Finite Group Databasebeta

← C23×#12782 #19160408

Subgroups — C23×#12782 #19160408

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3544C211yesmaximal, characteristic
2C23×#2133C212yesmaximal, complemented
3C2×C23×#356C211yesmaximal, characteristic, complemented
4C23×#1912C212yesmaximal, complemented
5C23×#1886C211yesmaximal, characteristic, complemented
6C23×#563—21no—
7C23×#563C412yescomplemented
8Q82×C23C411yescharacteristic, complemented
9C23×#555C411yescharacteristic, complemented
10C23×#554—21no—
11C23×#552—22no—
12C23×#547—22no—
13C23×#544—22no—
14C23×#542—22no—
15C23×#536—21no—
16C23×#536C2212yes—
17C2×C23×C4⋊Q8C2211yescharacteristic
18C23×#529—21no—
19C23×#519C2211yescharacteristic
20C23×#421—22no—
21C23×#419—21no—
22C2×C23×#101C2211yescharacteristic
23C2×C23×#101C2212yes—
24C23×#356—21no—
25C23×#356—21no—
26C23×#356—22no—
27C23×#355—22no—
28C23×#339—21no—
29C23×H(3,ℤ/4)—22no—
30C2×#41×C23—21no—
31C2×#41×C23D412yes—
32C22×Q8×C23D411yescharacteristic
33C22×Q8×C23C2311yescharacteristic
34C2×M16×C23—22no—
35C2×M16×C23—21no—
36C23×C4⋊Q8—21no—
37C23×C4⋊Q8—22no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—22no—
40C23×C4⋊Q8C2×C411yescharacteristic
41C23×C4⋊Q8C2×C412yes—
42C23×C4⋊Q8C2×C411yescharacteristic
43C23×C4⋊Q8D411yescharacteristic
44C23×C4⋊Q8D411yescharacteristic
45C23×C4⋊Q8D412yes—
46C23×#126—22no—
47C23×#126—22no—
48C23×#126—22no—
49C23×#125—22no—
50C23×#125—22no—
51C23×#125—21no—
52C23×#125—22no—
53C23×#124—21no—
54C23×#124—21no—
55C23×#124—21no—
56C23×#124C2×C412yes—
57C23×#124D412yes—
58C23×#123—22no—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—22no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8—21no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8—22no—
70C23×C22⋊Q8—22no—
71C23×#121—21no—
72C23×#121—22no—
73C4×Q8×C23—22no—
74C4×Q8×C23—21no—
75C4×Q8×C23—42no—
76C4×D4×C23—22no—
77C4×D4×C23—22no—
78C4×D4×C23—21no—
79C23×#117—21no—
80C23×#117—22no—
81C23×#117—22no—
82C23×#117—22no—
83C23×#117—21no—
84C23×#117D412yes—
85C2×C4⋊C4×C23—22no—
86C2×C4⋊C4×C23—22no—
87C2×C42×C23D411yescharacteristic
88C23×#104—44no—
89C23×#103—42no—
90C23×#101—22no—
91C23×#101—22no—
92C23×#101—22no—
93#41×C23—42no—
94#41×C23—42no—
95#41×C23—81no—
96#41×C23—81no—
97C2×Q8×C23—21no—
98C2×Q8×C23—21no—
99C2×Q8×C23—21no—
100C2×Q8×C23—41no—
101C2×Q8×C23—41no—
102C2×Q8×C23—41no—
103C2×Q8×C23—41no—
104C2×Q8×C23C22⋊C411yescharacteristic
105C2×Q8×C23C22⋊C412yes—
106C2×Q8×C23C22⋊C411yescharacteristic
107C2×Q8×C23C22×C411yescharacteristic
108C2×Q8×C23C2×D411yescharacteristic
109C2×Q8×C23C2×D412yes—
110C2×D4×C23—21no—
111C2×D4×C23—22no—
112C2×D4×C23—21no—
113C2×D4×C23C22⋊C412yes—
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—22no—
118C22×C4×C23—22no—
119C22×C4×C23C2×D411yescharacteristic
120C22×C4×C23C2×D412yes—
121M16×C23—42no—
122M16×C23—41no—
123M16×C23—41no—
124M16×C23—42no—
125C2×C8×C23—41no—
126C2×C8×C23—42no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—22no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23—22no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—42no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—22no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—41no—
155C42×C23—21no—
156C42×C23—22no—
157C42×C23—21no—
158C42×C23—21no—
159C42×C23—24no—
160C42×C23—41no—
161C42×C23C22⋊C411yescharacteristic
162C42×C23C22⋊C411yescharacteristic
163C42×C23C22⋊C412yes—
164C42×C23C22⋊C412yes—
165#12782C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—22no—
167C23×C23#12011yescharacteristic
168Q8×C23—41no—
169Q8×C23—41no—
170Q8×C23—42no—
171Q8×C23—41no—
172Q8×C23—41no—
173Q8×C23—42no—
174Q8×C23—42no—
175Q8×C23—81no—
176Q8×C23—81no—
177D4×C23—42no—
178D4×C23—42no—
179D4×C23—42no—
180D4×C23—42no—
181D4×C23—81no—
182C2×C4×C23—21no—
183C2×C4×C23—22no—
184C2×C4×C23—21no—
185C2×C4×C23—22no—
186C2×C4×C23—21no—
187C2×C4×C23—22no—
188C2×C4×C23—21no—
189C2×C4×C23—22no—
190C2×C4×C23—21no—
191C2×C4×C23—22no—
192C2×C4×C23—21no—
193C2×C4×C23—42no—
194C2×C4×C23—44no—
195C2×C4×C23—41no—
196C2×C4×C23—41no—
197C2×C4×C23—42no—
198C2×C4×C23—82no—
199C2×C4×C23—82no—
200C2×C4×C23C2×C22⋊C411yescharacteristic
201C2×C4×C23C2×C22⋊C412yes—
202C2×C4×C23#12011yescharacteristic
203C2×C4×C23#12012yes—
204C8×C23—82no—
205C8×C23—81no—
206#3544C2×C2311yescharacteristic
207#2133C2×C2312yescomplemented
208C2×#356C2×C2311yescharacteristic, complemented
209#1912C2×C2312yescomplemented
