Finite Group Databasebeta

← C23×#11716 #19159342

Subgroups — C23×#11716 #19159342

Order 5888.

592 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2626C211yesmaximal, characteristic
2C2×C23×#342C211yesmaximal, characteristic
3C2×C23×#323C211yesmaximal, characteristic
4C23×#1352C212yesmaximal, complemented
5C23×#1351C212yesmaximal, complemented
6C2×C23×C22⋊Q8C411yescharacteristic, complemented
7C22×C22⋊C4×C23—21no—
8C22×C22⋊C4×C23C2211yescharacteristic
9C2×C23×C22⋊C8—21no—
10C23×#399C411yescharacteristic, complemented
11C23×#399C2211yescharacteristic
12C23×#396—21no—
13C23×#396—21no—
14C23×#393C411yescharacteristic, complemented
15C23×#393C2211yescharacteristic
16C23×#385—21no—
17C23×#380C411yescharacteristic, complemented
18C23×#379—21no—
19C23×#342—21no—
20C23×#342C2212yes—
21C23×#323—21no—
22C23×#323C2212yes—
23C25×C23D411yescharacteristic
24C22×Q8×C23C2×C411yescharacteristic, complemented
25C23×C4×C23—41no—
26C23×C4×C23C2×C411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C22×C8×C23—41no—
29C23×C22⋊Q8—41no—
30C23×C22⋊Q8—41no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—21no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C411yescharacteristic, complemented
36C2×C22⋊C4×C23—24no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—21no—
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—41no—
45C2×C22⋊C4×C23—41no—
46C2×C22⋊C4×C23—41no—
47C2×C22⋊C4×C23—42no—
48C2×C22⋊C4×C23C2×C411yescharacteristic
49C2×C22⋊C4×C23D411yescharacteristic
50C2×C22⋊C4×C23D411yescharacteristic
51C2×C22⋊C4×C23C2311yescharacteristic
52C23×C22⋊C8—22no—
53C23×C22⋊C8—41no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95C2×C411yescharacteristic, complemented
57C23×#95C2×C411yescharacteristic, complemented
58C24×C23—21no—
59C24×C23—21no—
60C24×C23—22no—
61C24×C23—81no—
62C24×C23—81no—
63C24×C23C22⋊C411yescharacteristic
64C24×C23C22⋊C411yescharacteristic
65C24×C23SD1612yes—
66C24×C23Q1612yes—
67C24×C23C2×D411yescharacteristic
68C2×Q8×C23—41no—
69C2×Q8×C23—81no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—21no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—21no—
78C22×C4×C23—44no—
79C22×C4×C23—42no—
80C22×C4×C23—81no—
81C22×C4×C23—81no—
82C22×C4×C23—82no—
83C22×C4×C23—82no—
84C22×C4×C23C22⋊C411yescharacteristic
85C22×C4×C23C22⋊C411yescharacteristic
86C22×C4×C23C22×C411yescharacteristic
87C22×C4×C23C2×D411yescharacteristic
88C2×C8×C23—42no—
89C2×C8×C23—82no—
90C4⋊C4×C23—41no—
91C4⋊C4×C23—42no—
92C4⋊C4×C23—81no—
93C4⋊C4×C23—81no—
94C4⋊C4×C23—81no—
95C22⋊C4×C23—24no—
96C22⋊C4×C23—41no—
97C22⋊C4×C23—41no—
98C22⋊C4×C23—41no—
99C22⋊C4×C23—41no—
100C22⋊C4×C23—44no—
101C22⋊C4×C23—41no—
102C22⋊C4×C23—41no—
103C22⋊C4×C23—42no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—81no—
106C22⋊C4×C23—81no—
107C22⋊C4×C23—81no—
108C22⋊C4×C23—81no—
109#11716C2311yesmaximal, characteristic, direct product factor, Sylow
110C23×C23—22no—
111C23×C23—22no—
112C23×C23—21no—
113C23×C23—21no—
114C23×C23—22no—
115C23×C23—22no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—42no—
121C23×C23—42no—
122C23×C23—41no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—81no—
127C23×C23—81no—
128C23×C23—82no—
129C23×C23—82no—
130C23×C23—84no—
131C23×C23#9911yescharacteristic
132C23×C23#9911yescharacteristic
133C23×C23#10312yes—
134C23×C23#10312yes—
135C23×C23#10412yes—
136C23×C23C2×C22⋊C411yescharacteristic
137C23×C23C2×SD1611yescharacteristic
138C23×C23C2×Q1611yescharacteristic
139Q8×C23—82no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—42no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—44no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
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—81no—
163C2×C4×C23—81no—
164C2×C4×C23—81no—
165C2×C4×C23—88no—
166C2×C4×C23—82no—
167C2×C4×C23—82no—
168C8×C23—82no—
169#2626C2×C2311yescharacteristic
170C2×#342C2×C2311yescharacteristic
171C2×#323C2×C2311yescharacteristic
172#1352C2×C2312yescomplemented
173#1351C2×C2312yescomplemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—21no—
181C22×C23—22no—
182C22×C23—21no—
183C22×C23—42no—
