Finite Group Databasebeta

← C23×#8542 #19156168

Subgroups — C23×#8542 #19156168

Order 5888.

1578 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1578 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C22×C23×#121C211yesmaximal, characteristic
2C23×#1819C212yesmaximal, complemented
3C23×#1781C212yesmaximal, complemented
4C23×#1781C212yesmaximal, complemented
5C23×D4×C23C411yescharacteristic, complemented
6C23×D4×C23C2211yescharacteristic
7C23×D4×C23C2211yescharacteristic
8C24×C4×C23C411yescharacteristic, complemented
9C2×C23×#121—24no—
10C2×C23×#121—24no—
11C2×C23×#121—24no—
12C22×C4⋊C4×C23C411yescharacteristic, complemented
13C22×C22⋊C4×C23C411yescharacteristic, complemented
14C22×C22⋊C4×C23C2211yescharacteristic
15C2×C23×#99—24no—
16C2×C23×#99—24no—
17C23×#379C2214yes—
18C23×#342—24no—
19C25×C23C2×C411yescharacteristic
20C25×C23C2×C411yescharacteristic
21C25×C23C2311yescharacteristic
22C22×D4×C23—22no—
23C22×D4×C23—22no—
24C22×D4×C23—22no—
25C22×D4×C23—22no—
26C22×D4×C23—22no—
27C22×D4×C23—22no—
28C22×D4×C23—22no—
29C22×D4×C23—44no—
30C22×D4×C23—44no—
31C22×D4×C23—44no—
32C22×D4×C23D414yes—
33C22×D4×C23D414yes—
34C23×C4×C23—22no—
35C23×C4×C23—22no—
36C23×C4×C23—22no—
37C23×C4×C23—44no—
38C23×C4×C23C2×C411yescharacteristic
39C23×C4×C23C2×C411yescharacteristic
40C23×C4×C23C2×C411yescharacteristic
41C23×C4×C23C2×C411yescharacteristic
42C23×#121—232no—
43C2×C4⋊C4×C23—22no—
44C2×C4⋊C4×C23—22no—
45C2×C4⋊C4×C23—22no—
46C2×C22⋊C4×C23—22no—
47C2×C22⋊C4×C23—22no—
48C2×C22⋊C4×C23—22no—
49C2×C22⋊C4×C23—22no—
50C2×C22⋊C4×C23—22no—
51C2×C22⋊C4×C23—24no—
52C2×C22⋊C4×C23—24no—
53C2×C22⋊C4×C23—24no—
54C2×C22⋊C4×C23D414yes—
55C23×#99—48no—
56C23×#99—48no—
57C24×C23—24no—
58C24×C23—21no—
59C24×C23—21no—
60C24×C23—21no—
61C24×C23—21no—
62C24×C23—42no—
63C24×C23—42no—
64C24×C23—42no—
65C24×C23—42no—
66C24×C23—42no—
67C24×C23—42no—
68C24×C23—42no—
69C24×C23—42no—
70C24×C23C22⋊C412yes—
71C24×C23C22⋊C412yes—
72C24×C23C22×C411yescharacteristic
73C24×C23C2×D412yes—
74C24×C23C2×D412yes—
75C24×C23C2×D412yes—
76C2×D4×C23—28no—
77C2×D4×C23—28no—
78C2×D4×C23—28no—
79C2×D4×C23—28no—
80C2×D4×C23—28no—
81C2×D4×C23—44no—
82C2×D4×C23—44no—
83C2×D4×C23—48no—
84C2×D4×C23—48no—
85C2×D4×C23—48no—
86C2×D4×C23—48no—
87C2×D4×C23—48no—
88C2×D4×C23—44no—
89C2×D4×C23—48no—
90C2×D4×C23—44no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—24no—
94C22×C4×C23—28no—
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—42no—
102C22×C4×C23—48no—
103C22×C4×C23—48no—
104C22×C4×C23—44no—
105C22×C4×C23—44no—
106C22×C4×C23—42no—
107C22×C4×C23—42no—
108C22×C4×C23—44no—
109C22×C4×C23C22⋊C412yes—
110C22×C4×C23C22⋊C412yes—
111C22×C4×C23C22⋊C412yes—
112C22×C4×C23C22⋊C412yes—
113C4⋊C4×C23—28no—
114C22⋊C4×C23—28no—
115C22⋊C4×C23—28no—
116C22⋊C4×C23—48no—
117C22⋊C4×C23—48no—
118C22⋊C4×C23—48no—
119#8542C2311yesmaximal, characteristic, direct product factor, Sylow
