Finite Group Databasebeta

← C23×#9516 #19157142

Subgroups — C23×#9516 #19157142

Order 5888.

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

normal subgroups only

This group has 626 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×#2975C211yesmaximal, characteristic, complemented
2C23×#2534C211yesmaximal, characteristic
3C23×#1465C211yesmaximal, characteristic, complemented
4C23×#1462C212yesmaximal, complemented
5C23×#1305C212yesmaximal, complemented
6C2×C23×#123C411yescharacteristic
7C2×C23×#120C411yescharacteristic
8C2×C4×D4×C23C2211yescharacteristic
9C22×C22⋊C4×C23C2211yescharacteristic
10C23×#436—24no—
11C23×#431—21no—
12C23×#407—21no—
13C23×#407C2212yes—
14C23×#407C2212yes—
15C23×#396—21no—
16C23×#394—21no—
17C23×#388—21no—
18C23×#386—21no—
19C23×#381C411yescharacteristic
20C23×#380C2211yescharacteristic
21C23×#379C411yescharacteristic
22C23×#324—21no—
23C23×#323—24no—
24C23×#323—21no—
25C25×C23C2×C411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23C2×C411yescharacteristic
28C23×C4×C23C2×C411yescharacteristic
29C23×C4×C23D411yescharacteristic
30C23×C4×C23C2311yescharacteristic
31C2×M16×C23—22no—
32C2×M16×C23—22no—
33C2×M16×C23—22no—
34C23×#123—42no—
35C23×#120—42no—
36C4×D4×C23—22no—
37C4×D4×C23—22no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23—21no—
40C2×C4⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—22no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—21no—
49C2×C22⋊C4×C23C2×C411yescharacteristic
50C2×C22⋊C4×C23C2×C411yescharacteristic
51C2×C22⋊C4×C23C2×C411yescharacteristic
52C2×C22⋊C4×C23D411yescharacteristic
53C2×C42×C23D411yescharacteristic
54C23×C4⋊C8—22no—
55C23×C22⋊C8—22no—
56C23×C22⋊C8—22no—
57C23×C22⋊C8—22no—
58C23×C22⋊C8—22no—
59C23×C22⋊C8D414yescomplemented
60C4×C8×C23—22no—
61C23×#95—21no—
62C23×#95—21no—
63C23×#95—21no—
64C24×C23—21no—
65C24×C23—22no—
66C24×C23—42no—
67C24×C23—41no—
68C24×C23—41no—
69C24×C23—81no—
70C24×C23C22⋊C411yescharacteristic
71C24×C23C22×C411yescharacteristic
72C2×D4×C23—22no—
73C2×D4×C23—22no—
74C2×D4×C23—41no—
75C2×D4×C23—42no—
76C2×D4×C23—42no—
77C2×D4×C23—81no—
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—22no—
84C22×C4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23—81no—
87C22×C4×C23—81no—
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22⋊C411yescharacteristic
91C22×C4×C23C2×D412yes—
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23#4112yes—
95M16×C23—82no—
96M16×C23—82no—
97M16×C23—82no—
98C2×C8×C23—24no—
99C2×C8×C23—42no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23—81no—
103C22⋊C4×C23—22no—
104C22⋊C4×C23—22no—
105C22⋊C4×C23—42no—
106C22⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—41no—
109C22⋊C4×C23—42no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C22⋊C4×C23—81no—
114C42×C23—21no—
115C42×C23—21no—
116#9516C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—22no—
120C23×C23—24no—
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—21no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23—41no—
131C23×C23—42no—
132C23×C23—41no—
133C23×C23—41no—
134C23×C23—41no—
135C23×C23—82no—
136C23×C23—82no—
137C23×C23—84no—
138C23×C23—84no—
139C23×C23C2×C22⋊C411yescharacteristic
140C23×C23C4×D412yes—
141D4×C23—44no—
142D4×C23—84no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—41no—
148C2×C4×C23—42no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—42no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—81no—
164C2×C4×C23—82no—
165C2×C4×C23#10012yes—
166C2×C4×C23#12011yescharacteristic
167C2×C4×C23#12112yes—
168C2×C4×C23#12311yescharacteristic
169C8×C23—44no—
170C8×C23—82no—
171#2975C2×C2311yescharacteristic, complemented
172#2534C2×C2311yescharacteristic
173#1465C2×C2311yescharacteristic, complemented
174#1462C2×C2312yescomplemented
175#1305C2×C2312yescomplemented
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—22no—
