56C4×D4×C23—21no—
57C4×D4×C23D412yes—
58C4×D4×C23C2312yes—
59C23×#117—21no—
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23—21no—
62C2×C4⋊C4×C23D411yescharacteristic
63C2×C22⋊C4×C23—21no—
64C2×C42×C23C2311yescharacteristic
65C23×#107—24no—
66C23×#107—22no—
67C23×#106—22no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8—21no—
70C23×C4⋊C8D412yes—
71C23×C4⋊C8C2312yes—
72C23×C4⋊C8C2312yes—
73C23×C4⋊C8C2312yes—
74C23×#103—24no—
75C23×#102—24no—
76C23×#102—22no—
77C23×#102—22no—
78C23×C22⋊C8—21no—
79C23×C22⋊C8—24no—
80C23×C22⋊C8D412yes—
81C4×C8×C23—21no—
82C4×C8×C23C2312yes—
83C24×C23—21no—
84C2×Q8×C23—22no—
85C2×Q8×C23—21no—
86C2×Q8×C23—21no—
87C2×Q8×C23—41no—
88C2×Q8×C23—82no—
89C2×D4×C23—21no—
90C2×D4×C23—22no—
91C2×D4×C23Q1614yescomplemented
92C2×D4×C23C2×D412yes—
93C22×C4×C23—21no—
94C22×C4×C23—22no—
95C22×C4×C23—22no—
96C22×C4×C23—41no—
97C22×C4×C23—82no—
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100C22×C4×C23#4111yescharacteristic
101Q16×C23—84no—
102M16×C23—42no—
103C2×C8×C23—22no—
104C2×C8×C23—22no—
105C2×C8×C23—22no—
106C2×C8×C23—42no—
107C2×C8×C23—84no—
108C2×C8×C23C2×D414yes—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—22no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—22no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C4⋊C4×C23—41no—
120C4⋊C4×C23C2×D412yes—
121C22⋊C4×C23—22no—
122C22⋊C4×C23—22no—
123C22⋊C4×C23—41no—
124C42×C23—41no—
125C42×C23C2×D411yescharacteristic
126C42×C23C2×D411yescharacteristic
127C42×C23#4111yescharacteristic
128C42×C23C2411yescharacteristic
129#19387C2311yesmaximal, characteristic, direct product factor, Sylow
130C23×C23—22no—
131C23×C23—24no—
132C23×C23—81no—
133C23×C23—81no—
134C23×C23#12111yescharacteristic
135Q8×C23—42no—
136Q8×C23—44no—
137Q8×C23—81no—
138Q8×C23—82no—
139D4×C23—22no—
140D4×C23—24no—
141D4×C23C2×Q1614yes—
142C2×C4×C23—22no—
143C2×C4×C23—22no—
144C2×C4×C23—41no—
145C2×C4×C23—44no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—82no—
150C2×C4×C23—81no—
151C2×C4×C23—82no—
152C2×C4×C23#12111yescharacteristic
153C2×C4×C23#12111yescharacteristic
154C2×C4×C23#12111yescharacteristic
155C2×C4×C23C2×Q1612yes—
156C2×C4×C23#13512yes—
157C2×C4×C23C22×D411yescharacteristic
158C2×C4×C23C22×D411yescharacteristic
159C2×C4×C23C2×#4111yescharacteristic
160C8×C23—24no—
161C8×C23—42no—
162C8×C23—42no—
163C8×C23—42no—
164#3209C2×C2311yescharacteristic, complemented
165#3077C2×C2311yescharacteristic
166C2×#474C2×C2311yescharacteristic
167#2972C2×C2311yescharacteristic
168#1683C2×C2314yescomplemented
169#1669C2×C2314yescomplemented
170#1571C2×C2311yescharacteristic, complemented
171#1549C2×C2311yescharacteristic, complemented
172#1542C2×C2311yescharacteristic, complemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—24no—
176C22×C23—42no—
177C22×C23—42no—
178C22×C23—84no—
179C22×C23#46712yes—
180C22×C23C2×#12111yescharacteristic
181C4×C23—21no—
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—42no—
185C4×C23—82no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23#46712yes—
189C4×C23C22×Q1611yescharacteristic
190C4×C23C2×#13511yescharacteristic
191C4×C23#57612yes—
192#533C22×C2311yescharacteristic
193C2×C4⋊Q8C22×C2311yescharacteristic
194C2×C4×D4C22×C2311yescharacteristic
195#494C22×C2312yescomplemented
196#487C22×C2312yescomplemented
197#478—22no—
198#476C22×C2312yes—
199#474—22no—
200#474C22×C2312yes—
201#474C22×C2314yes—
202#462—22no—
203#451—22no—
204C8×D4—22no—
205C8×D4C22×C2314yes—
206#431C22×C2311yescharacteristic
207#431C22×C2311yescharacteristic
208#428—21no—
209C2×#107—21no—
210#423C22×C2311yescharacteristic
211C2×C4⋊C8C22×C2311yescharacteristic
212#418—21no—
213C2×#102—21no—
214C2×C22⋊C8—22no—
215#335C22×C2314yescomplemented
216#333C22×C2312yescomplemented
217#332C22×C2312yescomplemented
