Finite Group Databasebeta

← C23×#13329 #19160955

Subgroups — C23×#13329 #19160955

Order 5888.

578 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×Q32C211yesmaximal, characteristic, complemented
2C22×C23×SD32C211yesmaximal, characteristic
3C2×C23×#495C211yesmaximal, characteristic
4C2×C23×#358C211yesmaximal, characteristic, complemented
5C2×C23×#357C211yesmaximal, characteristic
6C23×#1969C211yesmaximal, characteristic, complemented
7C23×#1374C211yesmaximal, characteristic, complemented
8C22×Q16×C23C2211yescharacteristic
9C22×Q16×C23C2211yescharacteristic
10C22×SD16×C23—21no—
11C22×D8×C23C2211yescharacteristic
12C2×C23×#124—21no—
13C2×C23×Q32—22no—
14C2×C23×Q32—22no—
15C2×C23×Q32—22no—
16C2×C23×SD32—22no—
17C2×C23×SD32—22no—
18C2×C23×SD32—22no—
19C22×C16×C23C2211yescharacteristic
20C22×C16×C23C2211yescharacteristic
21C23×#495—22no—
22C23×#495—22no—
23C2×C23×#107C2211yescharacteristic
24C2×C23×#103—21no—
25C2×C4×C8×C23C2211yescharacteristic
26C23×#391—21no—
27C23×#358—22no—
28C23×#357—22no—
29C22×Q8×C23—21no—
30C22×Q8×C23—21no—
31C22×D4×C23—21no—
32C2×Q16×C23—21no—
33C2×Q16×C23—21no—
34C2×Q16×C23—21no—
35C2×Q16×C23—21no—
36C2×Q16×C23—21no—
37C2×Q16×C23—21no—
38C2×Q16×C23—21no—
39C2×Q16×C23—21no—
40C2×Q16×C23—21no—
41C2×Q16×C23—21no—
42C2×Q16×C23D412yes—
43C2×Q16×C23D412yes—
44C2×SD16×C23—24no—
45C2×SD16×C23—42no—
46C2×SD16×C23—42no—
47C2×D8×C23—21no—
48C2×D8×C23—21no—
49C2×D8×C23—21no—
50C2×D8×C23—21no—
51C2×D8×C23—21no—
52C2×D8×C23D412yes—
53C22×C8×C23D411yescharacteristic
54C22×C8×C23C2311yescharacteristic
55C23×#124—24no—
56C23×#124—42no—
57C2×C4⋊C4×C23—21no—
58C2×C22⋊C4×C23—42no—
59C2×C42×C23D411yescharacteristic
60C23×Q32—28no—
61C23×SD32—28no—
62C2×C16×C23—21no—
63C2×C16×C23—21no—
64C2×C16×C23—21no—
65C2×C16×C23—21no—
66C2×C16×C23D412yes—
67C2×C16×C23D412yes—
68C23×#107—21no—
69C23×#107—21no—
70C23×#103—42no—
71C4×C8×C23—21no—
72C4×C8×C23—21no—
73C23×#95—42no—
74C24×C23—41no—
75C2×Q8×C23—22no—
76C2×Q8×C23—22no—
77C2×Q8×C23—41no—
78C2×Q8×C23—41no—
79C2×Q8×C23—41no—
80C2×Q8×C23—41no—
81C2×Q8×C23—41no—
82C2×Q8×C23—41no—
83C2×Q8×C23—41no—
84C2×Q8×C23—41no—
85C2×Q8×C23—41no—
86C2×Q8×C23—41no—
87C2×D4×C23—22no—
88C2×D4×C23—41no—
89C2×D4×C23—41no—
90C2×D4×C23—41no—
91C2×D4×C23—41no—
92C2×D4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23D812yes—
97C22×C4×C23C2×D411yescharacteristic
98Q16×C23—22no—
99Q16×C23—22no—
100Q16×C23—22no—
101Q16×C23—22no—
102Q16×C23—41no—
103Q16×C23—41no—
104Q16×C23—41no—
105Q16×C23—41no—
106SD16×C23—48no—
107D8×C23—22no—
108D8×C23—22no—
109D8×C23—41no—
110D8×C23—41no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23D812yes—
114C2×C8×C23C2×D411yescharacteristic
115C2×C8×C23C2×D411yescharacteristic
116C2×C8×C23C2×D411yescharacteristic
117C2×C8×C23C2×D411yescharacteristic
118C2×C8×C23C2×D411yescharacteristic
119C2×C8×C23#4111yescharacteristic
120C4⋊C4×C23—41no—