210#1886C2×C2311yescharacteristic, complemented
211C22×C23—41no—
212C22×C23—42no—
213C22×C23—42no—
214C22×C23—42no—
215C22×C23#37911yescharacteristic
216C22×C23B(4,𝔽2)11yescharacteristic
217C22×C23#45811yescharacteristic
218C4×C23—22no—
219C4×C23—22no—
220C4×C23—21no—
221C4×C23—22no—
222C4×C23—21no—
223C4×C23—42no—
224C4×C23—41no—
225C4×C23—42no—
226C4×C23—41no—
227C4×C23—41no—
228C4×C23—42no—
229C4×C23—82no—
230#563—21no—
231#563C4×C2312yescomplemented
232Q82C4×C2311yescharacteristic, complemented
233#555C4×C2311yescharacteristic, complemented
234#554—21no—
235#552—22no—
236#547—22no—
237#544—22no—
238#542—22no—
239#536—21no—
240#536C22×C2312yes—
241C2×C4⋊Q8C22×C2311yescharacteristic
242#529—21no—
243#519C22×C2311yescharacteristic
244#421—22no—
245#419—21no—
246C2×#101C22×C2312yes—
247C2×#101C22×C2311yescharacteristic
248#356—21no—
249#356—21no—
250#356—22no—
251#355—22no—
252#339—21no—
253H(3,ℤ/4)—22no—
254C2×C23—21no—
255C2×C23—21no—
256C2×C23—21no—
257C2×C23—82no—
258C2×C23#179011yescharacteristic
259C2×#41—21no—
260C2×#41D4×C2312yes—
261C22×Q8D4×C2311yescharacteristic
262C22×Q8C23×C2311yescharacteristic
263C2×M16—22no—
264C2×M16—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8C2×C4×C2311yescharacteristic
270C4⋊Q8C2×C4×C2311yescharacteristic
271C4⋊Q8C2×C4×C2312yes—
272C4⋊Q8D4×C2311yescharacteristic
273C4⋊Q8D4×C2311yescharacteristic
274C4⋊Q8D4×C2312yes—
275#126—22no—
276#126—22no—
277#126—22no—
278#125—21no—
279#125—22no—
280#125—22no—
281#125—22no—
282#124—21no—
283#124—21no—
284#124—21no—
285#124C2×C4×C2312yes—
286#124D4×C2312yes—
287#123—22no—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292C22⋊Q8—21no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300#121—21no—
301#121—22no—
302C4×Q8—21no—
303C4×Q8—22no—
304C4×Q8—42no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—21no—
308#117—22no—
309#117—21no—
310#117—22no—
311#117—22no—
312#117—21no—
313#117D4×C2312yes—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C42D4×C2311yescharacteristic
317#104—44no—
318#103—42no—
319#101—22no—
320#101—22no—
321#101—22no—
322C23#1278211yescharacteristic, direct product factor, Sylow
323#41—42no—
324#41—42no—
325#41—81no—
326#41—81no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C22⋊C4×C2311yescharacteristic
335C2×Q8C22⋊C4×C2312yes—
336C2×Q8C22⋊C4×C2311yescharacteristic
337C2×Q8C22×C4×C2311yescharacteristic
338C2×Q8C2×D4×C2311yescharacteristic
339C2×Q8C2×D4×C2312yes—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4C22⋊C4×C2312yes—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×D4×C2311yescharacteristic
350C22×C4C2×D4×C2312yes—
351M16—42no—
352M16—41no—
353M16—41no—
354M16—42no—
355C2×C8—41no—
356C2×C8—42no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C42—21no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—24no—
390C42—41no—
391C42C22⋊C4×C2311yescharacteristic
392C42C22⋊C4×C2311yescharacteristic
393C42C22⋊C4×C2312yes—
394C42C22⋊C4×C2312yes—
395C23—22no—
396C23C23×#12011yescharacteristic
397Q8—41no—
398Q8—41no—
399Q8—42no—
400Q8—41no—
401Q8—41no—
402Q8—42no—
403Q8—42no—
404Q8—81no—
405Q8—81no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—81no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4C2×C22⋊C4×C2311yescharacteristic
430C2×C4C2×C22⋊C4×C2312yes—
431C2×C4C23×#12011yescharacteristic
432C2×C4C23×#12012yes—
433C8—82no—
434C8—81no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22C23×#37911yescharacteristic
440C22C23×B(4,𝔽2)11yescharacteristic
441C22C23×#45811yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—22no—
446C4—21no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—82no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—82no—
458C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×Q8C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC23×#3544C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C22×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2133C2semidirect
3. C2×C23×#356C2semidirect
4. C23×#1912C2semidirect
5. C23×#1886C2semidirect
7. C23×#563C4semidirect
8. Q82×C23C4semidirect
9. C23×#555C4semidirect
165. #12782C23direct (N × H)
207. #2133C2×C23semidirect
208. C2×#356C2×C23semidirect
209. #1912C2×C23semidirect
210. #1886C2×C23semidirect
231. #563C4×C23semidirect
232. Q82C4×C23semidirect
233. #555C4×C23semidirect
322. C23#12782direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35442
C23×#21332
C2×C23×#3562
C23×#19122
C23×#18862
#1278223