120C23×C23—22no—
121C23×C23—22no—
122C23×C23—22no—
123C23×C23—24no—
124C23×C23—24no—
125C23×C23—22no—
126C23×C23—22no—
127C23×C23—24no—
128C23×C23—24no—
129C23×C23—24no—
130C23×C23—22no—
131C23×C23—24no—
132C23×C23—42no—
133C23×C23—42no—
134C23×C23—42no—
135C23×C23—42no—
136C23×C23—42no—
137C23×C23—42no—
138C23×C23—42no—
139C23×C23—42no—
140C23×C23—42no—
141C23×C23—416no—
142C23×C23—42no—
143C23×C23—42no—
144C23×C23—42no—
145C23×C23—42no—
146C23×C23—42no—
147C23×C23—42no—
148C23×C23—42no—
149C23×C23—42no—
150C23×C23—42no—
151C23×C23—42no—
152C23×C23—42no—
153C23×C23—88no—
154C23×C23—88no—
155C23×C23#9914yescomplemented
156C23×C23C2×C22⋊C411yescharacteristic
157C23×C23C2×C22⋊C411yescharacteristic
158C23×C23C2×C22⋊C411yescharacteristic
159C23×C23#12014yes—
160D4×C23—416no—
161D4×C23—416no—
162D4×C23—416no—
163C2×C4×C23—24no—
164C2×C4×C23—24no—
165C2×C4×C23—22no—
166C2×C4×C23—22no—
167C2×C4×C23—22no—
168C2×C4×C23—22no—
169C2×C4×C23—22no—
170C2×C4×C23—24no—
171C2×C4×C23—416no—
172C2×C4×C23—42no—
173C2×C4×C23—42no—
174C2×C4×C23—42no—
175C2×C4×C23—42no—
176C2×C4×C23—42no—
177C2×C4×C23—42no—
178C2×C4×C23—42no—
179C2×C4×C23—42no—
180C2×C4×C23—84no—
181C2×C4×C23—84no—
182C2×C4×C23—84no—
183C2×C4×C23#9914yescomplemented
184C22×#121C2×C2311yescharacteristic
185#1819C2×C2312yescomplemented
186#1781C2×C2312yescomplemented
187#1781C2×C2312yescomplemented
188C22×C23—28no—
189C22×C23—21no—
190C22×C23—21no—
191C22×C23—21no—
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—21no—
195C22×C23—24no—
196C22×C23—24no—
197C22×C23—24no—
198C22×C23—24no—
199C22×C23—44no—
200C22×C23—48no—
201C22×C23—44no—
202C22×C23—48no—
203C22×C23—42no—
204C22×C23—42no—
205C22×C23—42no—
206C22×C23—42no—
207C22×C23—42no—
208C22×C23—84no—
209C22×C23—84no—
210C22×C23—84no—
211C22×C23—84no—
212C22×C23—84no—
213C22×C23—84no—
214C22×C23#37911yescharacteristic
215C22×C23C2×#9912yes—
216C22×C23C2×#9912yes—
217C22×C23#41012yes—
218C4×C23—24no—
219C4×C23—42no—
220C4×C23—42no—
221C4×C23—42no—
222C4×C23—164no—
223C23×D4C4×C2311yescharacteristic, complemented
224C23×D4C22×C2311yescharacteristic
225C23×D4C22×C2311yescharacteristic
226C24×C4C4×C2311yescharacteristic, complemented
227C2×#121—24no—
228C2×#121—24no—
229C2×#121—24no—
230C22×C4⋊C4C4×C2311yescharacteristic, complemented
231C22×C22⋊C4C4×C2311yescharacteristic, complemented
232C22×C22⋊C4C22×C2311yescharacteristic
233C2×#99—24no—
234C2×#99—24no—
235#379C22×C2314yes—
236#342—24no—
237C2×C23—22no—
238C2×C23—22no—
239C2×C23—22no—
240C2×C23—44no—
241C2×C23—44no—
242C2×C23—82no—
243C2×C23#178111yescharacteristic
244C2×C23#178111yescharacteristic
245C2×C23#178211yescharacteristic
246C25C2×C4×C2311yescharacteristic
247C25C2×C4×C2311yescharacteristic
248C25C23×C2311yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—44no—
257C22×D4—44no—
258C22×D4—44no—
259C22×D4D4×C2314yes—
