Finite Group Databasebeta

← C23×#9332 #19156958

Subgroups — C23×#9332 #19156958

Order 5888.

464 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3194C211yesmaximal, characteristic
2C23×#2978C211yesmaximal, characteristic, complemented
3C23×#1516C211yesmaximal, characteristic, complemented
4C23×#1510C211yesmaximal, characteristic, complemented
5C23×#1471C211yesmaximal, characteristic, complemented
6C23×#1306C211yesmaximal, characteristic, complemented
7C23×#1293C211yesmaximal, characteristic, complemented
8C23×#525C2211yescharacteristic
9C2×C23×#121C2211yescharacteristic
10C23×#483C411yescharacteristic, complemented
11C23×#482—21no—
12C23×#480C411yescharacteristic, complemented
13C23×#469—21no—
14C23×#468—21no—
15C23×#450C411yescharacteristic, complemented
16C23×#449—21no—
17C23×#447C411yescharacteristic, complemented
18C23×#436—21no—
19C23×#436—21no—
20C23×#435—21no—
21C23×#435—21no—
22C23×#433—21no—
23C23×#408—21no—
24C23×#407C2211yescharacteristic
25C23×#407C2211yescharacteristic
26C23×#407C2211yescharacteristic
27C2×C23×C22⋊C8C2211yescharacteristic
28C23×#381C2211yescharacteristic
29C23×#328—21no—
30C23×#327—21no—
31C23×#327—21no—
32C23×#327—21no—
33C23×#323—21no—
34C23×#323—21no—
35C2×#41×C23D411yescharacteristic
36C22×D4×C23—21no—
37C22×D4×C23C2×C411yescharacteristic
38C23×C4×C23C2311yescharacteristic
39C2×SD16×C23—21no—
40C2×D8×C23—21no—
41C2×M16×C23—21no—
42C2×M16×C23—21no—
43C2×M16×C23—21no—
44C2×M16×C23—21no—
45C22×C8×C23—21no—
46C22×C8×C23—21no—
47C23×#123—21no—
48C23×C22⋊Q8C2×C411yescharacteristic
49C23×C22⋊Q8D411yescharacteristic
50C23×#121—21no—
51C23×#121—21no—
52C23×#121—21no—
53C23×#121—21no—
54C23×#121C2×C411yescharacteristic
55C23×#121D411yescharacteristic
56C23×#120—21no—
57C4×D4×C23—21no—
58C4×D4×C23—21no—
59C23×#117D411yescharacteristic
60C2×C4⋊C4×C23C2×C411yescharacteristic
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C23×#107—21no—
64C23×#106—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×#103—21no—
68C23×#102—21no—
69C23×#102—21no—
70C23×#102—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8C2×C411yescharacteristic
79C23×C22⋊C8C2×C411yescharacteristic
80C23×C22⋊C8D411yescharacteristic
81C23×C22⋊C8D411yescharacteristic
82C23×C22⋊C8D411yescharacteristic
83C23×C22⋊C8D411yescharacteristic
84C23×#97—21no—
85C4×C8×C23—21no—
86C23×#95—21no—
87C23×#95—21no—
88C24×C23—21no—
89C24×C23#4111yescharacteristic
90#41×C23—81no—
91C2×Q8×C23C22⋊C411yescharacteristic
92C2×D4×C23—21no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—21no—
97C2×D4×C23—41no—
98C2×D4×C23—41no—
99C2×D4×C23—81no—
100C2×D4×C23—82no—
101C2×D4×C23C22⋊C411yescharacteristic
102C22×C4×C23—21no—
103C22×C4×C23—21no—
104C22×C4×C23—41no—
105C22×C4×C23—81no—
106C22×C4×C23C22×C411yescharacteristic
107C22×C4×C23C2×D411yescharacteristic
108C22×C4×C23C2×D411yescharacteristic
109C22×C4×C23C2×D411yescharacteristic
110C22×C4×C23C2×D411yescharacteristic
111C22×C4×C23#4111yescharacteristic
112SD16×C23—81no—
113D8×C23—81no—
114M16×C23—81no—
115M16×C23—81no—
116M16×C23—81no—
117M16×C23—81no—
118C2×C8×C23—21no—
119C2×C8×C23—21no—
120C2×C8×C23—21no—
121C2×C8×C23—21no—
122C2×C8×C23—21no—
