Finite Group Databasebeta

← C23×#8874 #19156500

Subgroups — C23×#8874 #19156500

Order 5888.

464 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2976C211yesmaximal, characteristic
2C23×#1871C211yesmaximal, characteristic
3C23×#1855C211yesmaximal, characteristic, complemented
4C23×#1851C211yesmaximal, characteristic, complemented
5C23×#1501C211yesmaximal, characteristic
6C23×#1297C211yesmaximal, characteristic, complemented
7C23×#1282C211yesmaximal, characteristic, complemented
8C22×M16×C23C2211yescharacteristic
9C23×#525C2211yescharacteristic
10C23×#436—21no—
11C23×#435—21no—
12C23×#435—21no—
13C8×D4×C23—21no—
14C23×#433—21no—
15C2×C23×#108—21no—
16C2×C23×C4⋊C8—21no—
17C2×C23×#104—21no—
18C23×#408—21no—
19C23×#407C2211yescharacteristic
20C2×C23×C22⋊C8C2211yescharacteristic
21C2×C23×C22⋊C8C2211yescharacteristic
22C4×C22⋊C4×C23C2211yescharacteristic
23C4×C22⋊C4×C23C2211yescharacteristic
24C23×#341—21no—
25C23×#339—21no—
26C23×#336—21no—
27C23×#328C411yescharacteristic, complemented
28C23×#328C411yescharacteristic, complemented
29C23×#327C411yescharacteristic, complemented
30C23×#327C411yescharacteristic, complemented
31C2×#41×C23D411yescharacteristic
32C23×C4×C23C2311yescharacteristic
33C2×M16×C23—21no—
34C2×M16×C23—21no—
35C2×M16×C23—21no—
36C2×M16×C23—42no—
37C2×M16×C23D411yescharacteristic
38C2×M16×C23D411yescharacteristic
39C22×C8×C23—21no—
40C22×C8×C23—21no—
41C22×C8×C23—21no—
42C22×C8×C23D411yescharacteristic
43C22×C8×C23D411yescharacteristic
44C23×#123—21no—
45C23×C22⋊Q8C2×C411yescharacteristic
46C23×C22⋊Q8Q811yescharacteristic
47C23×#121C2×C411yescharacteristic
48C23×#121Q811yescharacteristic
49C23×#120—21no—
50C4×D4×C23—21no—
51C4×D4×C23—21no—
52C23×#117D411yescharacteristic
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C42×C23—21no—
56C2×C42×C23—21no—
57C23×#108—42no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—42no—
61C23×#104—42no—
62C23×C22⋊C8—21no—
63C23×C22⋊C8—21no—
64C23×C22⋊C8—21no—
65C23×C22⋊C8—21no—
66C23×C22⋊C8—21no—
67C23×C22⋊C8—21no—
68C23×C22⋊C8—21no—
69C23×C22⋊C8C2×C411yescharacteristic
70C23×C22⋊C8C2×C411yescharacteristic
71C23×#97—21no—
72C4×C8×C23—21no—
73C23×#95—21no—
74C23×#95C2×C411yescharacteristic
75C23×#95C2×C411yescharacteristic
76C24×C23#4111yescharacteristic
77#41×C23—41no—
78#41×C23—41no—
79C2×Q8×C23C4⋊C411yescharacteristic
80C2×D4×C23—21no—
81C2×D4×C23—21no—
82C2×D4×C23—21no—
83C2×D4×C23C4⋊C411yescharacteristic
84C22×C4×C23—21no—
85C22×C4×C23—21no—
86C22×C4×C23—21no—
87C22×C4×C23—41no—
88C22×C4×C23—41no—
89C22×C4×C23—41no—
90C22×C4×C23C22×C411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23C2×Q811yescharacteristic
95C22×C4×C23#4111yescharacteristic
96M16×C23—22no—
97M16×C23—22no—
98M16×C23—41no—
99M16×C23—41no—
100M16×C23—41no—
101M16×C23—41no—
102M16×C23—81no—
103C2×C8×C23—22no—
104C2×C8×C23—21no—
105C2×C8×C23—21no—
106C2×C8×C23—21no—
107C2×C8×C23—21no—
108C2×C8×C23—21no—
109C2×C8×C23—21no—
110C2×C8×C23—41no—
111C2×C8×C23—41no—
112C2×C8×C23—41no—
113C2×C8×C23—41no—
114C2×C8×C23—41no—
115C2×C8×C23—41no—
116C2×C8×C23—81no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23C4⋊C411yescharacteristic
120C4⋊C4×C23C4⋊C411yescharacteristic
