Finite Group Databasebeta

← C23×#9496 #19157122

Subgroups — C23×#9496 #19157122

Order 5888.

900 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2405C211yesmaximal, characteristic
2C23×#2312C211yesmaximal, characteristic, complemented
3C23×#1819C211yesmaximal, characteristic, complemented
4C23×#1781C212yesmaximal
5C23×#1304C212yesmaximal, complemented
6C23×D4×C23C2211yescharacteristic
7C2×C23×#123—21no—
8C22×C22⋊C4×C23C2211yescharacteristic
9C2×C23×#99—24no—
10C23×#394—22no—
11C23×#388—24no—
12C23×#386—21no—
13C23×#385C412yescomplemented
14C23×#380—21no—
15C23×#379C412yescomplemented
16C23×#379C2212yes—
17C23×#379C2211yescharacteristic
18C23×#379C2212yes—
19C4×C22⋊C4×C23—21no—
20C23×#342—22no—
21C25×C23C2×C411yescharacteristic
22C25×C23C2311yescharacteristic
23C22×D4×C23—21no—
24C22×D4×C23—22no—
25C22×D4×C23—21no—
26C22×D4×C23—22no—
27C22×D4×C23—44no—
28C23×C4×C23C2×C411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C23×C4×C23D411yescharacteristic
31C23×#123—44no—
32C2×C4⋊C4×C23—22no—
33C2×C4⋊C4×C23—22no—
34C2×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23—22no—
41C2×C22⋊C4×C23—22no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—22no—
44C2×C22⋊C4×C23—22no—
45C2×C22⋊C4×C23C2×C412yes—
46C2×C22⋊C4×C23C2×C412yes—
47C2×C22⋊C4×C23D412yes—
48C2×C22⋊C4×C23D414yes—
49C2×C42×C23—22no—
50C23×#99—48no—
51C23×#95—22no—
52C23×#95—22no—
53C23×#95—22no—
54C24×C23—21no—
55C24×C23—22no—
56C24×C23—42no—
57C24×C23—42no—
58C24×C23—42no—
59C24×C23—42no—
60C24×C23—81no—
61C24×C23—81no—
62C24×C23C22×C411yescharacteristic
63C24×C23C2×D411yescharacteristic
64C24×C23C2×D411yescharacteristic
65C24×C23C2×D412yes—
66C24×C23#4112yes—
67C2×D4×C23—28no—
68C2×D4×C23—24no—
69C2×D4×C23—42no—
70C2×D4×C23—44no—
71C2×D4×C23—42no—
72C2×D4×C23—44no—
73C2×D4×C23—42no—
74C2×D4×C23—44no—
75C2×D4×C23—41no—
76C2×D4×C23—42no—
77C2×D4×C23—41no—
78C22×C4×C23—21no—
79C22×C4×C23—22no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—21no—
84C22×C4×C23—22no—
85C22×C4×C23—24no—
86C22×C4×C23—42no—
87C22×C4×C23—42no—
88C22×C4×C23—42no—
89C22×C4×C23—81no—
90C22×C4×C23C22⋊C412yes—
91C22×C4×C23C22⋊C412yes—
92C4⋊C4×C23—44no—
93C4⋊C4×C23—82no—
94C22⋊C4×C23—24no—
95C22⋊C4×C23—24no—
96C22⋊C4×C23—44no—
97C22⋊C4×C23—44no—
98C22⋊C4×C23—42no—
99C22⋊C4×C23—44no—
100C22⋊C4×C23—42no—
101C22⋊C4×C23—44no—
102C22⋊C4×C23—44no—
103C22⋊C4×C23—82no—
104C22⋊C4×C23—81no—
105C22⋊C4×C23—82no—
106C22⋊C4×C23—82no—
107C42×C23—82no—
108#9496C2311yesmaximal, characteristic, direct product factor, Sylow
109C23×C23—22no—
110C23×C23—22no—
111C23×C23—22no—
112C23×C23—22no—
113C23×C23—24no—
114C23×C23—22no—
115C23×C23—24no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—42no—
120C23×C23—41no—
121C23×C23—42no—
122C23×C23—41no—
123C23×C23—42no—
124C23×C23—42no—
125C23×C23—42no—
126C23×C23—42no—
127C23×C23—41no—
128C23×C23—42no—
129C23×C23—41no—
130C23×C23—42no—
131C23×C23—82no—
132C23×C23—82no—
133C23×C23—88no—
134C23×C23—88no—
135C23×C23C2×C22⋊C411yescharacteristic
136C23×C23C4×D412yes—
137C23×C23#12011yescharacteristic
138C23×C23#12112yes—
139C23×C23#12311yescharacteristic
140D4×C23—48no—
141D4×C23—44no—
142D4×C23—44no—
143C2×C4×C23—22no—
144C2×C4×C23—24no—
145C2×C4×C23—22no—
146C2×C4×C23—22no—
147C2×C4×C23—42no—
148C2×C4×C23—41no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—44no—
156C2×C4×C23—42no—
157C2×C4×C23—44no—
158C2×C4×C23—42no—
159C2×C4×C23—44no—
160C2×C4×C23—81no—
161C2×C4×C23—81no—
