Finite Group Databasebeta

← C23×#8574 #19156200

Subgroups — C23×#8574 #19156200

Order 5888.

614 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C23×C22⋊Q8C211yesmaximal, characteristic
2C23×#1797C211yesmaximal, characteristic
3C2×C23×#328C212yesmaximal
4C23×#1434C211yesmaximal, characteristic
5C23×#1294C212yesmaximal
6C2×C23×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8×C23C411yescharacteristic, complemented
8C22×C42×C23C2211yescharacteristic
9C2×C23×C4⋊C8—21no—
10C2×C23×C22⋊C8C2212yes—
11C23×#391C411yescharacteristic, complemented
12C23×#389C411yescharacteristic, complemented
13C23×#387—21no—
14C23×#385—21no—
15C23×#380C2211yescharacteristic
16C4×C4⋊C4×C23—21no—
17C4×C4⋊C4×C23C411yescharacteristic, complemented
18C4×C22⋊C4×C23—21no—
19C2×C23×#95—21no—
20C2×C23×#95—21no—
21C2×C23×#95C2212yes—
22C23×#336—21no—
23C23×#328—24no—
24C22×Q8×C23C2×C411yescharacteristic
25C23×C4×C23—22no—
26C23×C4×C23D411yescharacteristic
27C23×C4×C23D411yescharacteristic
28C23×C4×C23C2311yescharacteristic
29C22×C8×C23—22no—
30C23×C22⋊Q8—21no—
31C23×C22⋊Q8—21no—
32C23×C22⋊Q8D412yes—
33C23×C22⋊Q8Q812yes—
34C4×Q8×C23—42no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—21no—
37C2×C4⋊C4×C23—21no—
38C2×C4⋊C4×C23C2×C411yescharacteristic
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C4⋊C4×C23C2×C411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C42×C23—21no—
44C2×C42×C23—21no—
45C2×C42×C23—42no—
46C2×C42×C23C2×C411yescharacteristic
47C2×C42×C23C2×C411yescharacteristic
48C2×C42×C23D411yescharacteristic
49C2×C42×C23D411yescharacteristic
50C23×C4⋊C8—42no—
51C23×C22⋊C8—22no—
52C23×C22⋊C8—22no—
53C23×#95—24no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—22no—
60C23×#95—22no—
61C23×#95—42no—
62C24×C23C2×D411yescharacteristic
63C2×Q8×C23—21no—
64C2×Q8×C23—41no—
65C2×Q8×C23—41no—
66C2×Q8×C23C4⋊C412yes—
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23—22no—
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—22no—
73C22×C4×C23—24no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—41no—
77C22×C4×C23—41no—
78C22×C4×C23—41no—
79C22×C4×C23—41no—
80C22×C4×C23—42no—
81C22×C4×C23—42no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23C22⋊C411yescharacteristic
87C22×C4×C23C22⋊C411yescharacteristic
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22×C411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23C2×Q811yescharacteristic
94C22×C4×C23#4111yescharacteristic
95C22×C4×C23#4111yescharacteristic
96C2×C8×C23—42no—
97C2×C8×C23—42no—
98C2×C8×C23—42no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—42no—
105C4⋊C4×C23—42no—
106C4⋊C4×C23C4⋊C412yes—
107C22⋊C4×C23—22no—
108C22⋊C4×C23—22no—
109C22⋊C4×C23—42no—
110C42×C23—22no—
111C42×C23—22no—
112C42×C23—41no—
113C42×C23—41no—
114C42×C23—42no—
115C42×C23—42no—
116#8574C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—21no—
118C23×C23—21no—
119C23×C23—21no—
120C23×C23—21no—
121C23×C23C2×C22⋊C411yescharacteristic
122C23×C23#12111yescharacteristic
123C23×C23C22⋊Q811yescharacteristic
124C23×C23#13512yes—
125C23×C23#13712yes—
126Q8×C23—42no—
127Q8×C23—42no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—22no—
137C2×C4×C23—22no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—44no—
149C2×C4×C23—44no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—44no—
155C2×C4×C23—44no—
156C2×C4×C23—44no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—41no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—82no—
164C2×C4×C23—82no—
165C2×C4×C23—82no—
166C2×C4×C23—82no—
167C2×C4×C23#9912yes—
168C2×C4×C23#10412yes—
169C2×C4×C23C2×C4⋊C411yescharacteristic
170C2×C4×C23#11711yescharacteristic
171C2×C4×C23#12111yescharacteristic
172C2×C4×C23C22⋊Q811yescharacteristic
173C8×C23—82no—
174C4×C22⋊Q8C2×C2311yescharacteristic
175#1797C2×C2311yescharacteristic
176C2×#328C2×C2312yes—
177#1434C2×C2311yescharacteristic
178#1294C2×C2312yes—
179C22×C23—21no—
180C22×C23—21no—
181C22×C23—21no—
182C22×C23—21no—
183C22×C23—22no—
184C22×C23—22no—
185C22×C23—22no—
186C22×C23—22no—
187C22×C23#38011yescharacteristic
188C22×C23C2×#9911yescharacteristic
189C22×C23#41011yescharacteristic
190C22×C23#41611yescharacteristic
