Finite Group Databasebeta

← C23×#33911 #19181537

Subgroups — C23×#33911 #19181537

Order 5888.

472 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3338C211yesmaximal, characteristic
2C23×#3336C211yesmaximal, characteristic
3C23×#2255C211yesmaximal, characteristic, complemented
4C23×#2237C211yesmaximal, characteristic, complemented
5C23×#2236C211yesmaximal, characteristic, complemented
6C23×#2235C211yesmaximal, characteristic, complemented
7C23×#2235C211yesmaximal, characteristic, complemented
8C23×#2229C211yesmaximal, characteristic, complemented
9C23×#2225C211yesmaximal, characteristic, complemented
10C23×#2219C211yesmaximal, characteristic, complemented
11C23×#2216C211yesmaximal, characteristic, complemented
12C23×#2176C211yesmaximal, characteristic, complemented
13C23×#2168C211yesmaximal, characteristic
14C23×#2159C211yesmaximal, characteristic, complemented
15C2×C23×#367C211yesmaximal, characteristic, complemented
16C23×#554—21no—
17C23×#552—21no—
18C2×C23×Q32C2211yescharacteristic
19C2×C23×M32C2211yescharacteristic
20C22×C16×C23C2211yescharacteristic
21C23×#500C2211yescharacteristic
22C23×#499C2211yescharacteristic
23C23×#485—21no—
24C23×#484—21no—
25C23×#484—21no—
26C23×#483—21no—
27C23×#479—21no—
28C23×#477—21no—
29C23×#467C2211yescharacteristic
30C23×#467C2211yescharacteristic
31C23×#467C2211yescharacteristic
32C23×#467C2211yescharacteristic
33C4×Q16×C23C2211yescharacteristic
34C4×Q16×C23C2211yescharacteristic
35C8×D4×C23C2211yescharacteristic
36C23×#427C2211yescharacteristic
37C2×C23×#107C2211yescharacteristic
38C2×C23×#107C2211yescharacteristic
39C2×C23×#107C2211yescharacteristic
40C23×#367C2211yescharacteristic
41C23×#367C2211yescharacteristic
42C23×#367C2211yescharacteristic
43C23×#367C2211yescharacteristic
44C23×#367C2211yescharacteristic
45C23×#367C2211yescharacteristic
46C23×#366C2211yescharacteristic
47C23×#366C2211yescharacteristic
48C23×#366C2211yescharacteristic
49C23×C4⋊C16C2211yescharacteristic
50C23×#358C2211yescharacteristic
51C23×#358C2211yescharacteristic
52C23×#358C2211yescharacteristic
53C23×#358C2211yescharacteristic
54C23×#358C2211yescharacteristic
55C23×#358C2211yescharacteristic
56C23×C22⋊C16C2211yescharacteristic
57C23×C22⋊C16C2211yescharacteristic
58C23×#346C2211yescharacteristic
59C2×Q16×C23C2311yescharacteristic
60C2×Q16×C23C2311yescharacteristic
61C22×C8×C23C2311yescharacteristic
62C22×C8×C23C2311yescharacteristic
63C23×C4⋊Q8—21no—
64C23×#126—41no—
65C23×#125—21no—
66C23×#125—41no—
67C23×#125—41no—
68C23×#123—41no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8—21no—
71C23×C22⋊Q8—21no—
72C23×C22⋊Q8—21no—
73C23×C22⋊Q8—41no—
74C23×C22⋊Q8—41no—
75C4×Q8×C23—21no—
76C4×Q8×C23—21no—
77C4×D4×C23—41no—
78C4×D4×C23D411yescharacteristic
79C23×#117—21no—
80C23×#117—41no—
81C2×C4⋊C4×C23—21no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—21no—
84C23×Q32—22no—
85C23×M32—22no—
86C2×C16×C23—22no—
87C2×C16×C23C2311yescharacteristic
88C2×C16×C23C2311yescharacteristic
89C2×C16×C23C2311yescharacteristic
90C2×C16×C23C2311yescharacteristic