121C22⋊C4×C23—21no—
122C22⋊C4×C23—21no—
123C22⋊C4×C23—21no—
124C22⋊C4×C23—21no—
125C22⋊C4×C23—21no—
126C22⋊C4×C23—22no—
127C22⋊C4×C23—42no—
128C22⋊C4×C23—41no—
129C42×C23—21no—
130C42×C23—41no—
131C42×C23—41no—
132C42×C23—41no—
133C42×C23—41no—
134#8874C2311yesmaximal, characteristic, direct product factor, Sylow
135C23×C23—21no—
136C23×C23—21no—
137C23×C23—21no—
138C23×C23—41no—
139C23×C23—41no—
140C23×C23C4×D411yescharacteristic
141C23×C23C22⋊Q811yescharacteristic
142C23×C23#12311yescharacteristic
143Q8×C23—41no—
144D4×C23—41no—
145D4×C23—41no—
146D4×C23—41no—
147D4×C23—81no—
148D4×C23—81no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—21no—
154C2×C4×C23—21no—
155C2×C4×C23—22no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
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—41no—
167C2×C4×C23—42no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—81no—
171C2×C4×C23—81no—
172C2×C4×C23C2×C4⋊C411yescharacteristic
173C2×C4×C23C4×D411yescharacteristic
174C2×C4×C23#12011yescharacteristic
175C2×C4×C23C22⋊Q811yescharacteristic
176C8×C23—41no—
177C8×C23—41no—
178C8×C23—42no—
179C8×C23—41no—
180C8×C23—41no—
181C8×C23—41no—
182C8×C23—41no—
183#2976C2×C2311yescharacteristic
184#1871C2×C2311yescharacteristic
185#1855C2×C2311yescharacteristic, complemented
186#1851C2×C2311yescharacteristic, complemented
187#1501C2×C2311yescharacteristic
188#1297C2×C2311yescharacteristic, complemented
189#1282C2×C2311yescharacteristic, complemented
190C22×C23—21no—
191C22×C23—21no—
192C22×C23—21no—
193C22×C23—21no—
194C22×C23—21no—
195C22×C23—21no—
196C22×C23—21no—
197C22×C23—21no—
198C22×C23—41no—
199C22×C23—41no—
200C22×C23—41no—
201C22×C23—81no—
202C22×C23—81no—
203C22×C23#38511yescharacteristic
204C22×C23#44311yescharacteristic
205C22×C23#44411yescharacteristic
206C4×C23—21no—
207C4×C23—21no—
208C4×C23—21no—
209C4×C23—42no—
210C4×C23—41no—
211C4×C23—41no—
212C4×C23—41no—
213C4×C23—81no—
214C4×C23—82no—
215C4×C23B(4,𝔽2)11yescharacteristic
216C4×C23#45811yescharacteristic
217C22×M16C22×C2311yescharacteristic
218#525C22×C2311yescharacteristic
219#436—21no—
220#435—21no—
221#435—21no—
222C8×D4—21no—
223#433—21no—
224C2×#108—21no—
225C2×C4⋊C8—21no—
226C2×#104—21no—
227#408—21no—
228#407C22×C2311yescharacteristic
229C2×C22⋊C8C22×C2311yescharacteristic
230C2×C22⋊C8C22×C2311yescharacteristic
231C4×C22⋊C4C22×C2311yescharacteristic
232C4×C22⋊C4C22×C2311yescharacteristic
233#341—21no—
234#339—21no—
235#336—21no—
236#328C4×C2311yescharacteristic, complemented
237#328C4×C2311yescharacteristic, complemented
238#327C4×C2311yescharacteristic, complemented
239#327C4×C2311yescharacteristic, complemented
240C2×C23—21no—
241C2×C23—21no—
242C2×C23—41no—
243C2×C23—41no—
244C2×C23—81no—
245C2×C23#186111yescharacteristic
246C2×C23#186611yescharacteristic
247C2×C23#186711yescharacteristic
248C2×#41D4×C2311yescharacteristic
249C23×C4C23×C2311yescharacteristic
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—42no—
254C2×M16D4×C2311yescharacteristic
255C2×M16D4×C2311yescharacteristic
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8D4×C2311yescharacteristic
260C22×C8D4×C2311yescharacteristic