121C4⋊C4×C23—41no—
122C22⋊C4×C23—48no—
123C42×C23—21no—
124C42×C23—21no—
125C16×C23—22no—
126C16×C23—22no—
127#13329C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—42no—
129C23×C23—81no—
130C23×C23—81no—
131C23×C23—82no—
132C23×C23—82no—
133C23×C23C2×D811yescharacteristic
134Q8×C23—42no—
135Q8×C23—42no—
136Q8×C23—42no—
137Q8×C23—42no—
138Q8×C23—81no—
139Q8×C23—81no—
140Q8×C23—81no—
141Q8×C23—81no—
142D4×C23—42no—
143D4×C23—42no—
144D4×C23—81no—
145D4×C23—81no—
146C2×C4×C23—22no—
147C2×C4×C23—22no—
148C2×C4×C23—22no—
149C2×C4×C23—42no—
150C2×C4×C23—42no—
151C2×C4×C23—81no—
152C2×C4×C23—81no—
153C2×C4×C23—81no—
154C2×C4×C23—81no—
155C2×C4×C23—81no—
156C2×C4×C23—82no—
157C2×C4×C23#12011yescharacteristic
158C2×C4×C23#12111yescharacteristic
159C2×C4×C23#12711yescharacteristic
160C2×C4×C23C2×D811yescharacteristic
161C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
163C8×C23—22no—
164C8×C23#12011yescharacteristic
165C8×C23#12011yescharacteristic
166C8×C23#12111yescharacteristic
167C8×C23#12111yescharacteristic
168C22×Q32C2×C2311yescharacteristic, complemented
169C22×SD32C2×C2311yescharacteristic
170C2×#495C2×C2311yescharacteristic
171C2×#358C2×C2311yescharacteristic, complemented
172C2×#357C2×C2311yescharacteristic
173#1969C2×C2311yescharacteristic, complemented
174#1374C2×C2311yescharacteristic, complemented
175C22×C23—84no—
176C22×C23—82no—
177C22×C23—162no—
178C22×C23—161no—
179C22×C23—161no—
180C22×C23#44711yescharacteristic
181C22×C23#45911yescharacteristic
182C22×C23#49311yescharacteristic
183C22×C23#50812yes—
184C22×C23#51012yes—
185C4×C23—42no—
186C4×C23—82no—
187C4×C23—82no—
188C4×C23—161no—
189C4×C23#39211yescharacteristic
190C4×C23#44711yescharacteristic
191C4×C23#45911yescharacteristic
192C4×C23#49611yescharacteristic
193C22×Q16C22×C2311yescharacteristic
194C22×Q16C22×C2311yescharacteristic
195C22×SD16—21no—
196C22×D8C22×C2311yescharacteristic
197C2×#124—21no—
198C2×Q32—22no—
199C2×Q32—22no—
200C2×Q32—22no—
201C2×SD32—22no—
202C2×SD32—22no—
203C2×SD32—22no—
204C22×C16C22×C2311yescharacteristic
205C22×C16C22×C2311yescharacteristic
206#495—22no—
207#495—22no—
208C2×#107C22×C2311yescharacteristic
209C2×#103—21no—
210C2×C4×C8C22×C2311yescharacteristic
211#391—21no—
212#358—22no—
213#357—22no—
214C2×C23—162no—
215C2×C23#201111yescharacteristic
216C2×C23#218912yes—
217C2×C23#221112yes—
218C2×C23#224911yescharacteristic
219C2×C23#225111yescharacteristic
220C22×Q8—21no—
221C22×Q8—21no—
222C22×D4—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—21no—
227C2×Q16—21no—
228C2×Q16—21no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16D4×C2312yes—
234C2×Q16D4×C2312yes—
235C2×SD16—24no—
236C2×SD16—42no—
237C2×SD16—42no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—21no—
241C2×D8—21no—
242C2×D8—21no—
243C2×D8D4×C2312yes—
244C22×C8D4×C2311yescharacteristic