91C23×#107—21no—
92C23×#107—21no—
93C23×#107—21no—
94C23×#107—21no—
95C23×#107C2311yescharacteristic
96C23×#107C2311yescharacteristic
97C23×#107C2311yescharacteristic
98C23×#107C2311yescharacteristic
99C23×#107C2311yescharacteristic
100C23×#107C2311yescharacteristic
101C23×#106—21no—
102C23×#106—21no—
103C23×C4⋊C8D411yescharacteristic
104C23×#103—21no—
105C23×#103—21no—
106C23×#103—21no—
107C23×#103—21no—
108C23×#103—21no—
109C23×#103—21no—
110C23×C22⋊C8D411yescharacteristic
111C23×C22⋊C8D411yescharacteristic
112C4×C8×C23C2311yescharacteristic
113C2×Q8×C23—21no—
114C2×Q8×C23—21no—
115C2×Q8×C23—41no—
116C2×D4×C23D811yescharacteristic
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23C2×D411yescharacteristic
121C22×C4×C23C2×D411yescharacteristic
122Q16×C23—21no—
123Q16×C23—21no—
124Q16×C23—21no—
125Q16×C23—21no—
126C2×C8×C23—21no—
127C2×C8×C23—21no—
128C2×C8×C23—21no—
129C2×C8×C23—21no—
130C2×C8×C23C2×D411yescharacteristic
131C2×C8×C23C2×D411yescharacteristic
132C2×C8×C23C2×D411yescharacteristic
133C2×C8×C23C2411yescharacteristic
134C4⋊C4×C23—21no—
135C4⋊C4×C23—21no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—21no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—21no—
140C4⋊C4×C23—41no—
141C4⋊C4×C23—41no—
142C4⋊C4×C23—41no—
143C4⋊C4×C23—41no—
144C4⋊C4×C23—41no—
145C4⋊C4×C23—41no—
146C4⋊C4×C23—41no—
147C4⋊C4×C23—41no—
148C4⋊C4×C23—41no—
149C4⋊C4×C23—41no—
150C4⋊C4×C23—41no—
151C4⋊C4×C23—41no—
152C4⋊C4×C23D811yescharacteristic
153C22⋊C4×C23—41no—
154C22⋊C4×C23—41no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23D811yescharacteristic
159C22⋊C4×C23D811yescharacteristic
160C42×C23—41no—
161C42×C23—41no—
162C42×C23—41no—
163C42×C23C2×D411yescharacteristic
164C16×C23—21no—
165C16×C23—21no—
166C16×C23—21no—
167C16×C23#4112yes—
168#33911C2311yesmaximal, characteristic, direct product factor, Sylow
169C23×C23C2×D811yescharacteristic
170C23×C23C2×D811yescharacteristic
171Q8×C23—41no—
172Q8×C23—41no—
173Q8×C23—41no—
174Q8×C23—41no—
175Q8×C23—161no—
176D4×C23—22no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—21no—
180C2×C4×C23—21no—
181C2×C4×C23—41no—
182C2×C4×C23—41no—
183C2×C4×C23—41no—
184C2×C4×C23—41no—
185C2×C4×C23—41no—
186C2×C4×C23—41no—
187C2×C4×C23—41no—
188C2×C4×C23—41no—
189C2×C4×C23—82no—
190C2×C4×C23—161no—
191C2×C4×C23—161no—
192C2×C4×C23C2×D811yescharacteristic
193C2×C4×C23C2×D811yescharacteristic
194C2×C4×C23C2×D811yescharacteristic
195C2×C4×C23C2×D811yescharacteristic
196C2×C4×C23C22×D411yescharacteristic
197C8×C23—21no—
198C8×C23—21no—
199C8×C23—21no—
200C8×C23C2×#4111yescharacteristic
201C8×C232−1+411yescharacteristic
202#3338C2×C2311yescharacteristic
203#3336C2×C2311yescharacteristic
204#2255C2×C2311yescharacteristic, complemented
205#2237C2×C2311yescharacteristic, complemented
206#2236C2×C2311yescharacteristic, complemented
207#2235C2×C2311yescharacteristic, complemented
208#2235C2×C2311yescharacteristic, complemented
209#2229C2×C2311yescharacteristic, complemented
210#2225C2×C2311yescharacteristic, complemented
211#2219C2×C2311yescharacteristic, complemented