123C2×C8×C23—21no—
124C2×C8×C23—41no—
125C2×C8×C23—81no—
126C2×C8×C23—81no—
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—41no—
139C22⋊C4×C23—41no—
140C22⋊C4×C23—81no—
141C42×C23—21no—
142#9332C2311yesmaximal, characteristic, direct product factor, Sylow
143C23×C23—21no—
144C23×C23—41no—
145C23×C23—41no—
146C23×C23—41no—
147C23×C23—42no—
148C23×C23—41no—
149C23×C23—81no—
150C23×C23—81no—
151C23×C23C4×D411yescharacteristic
152C23×C23#12111yescharacteristic
153C23×C23#12311yescharacteristic
154Q8×C23—41no—
155D4×C23—41no—
156D4×C23—41no—
157D4×C23—41no—
158D4×C23—81no—
159D4×C23—81no—
160D4×C23—81no—
161D4×C23—81no—
162D4×C23—82no—
163D4×C23—82no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
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—81no—
177C2×C4×C23—81no—
178C2×C4×C23—81no—
179C2×C4×C23—81no—
180C2×C4×C23C2×C22⋊C411yescharacteristic
181C2×C4×C23C4×D411yescharacteristic
182C2×C4×C23#12011yescharacteristic
183C2×C4×C23#12111yescharacteristic
184C8×C23—41no—
185C8×C23—41no—
186C8×C23—41no—
187C8×C23—41no—
188C8×C23—41no—
189C8×C23—41no—
190C8×C23—81no—
191#3194C2×C2311yescharacteristic
192#2978C2×C2311yescharacteristic, complemented
193#1516C2×C2311yescharacteristic, complemented
194#1510C2×C2311yescharacteristic, complemented
195#1471C2×C2311yescharacteristic, complemented
196#1306C2×C2311yescharacteristic, complemented
197#1293C2×C2311yescharacteristic, complemented
198C22×C23—21no—
199C22×C23—21no—
200C22×C23—21no—
201C22×C23—21no—
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—21no—
207C22×C23—41no—
208C22×C23—41no—
209C22×C23—41no—
210C22×C23—82no—
211C22×C23—81no—
212C22×C23—81no—
213C22×C23—81no—
214C22×C23—162no—
215C22×C23#38611yescharacteristic
216C4×C23—21no—
217C4×C23—21no—
218C4×C23—21no—
219C4×C23—21no—
220C4×C23—41no—
221C4×C23—41no—
222C4×C23—41no—
223C4×C23—81no—
224C4×C23—161no—
225#525C22×C2311yescharacteristic
226C2×#121C22×C2311yescharacteristic
227#483C4×C2311yescharacteristic, complemented
228#482—21no—
229#480C4×C2311yescharacteristic, complemented
230#469—21no—
231#468—21no—
232#450C4×C2311yescharacteristic, complemented
233#449—21no—
234#447C4×C2311yescharacteristic, complemented
235#436—21no—
236#436—21no—
237#435—21no—
238#435—21no—
239#433—21no—
240#408—21no—
241#407C22×C2311yescharacteristic
242#407C22×C2311yescharacteristic
243#407C22×C2311yescharacteristic
244C2×C22⋊C8C22×C2311yescharacteristic
245#381C22×C2311yescharacteristic
246#328—21no—
247#327—21no—
248#327—21no—
249#327—21no—
250#323—21no—
251#323—21no—
252C2×C23—41no—
253C2×C23—41no—
254C2×C23—41no—
255C2×C23—81no—
256C2×C23—161no—
257C2×C23#188111yescharacteristic
258C2×C23#190311yescharacteristic
259C2×C23#191311yescharacteristic
260C2×#41D4×C2311yescharacteristic
261C22×D4—21no—
262C22×D4C2×C4×C2311yescharacteristic
263C23×C4C23×C2311yescharacteristic
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16—21no—
267C2×M16—21no—
268C2×M16—21no—
269C2×M16—21no—
270C22×C8—21no—
271C22×C8—21no—
272#123—21no—
273C22⋊Q8C2×C4×C2311yescharacteristic
274C22⋊Q8D4×C2311yescharacteristic
275#121—21no—
276#121—21no—
277#121—21no—
278#121—21no—
279#121C2×C4×C2311yescharacteristic
