Finite Group Databasebeta

← C23×#9341 #19156967

Subgroups — C23×#9341 #19156967

Order 5888.

534 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×#3193C211yesmaximal, characteristic
2C23×#2377C211yesmaximal, characteristic
3C23×#1996C211yesmaximal, characteristic, complemented
4C23×#1511C211yesmaximal, characteristic, complemented
5C23×#1507C211yesmaximal, characteristic
6C23×#1305C211yesmaximal, characteristic, complemented
7C23×#1284C211yesmaximal, characteristic, complemented
8C2×C23×C22⋊Q8C2211yescharacteristic
9C2×C23×#121C2211yescharacteristic
10C23×#484—21no—
11C23×#480C412yescomplemented
12C23×#468—22no—
13C23×#450—21no—
14C23×#448C412yescomplemented
15C23×#407C2211yescharacteristic
16C23×#407C2211yescharacteristic
17C23×#390—21no—
18C23×#388—21no—
19C23×#387—21no—
20C23×#386—21no—
21C23×#386—21no—
22C23×#385—21no—
23C23×#381C2211yescharacteristic
24C23×#380C2211yescharacteristic
25C4×C22⋊C4×C23C2211yescharacteristic
26C23×#342—21no—
27C23×#342—21no—
28C23×#328—21no—
29C23×#328—21no—
30C23×#327—22no—
31C22×Q8×C23C2×C411yescharacteristic, complemented
32C22×D4×C23—21no—
33C22×D4×C23D411yescharacteristic
34C23×C4×C23—21no—
35C23×C4×C23C2311yescharacteristic
36C2×SD16×C23—22no—
37C2×M16×C23—21no—
38C2×M16×C23—21no—
39C23×C22⋊Q8—21no—
40C23×C22⋊Q8—21no—
41C23×C22⋊Q8—21no—
42C23×C22⋊Q8—21no—
43C23×#121—21no—
44C23×#121—21no—
45C23×#121C2×C412yes—
46C23×#121D412yes—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23C2×C411yescharacteristic, complemented
51C2×C4⋊C4×C23D411yescharacteristic
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23D411yescharacteristic
60C2×C22⋊C4×C23D411yescharacteristic
61C2×C42×C23—21no—
62C2×C42×C23—21no—
63C23×#107—22no—
64C23×#103—22no—
65C23×#102—22no—
66C23×C22⋊C8—21no—
67C23×C22⋊C8—21no—
68C23×C22⋊C8—21no—
69C23×C22⋊C8C2×C412yescomplemented
70C23×#95—21no—
71C23×#95—21no—
72C23×#95—21no—
73C23×#95—21no—
74C23×#95—21no—
75C23×#95D411yescharacteristic
76C23×#95D411yescharacteristic
77C24×C23—21no—
78C24×C23C2×D411yescharacteristic
79C2×Q8×C23—21no—
80C2×Q8×C23—21no—
81C2×Q8×C23—81no—
82C2×D4×C23—22no—
83C2×D4×C23—22no—
84C2×D4×C23—21no—
85C2×D4×C23—81no—
86C2×D4×C23—81no—
87C2×D4×C23—81no—
88C2×D4×C23C22⋊C412yes—
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—21no—
93C22×C4×C23—21no—
94C22×C4×C23—21no—
95C22×C4×C23—21no—
96C22×C4×C23—41no—
97C22×C4×C23—81no—
98C22×C4×C23—81no—
99C22×C4×C23—81no—
100C22×C4×C23—81no—
101C22×C4×C23C22×C411yescharacteristic
102C22×C4×C23C2×D411yescharacteristic
103C22×C4×C23C2×D411yescharacteristic
104C22×C4×C23C2×D411yescharacteristic
105C22×C4×C23#4111yescharacteristic
106C22×C4×C23#4111yescharacteristic
107SD16×C23—82no—
108M16×C23—81no—
109M16×C23—81no—
110C2×C8×C23—22no—
111C2×C8×C23—41no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—81no—
118C4⋊C4×C23—81no—
119C4⋊C4×C23C22⋊C412yes—
120C22⋊C4×C23—22no—
121C22⋊C4×C23—22no—
122C22⋊C4×C23—22no—
123C22⋊C4×C23—41no—
124C22⋊C4×C23—41no—
125C22⋊C4×C23—41no—
126C22⋊C4×C23—41no—
127C22⋊C4×C23—41no—
128C22⋊C4×C23—81no—
129C22⋊C4×C23—81no—
130C22⋊C4×C23—81no—
131C22⋊C4×C23—81no—
132C42×C23—81no—
133C42×C23—81no—
134#9341C2311yesmaximal, characteristic, direct product factor, Sylow
135C23×C23—21no—
136C23×C23—21no—
137C23×C23—22no—
138C23×C23—41no—
139C23×C23—41no—
140C23×C23—41no—
141C23×C23—41no—
142C23×C23—41no—
143C23×C23—41no—
144C23×C23—41no—
145C23×C23—41no—
146C23×C23C4×D411yescharacteristic
147C23×C23#12011yescharacteristic
148C23×C23#12111yescharacteristic
149Q8×C23—42no—
150Q8×C23—81no—
151D4×C23—42no—
152D4×C23—81no—
153D4×C23—82no—
154D4×C23—82no—
155C2×C4×C23—22no—
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—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—42no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—41no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—41no—
179C2×C4×C23—81no—
180C2×C4×C23—81no—
181C2×C4×C23—84no—
182C2×C4×C23—81no—
183C2×C4×C23—81no—
184C2×C4×C23C2×C22⋊C411yescharacteristic
185C2×C4×C23C4×D411yescharacteristic
186C2×C4×C23#12111yescharacteristic
187C2×C4×C23#12311yescharacteristic
188C8×C23—42no—
189C8×C23—81no—
190#3193C2×C2311yescharacteristic
191#2377C2×C2311yescharacteristic
192#1996C2×C2311yescharacteristic, complemented
193#1511C2×C2311yescharacteristic, complemented
194#1507C2×C2311yescharacteristic
195#1305C2×C2311yescharacteristic, complemented
