Finite Group Databasebeta

← C23×#18392 #19166018

Subgroups — C23×#18392 #19166018

Order 5888.

934 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3507C211yesmaximal, characteristic
2C23×#3209C211yesmaximal, characteristic, complemented
3C23×#3206C211yesmaximal, characteristic
4C23×#2567C211yesmaximal, characteristic, complemented
5C23×#1603C218yesmaximal, complemented
6C23×#1503C211yesmaximal, characteristic, complemented
7C23×#1502C211yesmaximal, characteristic
8C23×#1461C211yesmaximal, characteristic, complemented
9C23×#559—21no—
10C23×#557—21no—
11C23×#556—21no—
12C23×#556—22no—
13C23×#554—21no—
14C23×#552—22no—
15C23×#539C2214yes—
16C23×#536—21no—
17C23×#533C2211yescharacteristic
18C23×#532C2211yescharacteristic
19C2×C23×C4⋊Q8C2211yescharacteristic
20C23×#529—22no—
21C23×#520—21no—
22C2×C4×Q8×C23C2211yescharacteristic
23C23×#494—21no—
24C23×#492—21no—
25C23×#487—21no—
26C23×#487—21no—
27C23×#462—22no—
28C23×#461—22no—
29C23×#451C2214yes—
30C23×#448C2214yes—
31C23×#431C2211yescharacteristic
32C23×#431C2211yescharacteristic
33C23×#395C2214yes—
34C23×#391—21no—
35C23×#389—22no—
36C23×#384—21no—
37C4×C4⋊C4×C23C2211yescharacteristic
38C23×#328C2214yes—
39C23×#327C2214yes—
40C23×#324C2214yes—
41C2×#41×C23—21no—
42C22×Q8×C23D411yescharacteristic
43C22×Q8×C23C2312yes—
44C2×Q16×C23—24no—
45C2×SD16×C23—24no—
46C23×C4⋊Q8—22no—
47C23×C4⋊Q8—21no—
48C23×C4⋊Q8—21no—
49C23×C4⋊Q8—21no—
50C23×C4⋊Q8—21no—
51C23×C4⋊Q8D411yescharacteristic
52C23×C4⋊Q8D411yescharacteristic
53C23×C4⋊Q8D411yescharacteristic
54C23×#127D411yescharacteristic
55C23×#126—24no—
56C23×#126—24no—
57C23×#125—22no—
58C23×#125—24no—
59C23×#125—21no—
60C23×#124—22no—
61C23×#124—22no—
62C23×#124—21no—
63C23×#123—24no—
64C23×#123—24no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—24no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8C2312yes—
70C23×#121—21no—
71C23×#121C2312yes—
72C4×Q8×C23—22no—
73C4×Q8×C23—22no—
74C4×Q8×C23—21no—
75C4×Q8×C23—21no—
76C4×Q8×C23—21no—
77C4×Q8×C23—21no—
78C4×Q8×C23—21no—
79C4×Q8×C23D412yes—
80C4×D4×C23—22no—
81C4×D4×C23—22no—
82C4×D4×C23—24no—
83C4×D4×C23—21no—
84C4×D4×C23D412yes—
85C23×#117—22no—
86C23×#117—24no—
87C23×#117—21no—
88C2×C4⋊C4×C23—22no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23—21no—
91C2×C4⋊C4×C23—21no—
92C2×C4⋊C4×C23—24no—
93C2×C4⋊C4×C23D411yescharacteristic
94C2×C4⋊C4×C23C2312yes—
95C2×C42×C23—21no—
96C2×C42×C23D412yes—
97C2×C42×C23C2311yescharacteristic
98C23×C4⋊C8—21no—
99C23×C4⋊C8—21no—
100C23×#103—24no—
101C23×#103—24no—
102C23×C22⋊C8C2312yes—
103C23×C22⋊C8C2312yes—
104C4×C8×C23—21no—
105C4×C8×C23—21no—
106C23×#95—24no—
107C23×#95C2312yes—
108#41×C23—42no—
109#41×C23—42no—
110C2×Q8×C23—21no—
111C2×Q8×C23—21no—
112C2×Q8×C23—21no—
113C2×Q8×C23—21no—
114C2×Q8×C23—22no—
115C2×Q8×C23—22no—
116C2×Q8×C23—22no—
117C2×Q8×C23—22no—
118C2×Q8×C23—82no—
119C2×Q8×C23C2×D412yes—
120C2×D4×C23—22no—
121C2×D4×C23—21no—
122C2×D4×C23—21no—
123C2×D4×C23C2×D412yes—
124C22×C4×C23—22no—
125C22×C4×C23—24no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23C2×D412yes—
129C22×C4×C23C2×D411yescharacteristic
130C22×C4×C23C2×D412yes—
131C22×C4×C23C2×D411yescharacteristic
132C22×C4×C23C2411yescharacteristic
133Q16×C23—84no—
134SD16×C23—84no—
135C2×C8×C23—22no—
136C2×C8×C23—22no—
137C4⋊C4×C23—24no—
138C4⋊C4×C23—24no—