261#123—21no—
262C22⋊Q8C2×C4×C2311yescharacteristic
263C22⋊Q8Q8×C2311yescharacteristic
264#121C2×C4×C2311yescharacteristic
265#121Q8×C2311yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117D4×C2311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42—21no—
273C2×C42—21no—
274#108—42no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—42no—
278#104—42no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8C2×C4×C2311yescharacteristic
287C22⋊C8C2×C4×C2311yescharacteristic
288#97—21no—
289C4×C8—21no—
290#95—21no—
291#95C2×C4×C2311yescharacteristic
292#95C2×C4×C2311yescharacteristic
293C23#887411yescharacteristic, direct product factor, Sylow
294C24#41×C2311yescharacteristic
295#41—41no—
296#41—41no—
297C2×Q8C4⋊C4×C2311yescharacteristic
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C4⋊C4×C2311yescharacteristic
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C22×C4×C2311yescharacteristic
309C22×C4C2×D4×C2311yescharacteristic
310C22×C4C2×D4×C2311yescharacteristic
311C22×C4C2×D4×C2311yescharacteristic
312C22×C4C2×Q8×C2311yescharacteristic
313C22×C4#41×C2311yescharacteristic
314M16—22no—
315M16—22no—
316M16—41no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—81no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—81no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C4⋊C4×C2311yescharacteristic
338C4⋊C4C4⋊C4×C2311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—22no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C42—21no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C4×D4×C2311yescharacteristic
358C23C23×C22⋊Q811yescharacteristic
359C23C23×#12311yescharacteristic
360Q8—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—81no—
365D4—81no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4C2×C4⋊C4×C2311yescharacteristic
390C2×C4C4×D4×C2311yescharacteristic
391C2×C4C23×#12011yescharacteristic
392C2×C4C23×C22⋊Q811yescharacteristic
393C8—41no—
394C8—41no—
395C8—42no—
396C8—41no—
397C8—41no—
398C8—41no—
399C8—41no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—81no—
412C22—81no—
413C22C23×#38511yescharacteristic
414C22C23×#44311yescharacteristic
415C22C23×#44411yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—81no—
424C4—82no—
425C4C23×B(4,𝔽2)11yescharacteristic
426C4C23×#45811yescharacteristic
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—81no—
432C2C23×#186111yescharacteristic
433C2C23×#186611yescharacteristic
434C2C23×#186711yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
socleC22×C23#385
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1855C2semidirect
4. C23×#1851C2semidirect
6. C23×#1297C2semidirect
7. C23×#1282C2semidirect
27. C23×#328C4semidirect
28. C23×#328C4semidirect
29. C23×#327C4semidirect
30. C23×#327C4semidirect
134. #8874C23direct (N × H)
185. #1855C2×C23semidirect
186. #1851C2×C23semidirect
188. #1297C2×C23semidirect
189. #1282C2×C23semidirect
236. #328C4×C23semidirect
237. #328C4×C23semidirect
238. #327C4×C23semidirect
239. #327C4×C23semidirect
293. C23#8874direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#29762
C23×#18712
C23×#18552
C23×#18512
C23×#15012
C23×#12972
C23×#12822
#887423