191C22×C23#41911yescharacteristic
192C22×C23C2×#10411yescharacteristic
193C22×C23#42111yescharacteristic
194C22×C23#46412yes—
195C22×C23#47912yes—
196C4×C23—21no—
197C4×C23—21no—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—22no—
201C4×C23—22no—
202C4×C23—42no—
203C4×C23—42no—
204C4×C23—41no—
205C4×C23—41no—
206C4×C23—84no—
207C4×C23—81no—
208C4×C23—81no—
209C2×C22⋊Q8C22×C2311yescharacteristic
210C2×C4×Q8C4×C2311yescharacteristic, complemented
211C22×C42C22×C2311yescharacteristic
212C2×C4⋊C8—21no—
213C2×C22⋊C8C22×C2312yes—
214#391C4×C2311yescharacteristic, complemented
215#389C4×C2311yescharacteristic, complemented
216#387—21no—
217#385—21no—
218#380C22×C2311yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4C4×C2311yescharacteristic, complemented
221C4×C22⋊C4—21no—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95C22×C2312yes—
225#336—21no—
226#328—24no—
227C2×C23—22no—
228C2×C23—22no—
229C2×C23#150011yescharacteristic
230C2×C23#150811yescharacteristic
231C2×C23#151511yescharacteristic
232C2×C23#151611yescharacteristic
233C2×C23#179911yescharacteristic
234C2×C23#180511yescharacteristic
235C2×C23#180611yescharacteristic
236C22×Q8C2×C4×C2311yescharacteristic
237C23×C4—22no—
238C23×C4D4×C2311yescharacteristic
239C23×C4D4×C2311yescharacteristic
240C23×C4C23×C2311yescharacteristic
241C22×C8—22no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8D4×C2312yes—
245C22⋊Q8Q8×C2312yes—
246C4×Q8—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×C4×C2311yescharacteristic
251C2×C4⋊C4C2×C4×C2311yescharacteristic
252C2×C4⋊C4C2×C4×C2311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—42no—
258C2×C42C2×C4×C2311yescharacteristic
259C2×C42C2×C4×C2311yescharacteristic
260C2×C42D4×C2311yescharacteristic
261C2×C42D4×C2311yescharacteristic
262C4⋊C8—42no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265#95—24no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—22no—
272#95—22no—
273#95—42no—
274C23#857411yescharacteristic, direct product factor, Sylow
275C24C2×D4×C2311yescharacteristic
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊C4×C2312yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C22⋊C4×C2311yescharacteristic
300C22×C4C22⋊C4×C2311yescharacteristic
301C22×C4C22⋊C4×C2311yescharacteristic
302C22×C4C22⋊C4×C2311yescharacteristic
303C22×C4C22×C4×C2311yescharacteristic
304C22×C4C2×D4×C2311yescharacteristic
305C22×C4C2×D4×C2311yescharacteristic
306C22×C4C2×Q8×C2311yescharacteristic
307C22×C4#41×C2311yescharacteristic
308C22×C4#41×C2311yescharacteristic
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4C4⋊C4×C2312yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C2×C22⋊C4×C2311yescharacteristic
334C23C23×#12111yescharacteristic
335C23C23×C22⋊Q811yescharacteristic
336C23C23×#13512yes—
337C23C23×#13712yes—
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4C23×#9912yes—
380C2×C4C23×#10412yes—
381C2×C4C2×C4⋊C4×C2311yescharacteristic
382C2×C4C23×#11711yescharacteristic
383C2×C4C23×#12111yescharacteristic
384C2×C4C23×C22⋊Q811yescharacteristic
385C8—82no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22C23×#38011yescharacteristic
395C22C2×C23×#9911yescharacteristic
396C22C23×#41011yescharacteristic
397C22C23×#41611yescharacteristic
398C22C23×#41911yescharacteristic
399C22C2×C23×#10411yescharacteristic
400C22C23×#42111yescharacteristic
401C22C23×#46412yes—
402C22C23×#47912yes—
403C4—22no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—84no—
414C4—81no—
415C4—81no—
416C2—22no—
417C2—22no—
418C2C23×#150011yescharacteristic
419C2C23×#150811yescharacteristic
420C2C23×#151511yescharacteristic
421C2C23×#151611yescharacteristic
422C2C23×#179911yescharacteristic
423C2C23×#180511yescharacteristic
424C2C23×#180611yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8×C23C4semidirect
11. C23×#391C4semidirect
12. C23×#389C4semidirect
17. C4×C4⋊C4×C23C4semidirect
116. #8574C23direct (N × H)
210. C2×C4×Q8C4×C23semidirect
214. #391C4×C23semidirect
215. #389C4×C23semidirect
220. C4×C4⋊C4C4×C23semidirect
274. C23#8574direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C23×C22⋊Q82
C23×#17972
C2×C23×#3282
C23×#14342
C23×#12942
#857423