Finite Group Databasebeta

← C23×#61935 #19209561

Subgroups — C23×#61935 #19209561

Order 5888.

1746 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3548C211yesmaximal, characteristic, complemented
2C23×#3509C211yesmaximal, characteristic, complemented
3C23×#3397C214yesmaximal, complemented
4C23×#3392C214yesmaximal, complemented
5C23×#3372C214yesmaximal, complemented
6C23×#3360C214yesmaximal, complemented
7C23×#3221C214yesmaximal, complemented
8C23×#3206C212yesmaximal, complemented
9C23×#3126C214yesmaximal, complemented
10C23×#3106C212yesmaximal, complemented
11C23×#2986C211yesmaximal, characteristic, complemented
12C23×#562—22no—
13C23×#557C2214yes—
14C23×#556C2214yes—
15C23×#554C2214yes—
16C23×#552—24no—
17C23×#550C2214yes—
18C23×#547—22no—
19C23×#541—22no—
20C23×#540C2214yes—
21C23×#537—24no—
22C23×#536C2212yes—
23C23×#533—22no—
24C23×#533C2211yescharacteristic
25C23×#532C2212yes—
26C2×C23×C4⋊Q8C2211yescharacteristic
27C23×#519C2212yes—
28C4×#41×C23C2211yescharacteristic
29C23×#496C2214yes—
30C23×#492C2214yes—
31C23×#492C2214yes—
32C23×#491C2214yes—
33C23×#488C2214yes—
34C23×#487C2214yes—
35C23×#482C2214yes—
36C23×#479C2218yes—
37C23×#477C2212yes—
38C23×#476C2212yes—
39C23×#463C2218yes—
40C23×#461C2214yes—
41C23×#461C2214yes—
42C23×#461C2218yes—
43C23×#460C2212yes—
44C23×#460C2218yes—
45C23×#459C2212yes—
46C23×#449C2214yes—
47C23×#448C2218yes—
48C23×#446C2214yes—
49C8×Q8×C23C2214yes—
50C23×#440C2218yes—
51C4×SD16×C23C2218yes—
52C4×SD16×C23C2214yes—
53C4×D8×C23C2214yes—
54C23×#433C2214yes—
55C23×#431C2212yes—
56C2×#41×C23—24no—
57C2×#41×C23—22no—
58C2×#41×C23C2311yescharacteristic
59C22×Q8×C23C2312yes—
60C2×SD16×C23C2318yes—
61C2×SD16×C23C2314yes—
62C2×SD16×C23C2318yes—
63C2×D8×C23C2314yes—
64C23×C4⋊Q8—21no—
65C23×C4⋊Q8—21no—
66C23×C4⋊Q8D412yes—
67C23×C4⋊Q8C2314yes—
68C23×C4⋊Q8C2314yes—
69C23×C4⋊Q8C2312yes—
70C23×#127C2312yes—
71C23×#127C2314yes—
72C23×#126—28no—
73C23×#125—24no—
74C23×#125—28no—
75C23×#125D412yes—
76C23×#125C2314yes—
77C23×#124—22no—
78C23×#124C2314yes—
79C23×#123—28no—
80C23×#123—24no—
81C23×C22⋊Q8—24no—
82C23×C22⋊Q8—24no—
83C23×C22⋊Q8—28no—
84C23×C22⋊Q8—24no—
85C23×C22⋊Q8D414yes—
86C23×#121—22no—
87C23×#121—24no—
88C23×#121—28no—
89C23×#121—28no—
90C23×#121C2314yes—
91C4×Q8×C23—24no—
92C4×Q8×C23—24no—
93C4×Q8×C23C2318yes—
94C4×Q8×C23C2312yes—
95C4×Q8×C23C2314yes—
96C4×D4×C23—22no—
97C4×D4×C23—22no—
98C4×D4×C23C2314yes—
99C4×D4×C23C2318yes—
100C4×D4×C23C2312yes—
101C23×#117—24no—
102C23×#117—24no—
103C23×#117—21no—
104C23×#117C2311yescharacteristic
105C23×#117C2312yes—
106C2×C4⋊C4×C23—22no—
107C2×C4⋊C4×C23—24no—
108C2×C42×C23—21no—
109C2×C42×C23C2311yescharacteristic
110C23×#107C2314yes—
111C23×#106C2314yes—
112C23×C4⋊C8C2312yes—