196#1284C2×C2311yescharacteristic, complemented
197C22×C23—21no—
198C22×C23—21no—
199C22×C23—21no—
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—42no—
206C22×C23—42no—
207C22×C23—42no—
208C22×C23—81no—
209C22×C23—81no—
210C22×C23—81no—
211C22×C23—81no—
212C22×C23—81no—
213C22×C23—81no—
214C22×C23#38611yescharacteristic
215C22×C23#45511yescharacteristic
216C22×C23#45611yescharacteristic
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—21no—
221C4×C23—42no—
222C4×C23—81no—
223C4×C23—81no—
224C4×C23—81no—
225C4×C23—81no—
226C4×C23—82no—
227C4×C23—81no—
228C4×C23—81no—
229C2×C22⋊Q8C22×C2311yescharacteristic
230C2×#121C22×C2311yescharacteristic
231#484—21no—
232#480C4×C2312yescomplemented
233#468—22no—
234#450—21no—
235#448C4×C2312yescomplemented
236#407C22×C2311yescharacteristic
237#407C22×C2311yescharacteristic
238#390—21no—
239#388—21no—
240#387—21no—
241#386—21no—
242#386—21no—
243#385—21no—
244#381C22×C2311yescharacteristic
245#380C22×C2311yescharacteristic
246C4×C22⋊C4C22×C2311yescharacteristic
247#342—21no—
248#342—21no—
249#328—21no—
250#328—21no—
251#327—22no—
252C2×C23—21no—
253C2×C23—21no—
254C2×C23—41no—
255C2×C23—41no—
256C2×C23—82no—
257C2×C23#188211yescharacteristic
258C2×C23#190411yescharacteristic
259C2×C23#191311yescharacteristic
260C22×Q8C2×C4×C2311yescharacteristic, complemented
261C22×D4—21no—
262C22×D4D4×C2311yescharacteristic
263C23×C4—21no—
264C23×C4C23×C2311yescharacteristic
265C2×SD16—22no—
266C2×M16—21no—
267C2×M16—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272#121—21no—
273#121—21no—
274#121C2×C4×C2312yes—
275#121D4×C2312yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C2×C4×C2311yescharacteristic, complemented
280C2×C4⋊C4D4×C2311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4D4×C2311yescharacteristic
289C2×C22⋊C4D4×C2311yescharacteristic
290C2×C42—21no—
291C2×C42—21no—
292#107—22no—
293#103—22no—
294#102—22no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8C2×C4×C2312yescomplemented
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95D4×C2311yescharacteristic
305#95D4×C2311yescharacteristic
306C23#934111yescharacteristic, direct product factor, Sylow
307C24—21no—
308C24C2×D4×C2311yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—81no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4—81no—
318C2×D4C22⋊C4×C2312yes—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C22×C4×C2311yescharacteristic
332C22×C4C2×D4×C2311yescharacteristic
333C22×C4C2×D4×C2311yescharacteristic
334C22×C4C2×D4×C2311yescharacteristic
335C22×C4#41×C2311yescharacteristic
336C22×C4#41×C2311yescharacteristic
337SD16—82no—
338M16—81no—
339M16—81no—
340C2×C8—22no—
341C2×C8—41no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—81no—
348C4⋊C4—81no—
349C4⋊C4C22⋊C4×C2312yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—81no—
359C22⋊C4—81no—
360C22⋊C4—81no—
361C22⋊C4—81no—
362C42—81no—
363C42—81no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C4×D4×C2311yescharacteristic
376C23C23×#12011yescharacteristic
377C23C23×#12111yescharacteristic
378Q8—42no—
379Q8—81no—
380D4—42no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4C2×C22⋊C4×C2311yescharacteristic
414C2×C4C4×D4×C2311yescharacteristic
415C2×C4C23×#12111yescharacteristic
416C2×C4C23×#12311yescharacteristic
417C8—42no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22C23×#38611yescharacteristic
437C22C23×#45511yescharacteristic
438C22C23×#45611yescharacteristic
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—81no—
451C2—21no—
452C2—21no—
453C2—41no—
454C2—41no—
455C2—82no—
456C2C23×#188211yescharacteristic
457C2C23×#190411yescharacteristic
458C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1996C2semidirect
4. C23×#1511C2semidirect
6. C23×#1305C2semidirect
7. C23×#1284C2semidirect
11. C23×#480C4semidirect
14. C23×#448C4semidirect
31. C22×Q8×C23C2×C4semidirect
50. C2×C4⋊C4×C23C2×C4semidirect
69. C23×C22⋊C8C2×C4semidirect
134. #9341C23direct (N × H)
192. #1996C2×C23semidirect
193. #1511C2×C23semidirect
195. #1305C2×C23semidirect
196. #1284C2×C23semidirect
232. #480C4×C23semidirect
235. #448C4×C23semidirect
260. C22×Q8C2×C4×C23semidirect
279. C2×C4⋊C4C2×C4×C23semidirect
298. C22⋊C8C2×C4×C23semidirect
306. C23#9341direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31932
C23×#23772
C23×#19962
C23×#15112
C23×#15072
C23×#13052
C23×#12842
#934123