212#2216C2×C2311yescharacteristic, complemented
213#2176C2×C2311yescharacteristic, complemented
214#2168C2×C2311yescharacteristic
215#2159C2×C2311yescharacteristic, complemented
216C2×#367C2×C2311yescharacteristic, complemented
217C22×C23—21no—
218C22×C23—21no—
219C22×C23—21no—
220C22×C23—21no—
221C22×C23—21no—
222C22×C23#51012yes—
223C22×C23C22×D811yescharacteristic
224C4×C23—21no—
225C4×C23—21no—
226C4×C23—21no—
227C4×C23—21no—
228C4×C23—81no—
229C4×C23—81no—
230C4×C23—81no—
231C4×C23—81no—
232C4×C23—81no—
233C4×C23—81no—
234C4×C23—81no—
235C4×C23—81no—
236C4×C23#54711yescharacteristic
237C4×C23#57811yescharacteristic
238#554—21no—
239#552—21no—
240C2×Q32C22×C2311yescharacteristic
241C2×M32C22×C2311yescharacteristic
242C22×C16C22×C2311yescharacteristic
243#500C22×C2311yescharacteristic
244#499C22×C2311yescharacteristic
245#485—21no—
246#484—21no—
247#484—21no—
248#483—21no—
249#479—21no—
250#477—21no—
251#467C22×C2311yescharacteristic
252#467C22×C2311yescharacteristic
253#467C22×C2311yescharacteristic
254#467C22×C2311yescharacteristic
255C4×Q16C22×C2311yescharacteristic
256C4×Q16C22×C2311yescharacteristic
257C8×D4C22×C2311yescharacteristic
258#427C22×C2311yescharacteristic
259C2×#107C22×C2311yescharacteristic
260C2×#107C22×C2311yescharacteristic
261C2×#107C22×C2311yescharacteristic
262#367C22×C2311yescharacteristic
263#367C22×C2311yescharacteristic
264#367C22×C2311yescharacteristic
265#367C22×C2311yescharacteristic
266#367C22×C2311yescharacteristic
267#367C22×C2311yescharacteristic
268#366C22×C2311yescharacteristic
269#366C22×C2311yescharacteristic
270#366C22×C2311yescharacteristic
271C4⋊C16C22×C2311yescharacteristic
272#358C22×C2311yescharacteristic
273#358C22×C2311yescharacteristic
274#358C22×C2311yescharacteristic
275#358C22×C2311yescharacteristic
276#358C22×C2311yescharacteristic
277#358C22×C2311yescharacteristic
278C22⋊C16C22×C2311yescharacteristic
279C22⋊C16C22×C2311yescharacteristic
280#346C22×C2311yescharacteristic
281C2×C23—22no—
282C2×C23—41no—
283C2×C23#333411yescharacteristic
284C2×C23C2×#51011yescharacteristic
285C2×C23#341611yescharacteristic
286C2×Q16C23×C2311yescharacteristic
287C2×Q16C23×C2311yescharacteristic
288C22×C8C23×C2311yescharacteristic
289C22×C8C23×C2311yescharacteristic
290C4⋊Q8—21no—
291#126—41no—
292#125—21no—
293#125—41no—
294#125—41no—
295#123—41no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—41no—
301C22⋊Q8—41no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×D4—41no—
305C4×D4D4×C2311yescharacteristic
306#117—21no—
307#117—41no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311Q32—22no—
312M32—22no—
313C2×C16—22no—
314C2×C16C23×C2311yescharacteristic
315C2×C16C23×C2311yescharacteristic
316C2×C16C23×C2311yescharacteristic
317C2×C16C23×C2311yescharacteristic
318#107—21no—
319#107—21no—
320#107—21no—
321#107—21no—
322#107C23×C2311yescharacteristic
323#107C23×C2311yescharacteristic
324#107C23×C2311yescharacteristic
325#107C23×C2311yescharacteristic
326#107C23×C2311yescharacteristic
327#107C23×C2311yescharacteristic