139C4⋊C4×C23—24no—
140C4⋊C4×C23—24no—
141C4⋊C4×C23—21no—
142C4⋊C4×C23—21no—
143C4⋊C4×C23—21no—
144C4⋊C4×C23—21no—
145C4⋊C4×C23—22no—
146C4⋊C4×C23—21no—
147C4⋊C4×C23—21no—
148C4⋊C4×C23—22no—
149C4⋊C4×C23—22no—
150C4⋊C4×C23—22no—
151C4⋊C4×C23—41no—
152C4⋊C4×C23—41no—
153C4⋊C4×C23—41no—
154C4⋊C4×C23—41no—
155C4⋊C4×C23—84no—
156C4⋊C4×C23C2×D412yes—
157C4⋊C4×C23C2×D412yes—
158C22⋊C4×C23—24no—
159C22⋊C4×C23—24no—
160C22⋊C4×C23—24no—
161C22⋊C4×C23—22no—
162C22⋊C4×C23—22no—
163C22⋊C4×C23—22no—
164C42×C23—24no—
165C42×C23—22no—
166C42×C23—22no—
167C42×C23—22no—
168C42×C23—42no—
169C42×C23C2×D411yescharacteristic
170C42×C23C2×D411yescharacteristic
171C42×C23C2×D411yescharacteristic
172C42×C23C2×D411yescharacteristic
173#18392C2311yesmaximal, characteristic, direct product factor, Sylow
174C23×C23—22no—
175C23×C23C22×D411yescharacteristic
176Q8×C23—42no—
177Q8×C23—44no—
178Q8×C23—44no—
179Q8×C23—42no—
180Q8×C23—42no—
181Q8×C23—42no—
182Q8×C23—42no—
183D4×C23—42no—
184D4×C23—42no—
185D4×C23—42no—
186D4×C23—42no—
187D4×C23—42no—
188C2×C4×C23—22no—
189C2×C4×C23—24no—
190C2×C4×C23—22no—
191C2×C4×C23—22no—
192C2×C4×C23—22no—
193C2×C4×C23—22no—
194C2×C4×C23—22no—
195C2×C4×C23—22no—
196C2×C4×C23—22no—
197C2×C4×C23—41no—
198C2×C4×C23—42no—
199C2×C4×C23—42no—
200C2×C4×C23—44no—
201C2×C4×C23—42no—
202C2×C4×C23—44no—
203C2×C4×C23—42no—
204C2×C4×C23—84no—
205C2×C4×C23#12011yescharacteristic
206C2×C4×C23#12011yescharacteristic
207C2×C4×C23#12011yescharacteristic
208C2×C4×C23#12011yescharacteristic
209C2×C4×C23#13512yes—
210C2×C4×C23#13712yes—
211C2×C4×C23C22×D411yescharacteristic
212C2×C4×C23C22×D411yescharacteristic
213C8×C23—42no—
214C8×C23—42no—
215#3507C2×C2311yescharacteristic
216#3209C2×C2311yescharacteristic, complemented
217#3206C2×C2311yescharacteristic
218#2567C2×C2311yescharacteristic, complemented
219#1603C2×C2318yescomplemented
220#1503C2×C2311yescharacteristic, complemented
221#1502C2×C2311yescharacteristic
222#1461C2×C2311yescharacteristic, complemented
223C22×C23—21no—
224C22×C23—21no—
225C22×C23—42no—
226C22×C23—42no—
227C22×C23—42no—
228C22×C23C2×#12011yescharacteristic
229C22×C23C2×#13511yescharacteristic
230C22×C23C2×#13711yescharacteristic
231C4×C23—21no—
232C4×C23—21no—
233C4×C23—21no—
234C4×C23—44no—
235C4×C23—42no—
236C4×C23—42no—
237C4×C23—42no—
238C4×C23—42no—
239C4×C23—42no—
240C4×C23—82no—
241C4×C23#45212yes—
242C4×C23#45212yes—
243C4×C23#45612yes—
244#559—21no—
245#557—21no—
246#556—21no—
247#556—22no—
248#554—21no—
249#552—22no—
250#539C22×C2314yes—
251#536—21no—
252#533C22×C2311yescharacteristic
253#532C22×C2311yescharacteristic
254C2×C4⋊Q8C22×C2311yescharacteristic
255#529—22no—
256#520—21no—
257C2×C4×Q8C22×C2311yescharacteristic
258#494—21no—
259#492—21no—
260#487—21no—
261#487—21no—
262#462—22no—
263#461—22no—
264#451C22×C2314yes—
265#448C22×C2314yes—
266#431C22×C2311yescharacteristic
267#431C22×C2311yescharacteristic
268#395C22×C2314yes—
269#391—21no—
270#389—22no—
271#384—21no—
272C4×C4⋊C4C22×C2311yescharacteristic
273#328C22×C2314yes—
274#327C22×C2314yes—
275#324C22×C2314yes—
276C2×C23—21no—
277C2×C23—21no—
278C2×C23—82no—
279C2×C23C2×#45211yescharacteristic
280C2×C23C2×#45611yescharacteristic
281C2×C23#302711yescharacteristic
282C2×#41—21no—
283C22×Q8D4×C2311yescharacteristic
284C22×Q8C23×C2312yes—