113C23×C4⋊C8C2312yes—
114C23×C4⋊C8C2318yes—
115C23×#103C2318yes—
116C23×#103C2314yes—
117C23×#103C2318yes—
118C23×#102C2314yes—
119C23×#102C2314yes—
120C23×#102C2314yes—
121C23×C22⋊C8C2314yes—
122C23×#97C2314yes—
123C4×C8×C23C2314yes—
124C4×C8×C23C2314yes—
125#41×C23—42no—
126#41×C23—84no—
127#41×C23—88no—
128C2×Q8×C23—22no—
129C2×Q8×C23—22no—
130C2×Q8×C23—44no—
131C2×Q8×C23C2×D414yes—
132C2×Q8×C23C2414yes—
133C2×Q8×C23C2411yescharacteristic
134C2×D4×C23—21no—
135C2×D4×C23—24no—
136C2×D4×C23—22no—
137C2×D4×C23—28no—
138C2×D4×C23—22no—
139C2×D4×C23C2411yescharacteristic
140C2×D4×C23C2414yes—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—21no—
144C22×C4×C23—24no—
145C22×C4×C23—28no—
146C22×C4×C23C2×D412yes—
147C22×C4×C23C2411yescharacteristic
148SD16×C23—48no—
149SD16×C23—48no—
150SD16×C23—44no—
151D8×C23—44no—
152C2×C8×C23C2414yes—
153C2×C8×C23C2414yes—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—24no—
156C4⋊C4×C23—28no—
157C4⋊C4×C23—24no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—28no—
160C4⋊C4×C23—28no—
161C4⋊C4×C23C2×D414yes—
162C4⋊C4×C23C2×D414yes—
163C4⋊C4×C23C2×D418yes—
164C4⋊C4×C23C2412yes—
165C4⋊C4×C23C2414yes—
166C22⋊C4×C23—21no—
167C22⋊C4×C23—22no—
168C22⋊C4×C23—28no—
169C22⋊C4×C23—24no—
170C22⋊C4×C23—28no—
171C22⋊C4×C23—24no—
172C22⋊C4×C23C2×D414yes—
173C42×C23—21no—
174C42×C23—22no—
175C42×C23—28no—
176C42×C23C2×D411yescharacteristic
177C42×C23C2×D411yescharacteristic
178C42×C23C2412yes—
179C42×C23C2414yes—
180#61935C2311yesmaximal, characteristic, direct product factor, Sylow
181C23×C23—21no—
182C23×C23—24no—
183C23×C23C22×D411yescharacteristic
184Q8×C23—28no—
185Q8×C23—28no—
186Q8×C23—22no—
187Q8×C23—44no—
188Q8×C23—82no—
189D4×C23—22no—
190D4×C23—24no—
191D4×C23—24no—
192D4×C23—42no—
193D4×C23—44no—
194D4×C23—82no—
195D4×C23—88no—
196D4×C23—84no—
197C2×C4×C23—24no—
198C2×C4×C23—22no—
199C2×C4×C23—22no—
200C2×C4×C23—24no—
201C2×C4×C23—21no—
202C2×C4×C23—24no—
203C2×C4×C23—42no—
204C2×C4×C23—44no—
205C2×C4×C23—44no—
206C2×C4×C23—48no—
207C2×C4×C23—88no—
208C2×C4×C23C22×D411yescharacteristic
209C2×C4×C23C22×D418yes—
210C2×C4×C23C22×D412yes—
211C2×C4×C23C22×D412yes—
212C2×C4×C23C2511yescharacteristic
213C8×C23—24no—
214C8×C23—24no—
215#3548C2×C2311yescharacteristic, complemented
216#3509C2×C2311yescharacteristic, complemented
217#3397C2×C2314yescomplemented
218#3392C2×C2314yescomplemented
219#3372C2×C2314yescomplemented
220#3360C2×C2314yescomplemented
221#3221C2×C2314yescomplemented
222#3206C2×C2312yescomplemented
223#3126C2×C2314yescomplemented
224#3106C2×C2312yescomplemented
225#2986C2×C2311yescharacteristic, complemented
226C22×C23—21no—
227C22×C23—22no—
228C22×C23—41no—
229C22×C23—44no—
230C22×C23—44no—
231C22×C23—42no—
232C22×C23—44no—