328#106—21no—
329#106—21no—
330C4⋊C8D4×C2311yescharacteristic
331#103—21no—
332#103—21no—
333#103—21no—
334#103—21no—
335#103—21no—
336#103—21no—
337C22⋊C8D4×C2311yescharacteristic
338C22⋊C8D4×C2311yescharacteristic
339C4×C8C23×C2311yescharacteristic
340C23#3391111yescharacteristic, direct product factor, Sylow
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—41no—
344C2×D4D8×C2311yescharacteristic
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×D4×C2311yescharacteristic
349C22×C4C2×D4×C2311yescharacteristic
350Q16—21no—
351Q16—21no—
352Q16—21no—
353Q16—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8C2×D4×C2311yescharacteristic
359C2×C8C2×D4×C2311yescharacteristic
360C2×C8C2×D4×C2311yescharacteristic
361C2×C8C24×C2311yescharacteristic
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4D8×C2311yescharacteristic
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4D8×C2311yescharacteristic
387C22⋊C4D8×C2311yescharacteristic
388C42—41no—
389C42—41no—
390C42—41no—
391C42C2×D4×C2311yescharacteristic
392C16—21no—
393C16—21no—
394C16—21no—
395C16#41×C2312yes—
396C23C2×D8×C2311yescharacteristic
397C23C2×D8×C2311yescharacteristic
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—161no—
403D4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—161no—
418C2×C4—161no—
419C2×C4C2×D8×C2311yescharacteristic
420C2×C4C2×D8×C2311yescharacteristic
421C2×C4C2×D8×C2311yescharacteristic
422C2×C4C2×D8×C2311yescharacteristic
423C2×C4C22×D4×C2311yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8C2×#41×C2311yescharacteristic
428C8C23×2−1+411yescharacteristic
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22C23×#51012yes—
435C22C22×D8×C2311yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4C23×#54711yescharacteristic
449C4C23×#57811yescharacteristic
450C2—22no—
451C2—41no—
452C2C23×#333411yescharacteristic
453C2C2×C23×#51011yescharacteristic
454C2C23×#341611yescharacteristic
subgroupG/H
center, socleC22×C23C22×D8
commutator, Frattini, cosocleC2×C8C24×C23
prime radicalC2×D4×C23D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C4×D4×C23 ◁ C8×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#2255C2semidirect
4. C23×#2237C2semidirect
5. C23×#2236C2semidirect
6. C23×#2235C2semidirect
7. C23×#2235C2semidirect
8. C23×#2229C2semidirect
9. C23×#2225C2semidirect
10. C23×#2219C2semidirect
11. C23×#2216C2semidirect
12. C23×#2176C2semidirect
14. C23×#2159C2semidirect
15. C2×C23×#367C2semidirect
168. #33911C23direct (N × H)
204. #2255C2×C23semidirect
205. #2237C2×C23semidirect
206. #2236C2×C23semidirect
207. #2235C2×C23semidirect
208. #2235C2×C23semidirect
209. #2229C2×C23semidirect
210. #2225C2×C23semidirect
211. #2219C2×C23semidirect
212. #2216C2×C23semidirect
213. #2176C2×C23semidirect
215. #2159C2×C23semidirect
216. C2×#367C2×C23semidirect
340. C23#33911direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#33382
C23×#33362
C23×#22552
C23×#22372
C23×#22362
C23×#22352
C23×#22292
C23×#22252
C23×#22192
C23×#22162
C23×#21762
C23×#21682
C23×#21592
C2×C23×#3672
#3391123