245C22×C8C23×C2311yescharacteristic
246#124—24no—
247#124—42no—
248C2×C4⋊C4—21no—
249C2×C22⋊C4—42no—
250C2×C42D4×C2311yescharacteristic
251Q32—28no—
252SD32—28no—
253C2×C16—21no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16D4×C2312yes—
258C2×C16D4×C2312yes—
259#107—21no—
260#107—21no—
261#103—42no—
262C4×C8—21no—
263C4×C8—21no—
264#95—42no—
265C23#1332911yescharacteristic, direct product factor, Sylow
266C24—41no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4D8×C2312yes—
289C22×C4C2×D4×C2311yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—22no—
293Q16—22no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—41no—
298SD16—48no—
299D8—22no—
300D8—22no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8D8×C2312yes—
306C2×C8C2×D4×C2311yescharacteristic
307C2×C8C2×D4×C2311yescharacteristic
308C2×C8C2×D4×C2311yescharacteristic
309C2×C8C2×D4×C2311yescharacteristic
310C2×C8C2×D4×C2311yescharacteristic
311C2×C8#41×C2311yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—48no—
315C42—21no—
316C42—21no—
317C16—22no—
318C16—22no—
319C23—42no—
320C23—81no—
321C23—81no—
322C23—82no—
323C23—82no—
324C23C2×D8×C2311yescharacteristic
325Q8—42no—
326Q8—42no—
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—81no—
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4C23×#12011yescharacteristic
349C2×C4C23×#12111yescharacteristic
350C2×C4C23×#12711yescharacteristic
351C2×C4C2×D8×C2311yescharacteristic
352C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
353C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
354C8—22no—
355C8C23×#12011yescharacteristic
356C8C23×#12011yescharacteristic
357C8C23×#12111yescharacteristic
358C8C23×#12111yescharacteristic
359C22—84no—
360C22—82no—
361C22—162no—
362C22—161no—
363C22—161no—
364C22C23×#44711yescharacteristic
365C22C23×#45911yescharacteristic
366C22C23×#49311yescharacteristic
367C22C23×#50812yes—
368C22C23×#51012yes—
369C4—42no—
370C4—82no—
371C4—82no—
372C4—161no—
373C4C23×#39211yescharacteristic
374C4C23×#44711yescharacteristic
375C4C23×#45911yescharacteristic
376C4C23×#49611yescharacteristic
377C2—162no—
378C2C23×#201111yescharacteristic
379C2C23×#218912yes—
380C2C23×#221112yes—
381C2C23×#224911yescharacteristic
382C2C23×#225111yescharacteristic
subgroupG/H
center, socleC23×C23C2×D8
commutator, Frattini, cosocleC22×C8C23×C23
prime radicalC22×D8×C23C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ C2×C4×C8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×Q32C2semidirect
4. C2×C23×#358C2semidirect
6. C23×#1969C2semidirect
7. C23×#1374C2semidirect
127. #13329C23direct (N × H)
168. C22×Q32C2×C23semidirect
171. C2×#358C2×C23semidirect
173. #1969C2×C23semidirect
174. #1374C2×C23semidirect
265. C23#13329direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×Q322
C22×C23×SD322
C2×C23×#4952
C2×C23×#3582
C2×C23×#3572
C23×#19692
C23×#13742
#1332923