Finite Group Databasebeta

← C23×#8539 #19156165

Subgroups — C23×#8539 #19156165

Order 5888.

1008 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3485C211yesmaximal, characteristic
2C23×#1785C211yesmaximal, characteristic, complemented
3C23×#1780C211yesmaximal, characteristic, complemented
4C23×#1468C214yesmaximal, complemented
5C2×C23×H(5,𝔽2)C411yescharacteristic
6C22×#41×C23C2211yescharacteristic
7C23×#547—24no—
8C23×#546—24no—
9C23×#536—21no—
10C23×#534—21no—
11C23×#534C412yes—
12C23×#525—21no—
13C2×C23×#123—21no—
14C2×C23×#123C2212yes—
15C23×#518C411yescharacteristic
16C2×C23×#101—21no—
17C2×C23×#100—21no—
18C23×#407C2212yescomplemented
19C2×C23×C22⋊C8—21no—
20C2×C23×C22⋊C8—21no—
21C2×C23×C22⋊C8C2212yescomplemented
22C23×#324—24no—
23C23×#323—24no—
24C23×H(5,𝔽2)—42no—
25C23×H(5,𝔽2)—42no—
26C2×#41×C23—22no—
27C2×#41×C23—21no—
28C2×#41×C23—24no—
29C2×#41×C23—42no—
30C2×#41×C23—42no—
31C22×Q8×C23D411yescharacteristic
32C22×D4×C23—21no—
33C22×D4×C23—22no—
34C22×D4×C23C2×C411yescharacteristic
35C22×D4×C23C2×C411yescharacteristic
36C22×D4×C23C2×C412yes—
37C22×D4×C23D411yescharacteristic
38C23×C4×C23D411yescharacteristic
39C23×C4×C23D411yescharacteristic
40C23×C4×C23C2311yescharacteristic
41C2×M16×C23—22no—
42C22×C8×C23—22no—
43C23×C4⋊Q8—22no—
44C23×#124—22no—
45C23×#123—22no—
46C23×#123—24no—
47C23×#123—22no—
48C23×#123—22no—
49C23×#123—24no—
50C23×#123—22no—
51C23×#123—22no—
52C23×C22⋊Q8—22no—
53C23×C22⋊Q8—24no—
54C23×#121—24no—
55C23×#121—22no—
56C23×#121—24no—
57C23×#120—22no—
58C23×#120—22no—
59C23×#120—22no—
60C4×D4×C23—24no—
61C4×D4×C23—24no—
62C23×#117—21no—
63C2×C4⋊C4×C23—22no—
64C2×C4⋊C4×C23D414yes—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—21no—
67C2×C22⋊C4×C23C2×C412yes—
68C2×C22⋊C4×C23D414yes—
69C23×#101—42no—
70C23×#100—42no—
71C23×C22⋊C8—24no—
72C23×C22⋊C8—24no—
73C23×C22⋊C8—42no—
74C23×C22⋊C8—42no—
75C24×C23—21no—
76C24×C23C22⋊C411yescharacteristic
77C24×C23C22⋊C411yescharacteristic
78C24×C23C22⋊C412yes—
79C24×C23C22×C411yescharacteristic
80#41×C23—42no—
81#41×C23—416no—
82#41×C23—48no—
83#41×C23—42no—
84C2×Q8×C23—21no—
85C2×Q8×C23—22no—
86C2×Q8×C23—22no—
87C2×Q8×C23—41no—
88C2×D4×C23—22no—
89C2×D4×C23—22no—
90C2×D4×C23—22no—
91C2×D4×C23—21no—
92C2×D4×C23—22no—
93C2×D4×C23—22no—
94C2×D4×C23—22no—
95C2×D4×C23—22no—
96C2×D4×C23—24no—
97C2×D4×C23—22no—
98C2×D4×C23—22no—
99C2×D4×C23—41no—
100C2×D4×C23—41no—
101C2×D4×C23—41no—
102C2×D4×C23—48no—
103C2×D4×C23—42no—
104C2×D4×C23—42no—
105C2×D4×C23—42no—
106C2×D4×C23—42no—
107C22×C4×C23—21no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—22no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—22no—
114C22×C4×C23—41no—
115C22×C4×C23—41no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23C22⋊C411yescharacteristic
119C22×C4×C23C22⋊C411yescharacteristic
120C22×C4×C23C22⋊C414yes—
121C22×C4×C23C22⋊C412yes—
122C22×C4×C23C2×D411yescharacteristic
123C22×C4×C23C2×D411yescharacteristic