218#332C22×C2314yescomplemented
219C2×C23—22no—
220C2×C23—44no—
221C2×C23#304111yescharacteristic
222C2×C23C2×#46711yescharacteristic
223C2×C23#306911yescharacteristic
224C22×Q8—21no—
225C22×D4D4×C2311yescharacteristic
226C23×C4—21no—
227C2×Q16—24no—
228C2×M16—21no—
229C22×C8—21no—
230C22×C8—24no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8C23×C2312yes—
235C4⋊Q8C23×C2311yescharacteristic
236C4⋊Q8C23×C2311yescharacteristic
237#125—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4D4×C2312yes—
245C4×D4C23×C2312yes—
246#117—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4D4×C2311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C42C23×C2311yescharacteristic
252#107—24no—
253#107—22no—
254#106—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8D4×C2312yes—
258C4⋊C8C23×C2312yes—
259C4⋊C8C23×C2312yes—
260C4⋊C8C23×C2312yes—
261#103—24no—
262#102—24no—
263#102—22no—
264#102—22no—
265C22⋊C8—21no—
266C22⋊C8—24no—
267C22⋊C8D4×C2312yes—
268C4×C8—21no—
269C4×C8C23×C2312yes—
270C23#1938711yescharacteristic, direct product factor, Sylow
271C24—21no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—21no—
278C2×D4—22no—
279C2×D4Q16×C2314yescomplemented
280C2×D4C2×D4×C2312yes—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4C2×D4×C2311yescharacteristic
287C22×C4C2×D4×C2311yescharacteristic
288C22×C4#41×C2311yescharacteristic
289Q16—84no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—42no—
295C2×C8—84no—
296C2×C8C2×D4×C2314yes—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4C2×D4×C2312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—41no—
312C42—41no—
313C42C2×D4×C2311yescharacteristic
314C42C2×D4×C2311yescharacteristic
315C42#41×C2311yescharacteristic
316C42C24×C2311yescharacteristic
317C23—22no—
318C23—24no—
319C23—81no—
320C23—81no—
321C23C23×#12111yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—22no—
327D4—24no—
328D4C2×Q16×C2314yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4C23×#12111yescharacteristic
340C2×C4C23×#12111yescharacteristic
341C2×C4C23×#12111yescharacteristic
342C2×C4C2×Q16×C2312yes—
343C2×C4C23×#13512yes—
344C2×C4C22×D4×C2311yescharacteristic
345C2×C4C22×D4×C2311yescharacteristic
346C2×C4C2×#41×C2311yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—42no—
351C22—21no—
352C22—21no—
353C22—24no—
354C22—42no—
355C22—42no—
356C22—84no—
357C22C23×#46712yes—
358C22C2×C23×#12111yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—82no—
364C4—81no—
365C4—81no—
366C4C23×#46712yes—
367C4C22×Q16×C2311yescharacteristic
368C4C2×C23×#13511yescharacteristic
369C4C23×#57612yes—
370C2—22no—
371C2—44no—
372C2C23×#304111yescharacteristic
373C2C2×C23×#46711yescharacteristic
374C2C23×#306911yescharacteristic
subgroupG/H
center, socleC22×C23C2×#121
commutator, Frattini, cosocleC42C24×C23
prime radicalC22×D4×C23D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3209C2semidirect
5. C23×#1683C2semidirect
6. C23×#1669C2semidirect
7. C23×#1571C2semidirect
8. C23×#1549C2semidirect
9. C23×#1542C2semidirect
13. C23×#494C22semidirect
14. C23×#487C22semidirect
33. C23×#335C22semidirect
34. C23×#333C22semidirect
35. C23×#332C22semidirect
36. C23×#332C22semidirect
91. C2×D4×C23Q16semidirect
129. #19387C23direct (N × H)
164. #3209C2×C23semidirect
168. #1683C2×C23semidirect
169. #1669C2×C23semidirect
170. #1571C2×C23semidirect
171. #1549C2×C23semidirect
172. #1542C2×C23semidirect
195. #494C22×C23semidirect
196. #487C22×C23semidirect
215. #335C22×C23semidirect
216. #333C22×C23semidirect
217. #332C22×C23semidirect
218. #332C22×C23semidirect
270. C23#19387direct (N × H)
279. C2×D4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32092
C23×#30772
C2×C23×#4742
C23×#29722
C23×#16832
C23×#16692
C23×#15712
C23×#15492
C23×#15422
#1938723