162C2×C4×C23—82no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165#2405C2×C2311yescharacteristic
166#2312C2×C2311yescharacteristic, complemented
167#1819C2×C2311yescharacteristic, complemented
168#1781C2×C2312yes—
169#1304C2×C2312yescomplemented
170C22×C23—21no—
171C22×C23—21no—
172C22×C23—21no—
173C22×C23—21no—
174C22×C23—22no—
175C22×C23—22no—
176C22×C23—21no—
177C22×C23—22no—
178C22×C23—22no—
179C22×C23—21no—
180C22×C23—22no—
181C22×C23—21no—
182C22×C23—22no—
183C22×C23—44no—
184C22×C23—44no—
185C22×C23—44no—
186C22×C23—42no—
187C22×C23—84no—
188C22×C23—82no—
189C22×C23—82no—
190C22×C23—84no—
191C22×C23—84no—
192C22×C23—84no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23#38611yescharacteristic
196C22×C23#41012yes—
197C22×C23B(4,𝔽2)12yes—
198C4×C23—42no—
199C4×C23—42no—
200C4×C23—42no—
201C4×C23—84no—
202C4×C23—82no—
203C4×C23—162no—
204C23×D4C22×C2311yescharacteristic
205C2×#123—21no—
206C22×C22⋊C4C22×C2311yescharacteristic
207C2×#99—24no—
208#394—22no—
209#388—24no—
210#386—21no—
211#385C4×C2312yescomplemented
212#380—21no—
213#379C4×C2312yescomplemented
214#379C22×C2312yes—
215#379C22×C2311yescharacteristic
216#379C22×C2312yes—
217C4×C22⋊C4—21no—
218#342—22no—
219C2×C23—22no—
220C2×C23—22no—
221C2×C23—42no—
222C2×C23—41no—
223C2×C23—42no—
224C2×C23—82no—
225C2×C23#188911yescharacteristic
226C2×C23#189911yescharacteristic
227C2×C23#191311yescharacteristic
228C25C2×C4×C2311yescharacteristic
229C25C23×C2311yescharacteristic
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—44no—
235C23×C4C2×C4×C2311yescharacteristic
236C23×C4D4×C2311yescharacteristic
237C23×C4D4×C2311yescharacteristic
238#123—44no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C2×C4×C2312yes—
253C2×C22⋊C4C2×C4×C2312yes—
254C2×C22⋊C4D4×C2312yes—
255C2×C22⋊C4D4×C2314yes—
256C2×C42—22no—
257#99—48no—
258#95—22no—
259#95—22no—
260#95—22no—
261C23#949611yescharacteristic, direct product factor, Sylow
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—81no—
269C24—81no—
270C24C22×C4×C2311yescharacteristic
271C24C2×D4×C2311yescharacteristic
272C24C2×D4×C2311yescharacteristic
273C24C2×D4×C2312yes—
274C24#41×C2312yes—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4C22⋊C4×C2312yes—
299C22×C4C22⋊C4×C2312yes—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C22⋊C4—81no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C42—82no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23—88no—
341C23—88no—
342C23C2×C22⋊C4×C2311yescharacteristic
343C23C4×D4×C2312yes—
344C23C23×#12011yescharacteristic
345C23C23×#12112yes—
346C23C23×#12311yescharacteristic
347D4—48no—
348D4—44no—
349D4—44no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22C23×#38611yescharacteristic
398C22C23×#41012yes—
399C22C23×B(4,𝔽2)12yes—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—84no—
404C4—82no—
405C4—162no—
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2C23×#188911yescharacteristic
413C2C23×#189911yescharacteristic
414C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2312C2semidirect
3. C23×#1819C2semidirect
5. C23×#1304C2semidirect
13. C23×#385C4semidirect
15. C23×#379C4semidirect
108. #9496C23direct (N × H)
166. #2312C2×C23semidirect
167. #1819C2×C23semidirect
169. #1304C2×C23semidirect
211. #385C4×C23semidirect
213. #379C4×C23semidirect
261. C23#9496direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24052
C23×#23122
C23×#18192
C23×#17812
C23×#13042
#949623