260C22×D4D4×C2314yes—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—44no—
265C23×C4C2×C4×C2311yescharacteristic
266C23×C4C2×C4×C2311yescharacteristic
267C23×C4C2×C4×C2311yescharacteristic
268C23×C4C2×C4×C2311yescharacteristic
269#121—232no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4D4×C2314yes—
282#99—48no—
283#99—48no—
284C23#854211yescharacteristic, direct product factor, Sylow
285C24—24no—
286C24—21no—
287C24—21no—
288C24—21no—
289C24—21no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C22⋊C4×C2312yes—
299C24C22⋊C4×C2312yes—
300C24C22×C4×C2311yescharacteristic
301C24C2×D4×C2312yes—
302C24C2×D4×C2312yes—
303C24C2×D4×C2312yes—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—28no—
308C2×D4—28no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—48no—
318C2×D4—44no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—28no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4C22⋊C4×C2312yes—
338C22×C4C22⋊C4×C2312yes—
339C22×C4C22⋊C4×C2312yes—
340C22×C4C22⋊C4×C2312yes—
341C4⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—416no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—88no—
381C23—88no—
382C23C23×#9914yescomplemented
383C23C2×C22⋊C4×C2311yescharacteristic
384C23C2×C22⋊C4×C2311yescharacteristic
385C23C2×C22⋊C4×C2311yescharacteristic
386C23C23×#12014yes—
387D4—416no—
388D4—416no—
389D4—416no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—416no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4C23×#9914yescomplemented
411C22—28no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—24no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—44no—
423C22—48no—
424C22—44no—
425C22—48no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—84no—
432C22—84no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—84no—
437C22C23×#37911yescharacteristic
438C22C2×C23×#9912yes—
439C22C2×C23×#9912yes—
440C22C23×#41012yes—
441C4—24no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—164no—
446C2—22no—
447C2—22no—
448C2—22no—
449C2—44no—
450C2—44no—
451C2—82no—
452C2C23×#178111yescharacteristic
453C2C23×#178111yescharacteristic
454C2C23×#178211yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC22×C23×#121C2
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×#1819C2semidirect
3. C23×#1781C2semidirect
4. C23×#1781C2semidirect
5. C23×D4×C23C4semidirect
8. C24×C4×C23C4semidirect
12. C22×C4⋊C4×C23C4semidirect
13. C22×C22⋊C4×C23C4semidirect
119. #8542C23direct (N × H)
155. C23×C23#99semidirect
183. C2×C4×C23#99semidirect
185. #1819C2×C23semidirect
186. #1781C2×C23semidirect
187. #1781C2×C23semidirect
223. C23×D4C4×C23semidirect
226. C24×C4C4×C23semidirect
230. C22×C4⋊C4C4×C23semidirect
231. C22×C22⋊C4C4×C23semidirect
284. C23#8542direct (N × H)
382. C23C23×#99semidirect
410. C2×C4C23×#99semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1212
C23×#18192
C23×#17812
#854223