285C2×Q16—24no—
286C2×SD16—24no—
287C4⋊Q8—22no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8D4×C2311yescharacteristic
293C4⋊Q8D4×C2311yescharacteristic
294C4⋊Q8D4×C2311yescharacteristic
295#127D4×C2311yescharacteristic
296#126—24no—
297#126—24no—
298#125—22no—
299#125—24no—
300#125—21no—
301#124—22no—
302#124—22no—
303#124—21no—
304#123—24no—
305#123—24no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—24no—
309C22⋊Q8—21no—
310C22⋊Q8C23×C2312yes—
311#121—21no—
312#121C23×C2312yes—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8D4×C2312yes—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—24no—
324C4×D4—21no—
325C4×D4D4×C2312yes—
326#117—22no—
327#117—24no—
328#117—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4D4×C2311yescharacteristic
335C2×C4⋊C4C23×C2312yes—
336C2×C42—21no—
337C2×C42D4×C2312yes—
338C2×C42C23×C2311yescharacteristic
339C4⋊C8—21no—
340C4⋊C8—21no—
341#103—24no—
342#103—24no—
343C22⋊C8C23×C2312yes—
344C22⋊C8C23×C2312yes—
345C4×C8—21no—
346C4×C8—21no—
347#95—24no—
348#95C23×C2312yes—
349C23#1839211yescharacteristic, direct product factor, Sylow
350#41—42no—
351#41—42no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—82no—
361C2×Q8C2×D4×C2312yes—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4C2×D4×C2312yes—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4C2×D4×C2312yes—
371C22×C4C2×D4×C2311yescharacteristic
372C22×C4C2×D4×C2312yes—
373C22×C4C2×D4×C2311yescharacteristic
374C22×C4C24×C2311yescharacteristic
375Q16—84no—
376SD16—84no—
377C2×C8—22no—
378C2×C8—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—84no—
398C4⋊C4C2×D4×C2312yes—
399C4⋊C4C2×D4×C2312yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C42—24no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—42no—
411C42C2×D4×C2311yescharacteristic
412C42C2×D4×C2311yescharacteristic
413C42C2×D4×C2311yescharacteristic
414C42C2×D4×C2311yescharacteristic
415C23—22no—
416C23C22×D4×C2311yescharacteristic
417Q8—42no—
418Q8—44no—
419Q8—44no—
420Q8—42no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—42no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—84no—
446C2×C4C23×#12011yescharacteristic
447C2×C4C23×#12011yescharacteristic
448C2×C4C23×#12011yescharacteristic
449C2×C4C23×#12011yescharacteristic
450C2×C4C23×#13512yes—
451C2×C4C23×#13712yes—
452C2×C4C22×D4×C2311yescharacteristic
453C2×C4C22×D4×C2311yescharacteristic
454C8—42no—
455C8—42no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22C2×C23×#12011yescharacteristic
462C22C2×C23×#13511yescharacteristic
463C22C2×C23×#13711yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—82no—
474C4C23×#45212yes—
475C4C23×#45212yes—
476C4C23×#45612yes—
477C2—21no—
478C2—21no—
479C2—82no—
480C2C2×C23×#45211yescharacteristic
481C2C2×C23×#45611yescharacteristic
482C2C23×#302711yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3209C2semidirect
4. C23×#2567C2semidirect
5. C23×#1603C2semidirect
6. C23×#1503C2semidirect
8. C23×#1461C2semidirect
173. #18392C23direct (N × H)
216. #3209C2×C23semidirect
218. #2567C2×C23semidirect
219. #1603C2×C23semidirect
220. #1503C2×C23semidirect
222. #1461C2×C23semidirect
349. C23#18392direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35072
C23×#32092
C23×#32062
C23×#25672
C23×#16032
C23×#15032
C23×#15022
C23×#14612
#1839223