124C22×C4×C23C2×D414yes—
125M16×C23—42no—
126M16×C23—42no—
127C2×C8×C23—42no—
128C2×C8×C23—42no—
129C2×C8×C23—42no—
130C2×C8×C23—42no—
131C4⋊C4×C23—24no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—24no—
134C4⋊C4×C23—24no—
135C22⋊C4×C23—22no—
136C22⋊C4×C23—24no—
137C22⋊C4×C23—24no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—24no—
141C22⋊C4×C23—24no—
142C42×C23—22no—
143#8539C2311yesmaximal, characteristic, direct product factor, Sylow
144C23×C23—21no—
145C23×C23—21no—
146C23×C23—21no—
147C23×C23—21no—
148C23×C23—22no—
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—21no—
152C23×C23—42no—
153C23×C23—42no—
154C23×C23—42no—
155C23×C23—42no—
156C23×C23—42no—
157C23×C23—44no—
158C23×C23—82no—
159C23×C23#9912yes—
160C23×C23C2×C22⋊C412yes—
161C23×C23C2×C22⋊C411yescharacteristic
162Q8×C23—44no—
163Q8×C23—42no—
164D4×C23—28no—
165D4×C23—44no—
166D4×C23—42no—
167D4×C23—44no—
168D4×C23—44no—
169D4×C23—44no—
170D4×C23—42no—
171D4×C23—42no—
172D4×C23—84no—
173C2×C4×C23—21no—
174C2×C4×C23—21no—
175C2×C4×C23—21no—
176C2×C4×C23—21no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—24no—
180C2×C4×C23—21no—
181C2×C4×C23—22no—
182C2×C4×C23—21no—
183C2×C4×C23—21no—
184C2×C4×C23—44no—
185C2×C4×C23—44no—
186C2×C4×C23—44no—
187C2×C4×C23—44no—
188C2×C4×C23—42no—
189C2×C4×C23—44no—
190C2×C4×C23—42no—
191C2×C4×C23—82no—
192C2×C4×C23#9912yes—
193C2×C4×C23#12012yes—
194C2×C4×C23#12012yes—
195C8×C23—82no—
196C8×C23—82no—
197#3485C2×C2311yescharacteristic
198#1785C2×C2311yescharacteristic, complemented
199#1780C2×C2311yescharacteristic, complemented
200#1468C2×C2314yescomplemented
201C22×C23—22no—
202C22×C23—22no—
203C22×C23—22no—
204C22×C23—24no—
205C22×C23—42no—
206C22×C23—42no—
207C22×C23—42no—
208C22×C23—41no—
209C22×C23—42no—
210C22×C23—42no—
211C22×C23—44no—
212C22×C23—81no—
213C22×C23—81no—
214C22×C23—81no—
215C22×C23—81no—
216C22×C23—162no—
217C22×C23—162no—
218C22×C23#37911yescharacteristic
219C22×C23C2×#9911yescharacteristic
220C22×C23C2×#9911yescharacteristic
221C4×C23—22no—
222C4×C23—22no—
223C4×C23—22no—
224C4×C23—41no—
225C4×C23—42no—
226C4×C23—44no—
227C2×H(5,𝔽2)C4×C2311yescharacteristic
228C22×#41C22×C2311yescharacteristic
229#547—24no—
230#546—24no—
231#536—21no—
232#534—21no—
233#534C4×C2312yes—
234#525—21no—
235C2×#123—21no—
236C2×#123C22×C2312yes—
237#518C4×C2311yescharacteristic
238C2×#101—21no—
239C2×#100—21no—
240#407C22×C2312yescomplemented
241C2×C22⋊C8—21no—
242C2×C22⋊C8—21no—
243C2×C22⋊C8C22×C2312yescomplemented
244#324—24no—
245#323—24no—
246C2×C23—21no—
247C2×C23—21no—
248C2×C23—42no—
249C2×C23—42no—
250C2×C23—81no—
251C2×C23—82no—
252C2×C23#178111yescharacteristic
253C2×C23#179112yes—
254H(5,𝔽2)—42no—
255H(5,𝔽2)—42no—
256C2×#41—21no—
257C2×#41—22no—
258C2×#41—24no—
259C2×#41—42no—
260C2×#41—42no—
261C22×Q8D4×C2311yescharacteristic
262C22×D4—21no—
263C22×D4—22no—
264C22×D4C2×C4×C2311yescharacteristic