179C22×C23—22no—
180C22×C23—22no—
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—42no—
184C22×C23—42no—
185C22×C23—42no—
186C22×C23—42no—
187C22×C23—41no—
188C22×C23—41no—
189C22×C23—41no—
190C22×C23—81no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—84no—
194C22×C23—84no—
195C22×C23—81no—
196C22×C23—81no—
197C22×C23#38611yescharacteristic
198C22×C23C2×#10011yescharacteristic
199C22×C23#41311yescharacteristic
200C4×C23—22no—
201C4×C23—21no—
202C4×C23—22no—
203C4×C23—21no—
204C4×C23—41no—
205C4×C23—41no—
206C4×C23—81no—
207C4×C23—81no—
208C4×C23—81no—
209C2×#123C4×C2311yescharacteristic
210C2×#120C4×C2311yescharacteristic
211C2×C4×D4C22×C2311yescharacteristic
212C22×C22⋊C4C22×C2311yescharacteristic
213#436—24no—
214#431—21no—
215#407—21no—
216#407C22×C2312yes—
217#407C22×C2312yes—
218#396—21no—
219#394—21no—
220#388—21no—
221#386—21no—
222#381C4×C2311yescharacteristic
223#380C22×C2311yescharacteristic
224#379C4×C2311yescharacteristic
225#324—21no—
226#323—21no—
227#323—24no—
228C2×C23—21no—
229C2×C23—21no—
230C2×C23—42no—
231C2×C23—81no—
232C2×C23—82no—
233C2×C23#189111yescharacteristic
234C2×C23#190011yescharacteristic
235C2×C23#191311yescharacteristic
236C25C2×C4×C2311yescharacteristic
237C22×D4—21no—
238C22×D4C2×C4×C2311yescharacteristic
239C23×C4C2×C4×C2311yescharacteristic
240C23×C4D4×C2311yescharacteristic
241C23×C4C23×C2311yescharacteristic
242C2×M16—22no—
243C2×M16—22no—
244C2×M16—22no—
245#123—42no—
246#120—42no—
247C4×D4—22no—
248C4×D4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4D4×C2311yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4C2×C4×C2311yescharacteristic
261C2×C22⋊C4C2×C4×C2311yescharacteristic
262C2×C22⋊C4C2×C4×C2311yescharacteristic
263C2×C22⋊C4D4×C2311yescharacteristic
264C2×C42D4×C2311yescharacteristic
265C4⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8D4×C2314yescomplemented
271C4×C8—22no—
272#95—21no—
273#95—21no—
274#95—21no—
275C23#951611yescharacteristic, direct product factor, Sylow
276C24—21no—
277C24—22no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—81no—
282C24C22⋊C4×C2311yescharacteristic
283C24C22×C4×C2311yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊C4×C2311yescharacteristic
301C22×C4C22⋊C4×C2311yescharacteristic
302C22×C4C22⋊C4×C2311yescharacteristic
303C22×C4C2×D4×C2312yes—
304C22×C4C2×D4×C2311yescharacteristic
305C22×C4C2×D4×C2311yescharacteristic
306C22×C4#41×C2312yes—
307M16—82no—
308M16—82no—
309M16—82no—
310C2×C8—24no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—81no—
326C42—21no—
327C42—21no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—24no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—82no—
347C23—82no—
348C23—84no—
349C23—84no—
350C23C2×C22⋊C4×C2311yescharacteristic
351C23C4×D4×C2312yes—
352D4—44no—
353D4—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4C23×#10012yes—
377C2×C4C23×#12011yescharacteristic
378C2×C4C23×#12112yes—
379C2×C4C23×#12311yescharacteristic
380C8—44no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22C23×#38611yescharacteristic
404C22C2×C23×#10011yescharacteristic
405C22C23×#41311yescharacteristic
406C4—22no—
407C4—21no—
408C4—22no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—81no—
419C2—82no—
420C2C23×#189111yescharacteristic
421C2C23×#190011yescharacteristic
422C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#120C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2975C2semidirect
3. C23×#1465C2semidirect
4. C23×#1462C2semidirect
5. C23×#1305C2semidirect
59. C23×C22⋊C8D4semidirect
116. #9516C23direct (N × H)
171. #2975C2×C23semidirect
173. #1465C2×C23semidirect
174. #1462C2×C23semidirect
175. #1305C2×C23semidirect
270. C22⋊C8D4×C23semidirect
275. C23#9516direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29752
C23×#25342
C23×#14652
C23×#14622
C23×#13052
#951623