Finite Group Databasebeta

← C23×#9038 #19156664

Subgroups — C23×#9038 #19156664

Order 5888.

468 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×#3091C211yesmaximal, characteristic
2C23×#2976C211yesmaximal, characteristic
3C23×#1826C211yesmaximal, characteristic, complemented
4C23×#1818C211yesmaximal, characteristic, complemented
5C23×#1501C211yesmaximal, characteristic
6C23×#1298C211yesmaximal, characteristic, complemented
7C23×#1282C211yesmaximal, characteristic, complemented
8C23×#525C2211yescharacteristic
9C2×C23×#117C2211yescharacteristic
10C23×#485—21no—
11C23×#484C411yescharacteristic, complemented
12C23×#483C411yescharacteristic, complemented
13C23×#482—21no—
14C23×#481C411yescharacteristic, complemented
15C23×#480C411yescharacteristic, complemented
16C23×#436—21no—
17C23×#435—21no—
18C23×#435—21no—
19C8×D4×C23—21no—
20C23×#433—21no—
21C23×#427—21no—
22C23×#416—21no—
23C23×#408—21no—
24C23×#407C2211yescharacteristic
25C2×C23×C22⋊C8C2211yescharacteristic
26C2×C23×C22⋊C8C2211yescharacteristic
27C2×C23×C22⋊C8C2211yescharacteristic
28C4×C22⋊C4×C23C2211yescharacteristic
29C23×#339—21no—
30C23×#339—21no—
31C23×#336—21no—
32C23×#336—21no—
33C23×#328—21no—
34C23×#327—21no—
35C2×#41×C23D411yescharacteristic
36C23×C4×C23C2311yescharacteristic
37C2×M16×C23—21no—
38C2×M16×C23D411yescharacteristic
39C2×M16×C23D411yescharacteristic
40C22×C8×C23—21no—
41C22×C8×C23—21no—
42C22×C8×C23—21no—
43C22×C8×C23D411yescharacteristic
44C22×C8×C23D411yescharacteristic
45C23×#123—21no—
46C23×C22⋊Q8C2×C411yescharacteristic
47C23×C22⋊Q8D411yescharacteristic
48C23×#121C2×C411yescharacteristic
49C23×#121D411yescharacteristic
50C23×#120—21no—
51C4×D4×C23—21no—
52C4×D4×C23—21no—
53C23×#117—21no—
54C23×#117—21no—
55C23×#117—21no—
56C23×#117—21no—
57C23×#117D411yescharacteristic
58C2×C4⋊C4×C23C2×C411yescharacteristic
59C2×C4⋊C4×C23C2×C411yescharacteristic
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C42×C23—21no—
63C2×C42×C23—21no—
64C23×#107—21no—
65C23×#107—21no—
66C23×#106—21no—
67C23×#106—21no—
68C23×C4⋊C8—21no—
69C23×C4⋊C8—21no—
70C23×#103—21no—
71C23×#103—21no—
72C23×#102—21no—
73C23×#102—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8—21no—
79C23×C22⋊C8—21no—
80C23×C22⋊C8—21no—
81C23×C22⋊C8—21no—
82C23×C22⋊C8—21no—
83C23×C22⋊C8C2×C411yescharacteristic
84C23×C22⋊C8C2×C411yescharacteristic
85C23×#97—21no—
86C4×C8×C23—21no—
87C23×#95—21no—
88C24×C23#4111yescharacteristic
89#41×C23—41no—
90#41×C23—41no—
91C2×Q8×C23C22⋊C411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23C22⋊C411yescharacteristic
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—21no—
99C22×C4×C23—41no—
100C22×C4×C23—41no—
101C22×C4×C23—41no—
102C22×C4×C23C22×C411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23C2×D411yescharacteristic
105C22×C4×C23C2×D411yescharacteristic
106C22×C4×C23C2×D411yescharacteristic
107C22×C4×C23#4111yescharacteristic
108M16×C23—41no—
109M16×C23—41no—
110M16×C23—81no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23—21no—
114C2×C8×C23—21no—
115C2×C8×C23—21no—
116C2×C8×C23—21no—
117C2×C8×C23—41no—
118C2×C8×C23—41no—
119C2×C8×C23—41no—
120C2×C8×C23—41no—
121C2×C8×C23—41no—
122C2×C8×C23—41no—
123C2×C8×C23—41no—