184C22×C23—42no—
185C22×C23—42no—
186C22×C23—41no—
187C22×C23—41no—
188C22×C23—44no—
189C22×C23—44no—
190C22×C23—41no—
191C22×C23—41no—
192C22×C23—41no—
193C22×C23—41no—
194C22×C23—81no—
195C22×C23—81no—
196C22×C23—82no—
197C22×C23—82no—
198C22×C23#32812yes—
199C22×C23#32812yes—
200C22×C23C2×#9911yescharacteristic
201C22×C23C2×#10311yescharacteristic
202C22×C23C2×#10411yescharacteristic
203C4×C23—41no—
204C4×C23—41no—
205C4×C23—41no—
206C4×C23—41no—
207C4×C23—81no—
208C4×C23—81no—
209C4×C23—84no—
210C4×C23—81no—
211C4×C23—81no—
212C2×C22⋊Q8C4×C2311yescharacteristic, complemented
213C22×C22⋊C4—21no—
214C22×C22⋊C4C22×C2311yescharacteristic
215C2×C22⋊C8—21no—
216#399C4×C2311yescharacteristic, complemented
217#399C22×C2311yescharacteristic
218#396—21no—
219#396—21no—
220#393C4×C2311yescharacteristic, complemented
221#393C22×C2311yescharacteristic
222#385—21no—
223#380C4×C2311yescharacteristic, complemented
224#379—21no—
225#342—21no—
226#342C22×C2312yes—
227#323—21no—
228#323C22×C2312yes—
229C2×C23—21no—
230C2×C23—22no—
231C2×C23—21no—
232C2×C23—22no—
233C2×C23—41no—
234C2×C23—42no—
235C2×C23—41no—
236C2×C23C2×#32811yescharacteristic
237C2×C23#212111yescharacteristic
238C2×C23#212711yescharacteristic
239C25D4×C2311yescharacteristic
240C22×Q8C2×C4×C2311yescharacteristic, complemented
241C23×C4—41no—
242C23×C4C2×C4×C2311yescharacteristic
243C23×C4D4×C2311yescharacteristic
244C22×C8—41no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×C4×C2311yescharacteristic, complemented
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C2×C4×C2311yescharacteristic
265C2×C22⋊C4D4×C2311yescharacteristic
266C2×C22⋊C4D4×C2311yescharacteristic
267C2×C22⋊C4C23×C2311yescharacteristic
268C22⋊C8—22no—
269C22⋊C8—41no—
270#95—21no—
271#95—21no—
272#95C2×C4×C2311yescharacteristic, complemented
273#95C2×C4×C2311yescharacteristic, complemented
274C23#1171611yescharacteristic, direct product factor, Sylow
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24C22⋊C4×C2311yescharacteristic
281C24C22⋊C4×C2311yescharacteristic
282C24SD16×C2312yes—
283C24Q16×C2312yes—
284C24C2×D4×C2311yescharacteristic
285C2×Q8—41no—
286C2×Q8—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C22⋊C4×C2311yescharacteristic
302C22×C4C22⋊C4×C2311yescharacteristic
303C22×C4C22×C4×C2311yescharacteristic
304C22×C4C2×D4×C2311yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23C23×#9911yescharacteristic
348C23C23×#9911yescharacteristic
349C23C23×#10312yes—
350C23C23×#10312yes—
351C23C23×#10412yes—
352C23C2×C22⋊C4×C2311yescharacteristic
353C23C2×SD16×C2311yescharacteristic
354C23C2×Q16×C2311yescharacteristic
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
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—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—88no—
382C2×C4—82no—
383C2×C4—82no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22C23×#32812yes—
410C22C23×#32812yes—
411C22C2×C23×#9911yescharacteristic
412C22C2×C23×#10311yescharacteristic
413C22C2×C23×#10411yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—84no—
421C4—81no—
422C4—81no—
423C2—21no—
424C2—22no—
425C2—21no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×C23×#32811yescharacteristic
431C2C23×#212111yescharacteristic
432C2C23×#212711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC25×C23D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1352C2semidirect
5. C23×#1351C2semidirect
6. C2×C23×C22⋊Q8C4semidirect
10. C23×#399C4semidirect
14. C23×#393C4semidirect
17. C23×#380C4semidirect
24. C22×Q8×C23C2×C4semidirect
35. C2×C4⋊C4×C23C2×C4semidirect
56. C23×#95C2×C4semidirect
57. C23×#95C2×C4semidirect
109. #11716C23direct (N × H)
172. #1352C2×C23semidirect
173. #1351C2×C23semidirect
212. C2×C22⋊Q8C4×C23semidirect
216. #399C4×C23semidirect
220. #393C4×C23semidirect
223. #380C4×C23semidirect
240. C22×Q8C2×C4×C23semidirect
251. C2×C4⋊C4C2×C4×C23semidirect
272. #95C2×C4×C23semidirect
273. #95C2×C4×C23semidirect
274. C23#11716direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#26262
C2×C23×#3422
C2×C23×#3232
C23×#13522
C23×#13512
#1171623