280#121D4×C2311yescharacteristic
281#120—21no—
282C4×D4—21no—
283C4×D4—21no—
284#117D4×C2311yescharacteristic
285C2×C4⋊C4C2×C4×C2311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288#107—21no—
289#106—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8C2×C4×C2311yescharacteristic
304C22⋊C8C2×C4×C2311yescharacteristic
305C22⋊C8D4×C2311yescharacteristic
306C22⋊C8D4×C2311yescharacteristic
307C22⋊C8D4×C2311yescharacteristic
308C22⋊C8D4×C2311yescharacteristic
309#97—21no—
310C4×C8—21no—
311#95—21no—
312#95—21no—
313C23#933211yescharacteristic, direct product factor, Sylow
314C24—21no—
315C24#41×C2311yescharacteristic
316#41—81no—
317C2×Q8C22⋊C4×C2311yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—81no—
326C2×D4—82no—
327C2×D4C22⋊C4×C2311yescharacteristic
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—81no—
332C22×C4C22×C4×C2311yescharacteristic
333C22×C4C2×D4×C2311yescharacteristic
334C22×C4C2×D4×C2311yescharacteristic
335C22×C4C2×D4×C2311yescharacteristic
336C22×C4C2×D4×C2311yescharacteristic
337C22×C4#41×C2311yescharacteristic
338SD16—81no—
339D8—81no—
340M16—81no—
341M16—81no—
342M16—81no—
343M16—81no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—41no—
351C2×C8—81no—
352C2×C8—81no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4C22⋊C4×C2311yescharacteristic
358C4⋊C4C22⋊C4×C2311yescharacteristic
359C22⋊C4—21no—
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—81no—
367C42—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—81no—
375C23—81no—
376C23C4×D4×C2311yescharacteristic
377C23C23×#12111yescharacteristic
378C23C23×#12311yescharacteristic
379Q8—41no—
380D4—41no—
381D4—41no—
382D4—41no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—82no—
388D4—82no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊C4×C2311yescharacteristic
406C2×C4C4×D4×C2311yescharacteristic
407C2×C4C23×#12011yescharacteristic
408C2×C4C23×#12111yescharacteristic
409C8—41no—
410C8—41no—
411C8—41no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—81no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—82no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—162no—
433C22C23×#38611yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4—161no—
443C2—41no—
444C2—41no—
445C2—41no—
446C2—81no—
447C2—161no—
448C2C23×#188111yescharacteristic
449C2C23×#190311yescharacteristic
450C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3194C2
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
2. C23×#2978C2semidirect
3. C23×#1516C2semidirect
4. C23×#1510C2semidirect
5. C23×#1471C2semidirect
6. C23×#1306C2semidirect
7. C23×#1293C2semidirect
10. C23×#483C4semidirect
12. C23×#480C4semidirect
15. C23×#450C4semidirect
17. C23×#447C4semidirect
142. #9332C23direct (N × H)
192. #2978C2×C23semidirect
193. #1516C2×C23semidirect
194. #1510C2×C23semidirect
195. #1471C2×C23semidirect
196. #1306C2×C23semidirect
197. #1293C2×C23semidirect
227. #483C4×C23semidirect
229. #480C4×C23semidirect
232. #450C4×C23semidirect
234. #447C4×C23semidirect
313. C23#9332direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31942
C23×#29782
C23×#15162
C23×#15102
C23×#14712
C23×#13062
C23×#12932
#933223