124C2×C8×C23—81no—
125C4⋊C4×C23—21no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—21no—
131C4⋊C4×C23C22⋊C411yescharacteristic
132C4⋊C4×C23C22⋊C411yescharacteristic
133C22⋊C4×C23—21no—
134C22⋊C4×C23—21no—
135C22⋊C4×C23—21no—
136C22⋊C4×C23—21no—
137C22⋊C4×C23—21no—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—41no—
140C22⋊C4×C23—41no—
141C22⋊C4×C23—41no—
142C42×C23—21no—
143C42×C23—41no—
144C42×C23—41no—
145C42×C23—41no—
146C42×C23—41no—
147#9038C2311yesmaximal, characteristic, direct product factor, Sylow
148C23×C23—21no—
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—41no—
152C23×C23—41no—
153C23×C23C4×D411yescharacteristic
154C23×C23#12111yescharacteristic
155C23×C23#12311yescharacteristic
156Q8×C23—41no—
157D4×C23—41no—
158D4×C23—41no—
159D4×C23—41no—
160D4×C23—81no—
161D4×C23—81no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—21no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—42no—
177C2×C4×C23—42no—
178C2×C4×C23—41no—
179C2×C4×C23—41no—
180C2×C4×C23—41no—
181C2×C4×C23—41no—
182C2×C4×C23—81no—
183C2×C4×C23—81no—
184C2×C4×C23C2×C22⋊C411yescharacteristic
185C2×C4×C23C4×D411yescharacteristic
186C2×C4×C23#12011yescharacteristic
187C2×C4×C23#12111yescharacteristic
188C8×C23—41no—
189C8×C23—41no—
190C8×C23—41no—
191C8×C23—41no—
192C8×C23—41no—
193C8×C23—41no—
194C8×C23—81no—
195#3091C2×C2311yescharacteristic
196#2976C2×C2311yescharacteristic
197#1826C2×C2311yescharacteristic, complemented
198#1818C2×C2311yescharacteristic, complemented
199#1501C2×C2311yescharacteristic
200#1298C2×C2311yescharacteristic, complemented
201#1282C2×C2311yescharacteristic, complemented
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—21no—
207C22×C23—21no—
208C22×C23—21no—
209C22×C23—21no—
210C22×C23—41no—
211C22×C23—41no—
212C22×C23—41no—
213C22×C23—81no—
214C22×C23—81no—
215C22×C23#38611yescharacteristic
216C22×C23#44311yescharacteristic
217C22×C23#44411yescharacteristic
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—41no—
222C4×C23—41no—
223C4×C23—41no—
224C4×C23—82no—
225C4×C23—81no—
226C4×C23—81no—
227C4×C23#45811yescharacteristic
228C4×C23#45811yescharacteristic
229#525C22×C2311yescharacteristic
230C2×#117C22×C2311yescharacteristic
231#485—21no—
232#484C4×C2311yescharacteristic, complemented
233#483C4×C2311yescharacteristic, complemented
234#482—21no—
235#481C4×C2311yescharacteristic, complemented
236#480C4×C2311yescharacteristic, complemented
237#436—21no—
238#435—21no—
239#435—21no—
240C8×D4—21no—
241#433—21no—
242#427—21no—
243#416—21no—
244#408—21no—
245#407C22×C2311yescharacteristic
246C2×C22⋊C8C22×C2311yescharacteristic
247C2×C22⋊C8C22×C2311yescharacteristic
248C2×C22⋊C8C22×C2311yescharacteristic
249C4×C22⋊C4C22×C2311yescharacteristic
250#339—21no—
251#339—21no—
252#336—21no—
253#336—21no—
254#328—21no—
255#327—21no—
256C2×C23—21no—
257C2×C23—21no—
258C2×C23—41no—
259C2×C23—41no—
260C2×C23—81no—
261C2×C23#187611yescharacteristic
262C2×C23#188111yescharacteristic
263C2×C23#189011yescharacteristic
264C2×#41D4×C2311yescharacteristic
265C23×C4C23×C2311yescharacteristic
266C2×M16—21no—
267C2×M16D4×C2311yescharacteristic
268C2×M16D4×C2311yescharacteristic
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8D4×C2311yescharacteristic