233C22×C23C23×D411yescharacteristic
234C4×C23—21no—
235C4×C23—22no—
236C4×C23—28no—
237C4×C23—41no—
238C4×C23—44no—
239C4×C23—42no—
240C4×C23—44no—
241C4×C23#57714yes—
242C4×C23#58512yes—
243#562—22no—
244#557C22×C2314yes—
245#556C22×C2314yes—
246#554C22×C2314yes—
247#552—24no—
248#550C22×C2314yes—
249#547—22no—
250#541—22no—
251#540C22×C2314yes—
252#537—24no—
253#536C22×C2312yes—
254#533—22no—
255#533C22×C2311yescharacteristic
256#532C22×C2312yes—
257C2×C4⋊Q8C22×C2311yescharacteristic
258#519C22×C2312yes—
259C4×#41C22×C2311yescharacteristic
260#496C22×C2314yes—
261#492C22×C2314yes—
262#492C22×C2314yes—
263#491C22×C2314yes—
264#488C22×C2314yes—
265#487C22×C2314yes—
266#482C22×C2314yes—
267#479C22×C2318yes—
268#477C22×C2312yes—
269#476C22×C2312yes—
270#463C22×C2318yes—
271#461C22×C2314yes—
272#461C22×C2318yes—
273#461C22×C2314yes—
274#460C22×C2312yes—
275#460C22×C2318yes—
276#459C22×C2312yes—
277#449C22×C2314yes—
278#448C22×C2318yes—
279#446C22×C2314yes—
280C8×Q8C22×C2314yes—
281#440C22×C2318yes—
282C4×SD16C22×C2318yes—
283C4×SD16C22×C2314yes—
284C4×D8C22×C2314yes—
285#433C22×C2314yes—
286#431C22×C2312yes—
287C2×C23—41no—
288C2×C23—81no—
289C2×C23—84no—
290C2×C23#348711yescharacteristic
291C2×C23C2×#57712yes—
292C2×#41—24no—
293C2×#41—22no—
294C2×#41C23×C2311yescharacteristic
295C22×Q8C23×C2312yes—
296C2×SD16C23×C2318yes—
297C2×SD16C23×C2314yes—
298C2×SD16C23×C2318yes—
299C2×D8C23×C2314yes—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8D4×C2312yes—
303C4⋊Q8C23×C2314yes—
304C4⋊Q8C23×C2314yes—
305C4⋊Q8C23×C2312yes—
306#127C23×C2312yes—
307#127C23×C2314yes—
308#126—28no—
309#125—24no—
310#125—28no—
311#125D4×C2312yes—
312#125C23×C2314yes—
313#124—22no—
314#124C23×C2314yes—
315#123—28no—
316#123—24no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—24no—
320C22⋊Q8—28no—
321C22⋊Q8D4×C2314yes—
322#121—22no—
323#121—24no—
324#121—28no—
325#121—28no—
326#121C23×C2314yes—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8C23×C2312yes—
330C4×Q8C23×C2318yes—
331C4×Q8C23×C2314yes—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4C23×C2312yes—
335C4×D4C23×C2314yes—
336C4×D4C23×C2318yes—
337#117—24no—
338#117—24no—
339#117—21no—
340#117C23×C2311yescharacteristic
341#117C23×C2312yes—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C42—21no—
345C2×C42C23×C2311yescharacteristic
346#107C23×C2314yes—
347#106C23×C2314yes—
348C4⋊C8C23×C2312yes—
349C4⋊C8C23×C2312yes—
350C4⋊C8C23×C2318yes—
351#103C23×C2318yes—
352#103C23×C2314yes—
353#103C23×C2318yes—
354#102C23×C2314yes—
355#102C23×C2314yes—
356#102C23×C2314yes—
357C22⋊C8C23×C2314yes—
358#97C23×C2314yes—
359C4×C8C23×C2314yes—
360C4×C8C23×C2314yes—
361C23#6193511yescharacteristic, direct product factor, Sylow
362#41—42no—