265C22×D4C2×C4×C2312yes—
266C22×D4C2×C4×C2311yescharacteristic
267C22×D4D4×C2311yescharacteristic
268C23×C4D4×C2311yescharacteristic
269C23×C4D4×C2311yescharacteristic
270C23×C4C23×C2311yescharacteristic
271C2×M16—22no—
272C22×C8—22no—
273C4⋊Q8—22no—
274#124—22no—
275#123—24no—
276#123—22no—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—24no—
282C22⋊Q8—22no—
283C22⋊Q8—24no—
284#121—22no—
285#121—24no—
286#121—24no—
287#120—22no—
288#120—22no—
289#120—22no—
290C4×D4—24no—
291C4×D4—24no—
292#117—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4D4×C2314yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4C2×C4×C2312yes—
298C2×C22⋊C4D4×C2314yes—
299#101—42no—
300#100—42no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C23#853911yescharacteristic, direct product factor, Sylow
306C24—21no—
307C24C22⋊C4×C2311yescharacteristic
308C24C22⋊C4×C2311yescharacteristic
309C24C22⋊C4×C2312yes—
310C24C22×C4×C2311yescharacteristic
311#41—42no—
312#41—416no—
313#41—48no—
314#41—42no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—48no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4C22⋊C4×C2311yescharacteristic
350C22×C4C22⋊C4×C2311yescharacteristic
351C22×C4C22⋊C4×C2314yes—
352C22×C4C22⋊C4×C2312yes—
353C22×C4C2×D4×C2311yescharacteristic
354C22×C4C2×D4×C2311yescharacteristic
355C22×C4C2×D4×C2314yes—
356M16—42no—
357M16—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C22⋊C4—22no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C42—22no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—82no—
389C23C23×#9912yes—
390C23C2×C22⋊C4×C2312yes—
391C23C2×C22⋊C4×C2311yescharacteristic
392Q8—44no—
393Q8—42no—
394D4—28no—
395D4—44no—
396D4—42no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—42no—
401D4—42no—
402D4—84no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4C23×#9912yes—
423C2×C4C23×#12012yes—
424C2×C4C23×#12012yes—
425C8—82no—
426C8—82no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—24no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—41no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—162no—
443C22—162no—
444C22C23×#37911yescharacteristic
445C22C2×C23×#9911yescharacteristic
446C22C2×C23×#9911yescharacteristic
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—44no—
453C2—21no—
454C2—21no—
455C2—42no—
456C2—42no—
457C2—81no—
458C2—82no—
459C2C23×#178111yescharacteristic
460C2C23×#179112yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3485C2
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
2. C23×#1785C2semidirect
3. C23×#1780C2semidirect
4. C23×#1468C2semidirect
18. C23×#407C22semidirect
21. C2×C23×C22⋊C8C22semidirect
143. #8539C23direct (N × H)
198. #1785C2×C23semidirect
199. #1780C2×C23semidirect
200. #1468C2×C23semidirect
240. #407C22×C23semidirect
243. C2×C22⋊C8C22×C23semidirect
305. C23#8539direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34852
C23×#17852
C23×#17802
C23×#14682
#853923