273C22×C8D4×C2311yescharacteristic
274#123—21no—
275C22⋊Q8C2×C4×C2311yescharacteristic
276C22⋊Q8D4×C2311yescharacteristic
277#121C2×C4×C2311yescharacteristic
278#121D4×C2311yescharacteristic
279#120—21no—
280C4×D4—21no—
281C4×D4—21no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—21no—
286#117D4×C2311yescharacteristic
287C2×C4⋊C4C2×C4×C2311yescharacteristic
288C2×C4⋊C4C2×C4×C2311yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293#107—21no—
294#107—21no—
295#106—21no—
296#106—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299#103—21no—
300#103—21no—
301#102—21no—
302#102—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8C2×C4×C2311yescharacteristic
313C22⋊C8C2×C4×C2311yescharacteristic
314#97—21no—
315C4×C8—21no—
316#95—21no—
317C23#903811yescharacteristic, direct product factor, Sylow
318C24#41×C2311yescharacteristic
319#41—41no—
320#41—41no—
321C2×Q8C22⋊C4×C2311yescharacteristic
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4C22⋊C4×C2311yescharacteristic
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4C22×C4×C2311yescharacteristic
333C22×C4C2×D4×C2311yescharacteristic
334C22×C4C2×D4×C2311yescharacteristic
335C22×C4C2×D4×C2311yescharacteristic
336C22×C4C2×D4×C2311yescharacteristic
337C22×C4#41×C2311yescharacteristic
338M16—41no—
339M16—41no—
340M16—81no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—81no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4C22⋊C4×C2311yescharacteristic
362C4⋊C4C22⋊C4×C2311yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—22no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C42—21no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—41no—
381C23—41no—
382C23C4×D4×C2311yescharacteristic
383C23C23×#12111yescharacteristic
384C23C23×#12311yescharacteristic
385Q8—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4C2×C22⋊C4×C2311yescharacteristic
414C2×C4C4×D4×C2311yescharacteristic
415C2×C4C23×#12011yescharacteristic
416C2×C4C23×#12111yescharacteristic
417C8—41no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—41no—
422C8—41no—
423C8—81no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22C23×#38611yescharacteristic
438C22C23×#44311yescharacteristic
439C22C23×#44411yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4C23×#45811yescharacteristic
450C4C23×#45811yescharacteristic
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—81no—
456C2C23×#187611yescharacteristic
457C2C23×#188111yescharacteristic
458C2C23×#189011yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
socleC22×C23#386
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1826C2semidirect
4. C23×#1818C2semidirect
6. C23×#1298C2semidirect
7. C23×#1282C2semidirect
11. C23×#484C4semidirect
12. C23×#483C4semidirect
14. C23×#481C4semidirect
15. C23×#480C4semidirect
147. #9038C23direct (N × H)
197. #1826C2×C23semidirect
198. #1818C2×C23semidirect
200. #1298C2×C23semidirect
201. #1282C2×C23semidirect
232. #484C4×C23semidirect
233. #483C4×C23semidirect
235. #481C4×C23semidirect
236. #480C4×C23semidirect
317. C23#9038direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30912
C23×#29762
C23×#18262
C23×#18182
C23×#15012
C23×#12982
C23×#12822
#903823