363#41—84no—
364#41—88no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—44no—
368C2×Q8C2×D4×C2314yes—
369C2×Q8C24×C2314yes—
370C2×Q8C24×C2311yescharacteristic
371C2×D4—21no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—28no—
375C2×D4—22no—
376C2×D4C24×C2311yescharacteristic
377C2×D4C24×C2314yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4C2×D4×C2312yes—
384C22×C4C24×C2311yescharacteristic
385SD16—48no—
386SD16—48no—
387SD16—44no—
388D8—44no—
389C2×C8C24×C2314yes—
390C2×C8C24×C2314yes—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4C2×D4×C2314yes—
399C4⋊C4C2×D4×C2314yes—
400C4⋊C4C2×D4×C2318yes—
401C4⋊C4C24×C2312yes—
402C4⋊C4C24×C2314yes—
403C22⋊C4—21no—
404C22⋊C4—22no—
405C22⋊C4—28no—
406C22⋊C4—24no—
407C22⋊C4—28no—
408C22⋊C4—24no—
409C22⋊C4C2×D4×C2314yes—
410C42—21no—
411C42—22no—
412C42—28no—
413C42C2×D4×C2311yescharacteristic
414C42C2×D4×C2311yescharacteristic
415C42C24×C2312yes—
416C42C24×C2314yes—
417C23—21no—
418C23—24no—
419C23C22×D4×C2311yescharacteristic
420Q8—28no—
421Q8—28no—
422Q8—22no—
423Q8—44no—
424Q8—82no—
425D4—22no—
426D4—24no—
427D4—24no—
428D4—42no—
429D4—44no—
430D4—82no—
431D4—88no—
432D4—84no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—88no—
444C2×C4C22×D4×C2311yescharacteristic
445C2×C4C22×D4×C2318yes—
446C2×C4C22×D4×C2312yes—
447C2×C4C22×D4×C2312yes—
448C2×C4C25×C2311yescharacteristic
449C8—24no—
450C8—24no—
451C22—21no—
452C22—22no—
453C22—41no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—44no—
458C22C23×D4×C2311yescharacteristic
459C4—21no—
460C4—22no—
461C4—28no—
462C4—41no—
463C4—44no—
464C4—44no—
465C4—42no—
466C4C23×#57714yes—
467C4C23×#58512yes—
468C2—41no—
469C2—81no—
470C2—84no—
471C2C23×#348711yescharacteristic
472C2C2×C23×#57712yes—
subgroupG/H
center, socleC22×C23C23×D4
commutator, Frattini, cosocleC2×C4C25×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×#533 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3548C2semidirect
2. C23×#3509C2semidirect
3. C23×#3397C2semidirect
4. C23×#3392C2semidirect
5. C23×#3372C2semidirect
6. C23×#3360C2semidirect
7. C23×#3221C2semidirect
8. C23×#3206C2semidirect
9. C23×#3126C2semidirect
10. C23×#3106C2semidirect
11. C23×#2986C2semidirect
180. #61935C23direct (N × H)
215. #3548C2×C23semidirect
216. #3509C2×C23semidirect
217. #3397C2×C23semidirect
218. #3392C2×C23semidirect
219. #3372C2×C23semidirect
220. #3360C2×C23semidirect
221. #3221C2×C23semidirect
222. #3206C2×C23semidirect
223. #3126C2×C23semidirect
224. #3106C2×C23semidirect
225. #2986C2×C23semidirect
361. C23#61935direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35482
C23×#35092
C23×#33972
C23×#33922
C23×#33722
C23×#33602
C23×#32212
C23×#32062
C23×#31262
C23×#31062
C23×#29862
#6193523