Finite Group Databasebeta

← C23×#8402 #19156028

Subgroups — C23×#8402 #19156028

Order 5888.

716 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3461C211yesmaximal, characteristic
2C23×#1985C211yesmaximal, characteristic, complemented
3C4×C23×#101C211yesmaximal, characteristic, complemented
4C23×#1524C212yesmaximal, complemented
5C23×#1483C212yesmaximal, complemented
6C23×#544C411yescharacteristic
7C23×#542—22no—
8C23×#541—22no—
9C23×#539—21no—
10C23×#539C412yes—
11C23×#537C411yescharacteristic
12C23×#536—21no—
13C23×#533—22no—
14C23×#529—22no—
15C23×#529—21no—
16C23×#529C2212yes—
17C4×#41×C23—21no—
18C2×C4×Q8×C23C2211yescharacteristic
19C2×C23×C4⋊C8C2211yescharacteristic
20C2×C23×#101—21no—
21C4×M16×C23C2212yes—
22C2×C23×#97C2211yescharacteristic
23C23×#343—21no—
24C23×#336—21no—
25C23×#336—21no—
26C23×#330—22no—
27C23×#329—22no—
28C23×Q8⋊C8—22no—
29C23×D4⋊C8—22no—
30C2×#41×C23—21no—
31C22×Q8×C23D411yescharacteristic
32C2×M16×C23—22no—
33C22×C8×C23—21no—
34C22×C8×C23—21no—
35C23×C4⋊Q8—22no—
36C23×C4⋊Q8—22no—
37C23×#126—22no—
38C23×#126—22no—
39C23×#126—22no—
40C23×#126—22no—
41C23×#125—21no—
42C23×#125—22no—
43C23×#125—22no—
44C23×#125—22no—
45C23×#125D412yes—
46C23×#124—22no—
47C23×#124—21no—
48C23×#124D412yes—
49C23×#123—22no—
50C23×#123—22no—
51C23×#123—22no—
52C23×#123—22no—
53C23×C22⋊Q8—22no—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×C22⋊Q8—21no—
58C23×C22⋊Q8—22no—
59C23×C22⋊Q8C2×C412yes—
60C23×#121—22no—
61C23×#121—21no—
62C23×#121C2×C412yes—
63C4×Q8×C23—21no—
64C4×Q8×C23—21no—
65C4×Q8×C23—21no—
66C4×Q8×C23—22no—
67C4×Q8×C23—22no—
68C4×Q8×C23D412yes—
69C4×D4×C23—21no—
70C4×D4×C23—21no—
71C4×D4×C23—21no—
72C4×D4×C23—22no—
73C4×D4×C23—22no—
74C4×D4×C23D412yes—
75C23×#117—22no—
76C23×#117—22no—
77C23×#117—21no—
78C23×#117—22no—
79C2×C4⋊C4×C23C2×C411yescharacteristic
80C2×C4⋊C4×C23C2×C411yescharacteristic
81C2×C4⋊C4×C23D411yescharacteristic
82C2×C42×C23—22no—
83C2×C42×C23D411yescharacteristic
84C2×C42×C23D411yescharacteristic
85C2×C42×C23C2311yescharacteristic
86C23×C4⋊C8—22no—
87C23×#101—24no—
88C23×#97—22no—
89C23×#97—22no—
90C4×C8×C23—22no—
91#41×C23—42no—
92#41×C23—42no—
93C2×Q8×C23—21no—
94C2×Q8×C23—21no—
95C2×Q8×C23—22no—
96C2×Q8×C23—22no—
97C2×Q8×C23—41no—
98C2×Q8×C23C22⋊C412yes—
99C2×D4×C23—21no—
100C2×D4×C23—22no—
101C2×D4×C23—21no—
102C2×D4×C23C22⋊C412yes—
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23C22⋊C411yescharacteristic
107C22×C4×C23C22⋊C411yescharacteristic
108C22×C4×C23C22⋊C411yescharacteristic
109C22×C4×C23C22⋊C411yescharacteristic
110C22×C4×C23C22×C411yescharacteristic
111C22×C4×C23C2×D411yescharacteristic
112C22×C4×C23C2×D411yescharacteristic
113M16×C23—24no—
114M16×C23—42no—
115C2×C8×C23—22no—
116C2×C8×C23—41no—
117C2×C8×C23—41no—
118C2×C8×C23—42no—
119C2×C8×C23—42no—
120C2×C8×C23—41no—
121C4⋊C4×C23—21no—
122C4⋊C4×C23—22no—
123C4⋊C4×C23—21no—
124C4⋊C4×C23—22no—
125C4⋊C4×C23—22no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—22no—
129C4⋊C4×C23—22no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23—22no—
136C4⋊C4×C23C22⋊C412yes—
137C4⋊C4×C23C22⋊C412yes—
138C22⋊C4×C23—22no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—22no—
142C22⋊C4×C23—22no—
143C22⋊C4×C23—22no—
144C22⋊C4×C23—22no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C42×C23—22no—
148C42×C23—22no—
149C42×C23—21no—
150C42×C23—21no—
151C42×C23—21no—
152C42×C23—22no—
153C42×C23—21no—
154C42×C23C2×D412yes—
155C42×C23C2×D412yes—
156#8402C2311yesmaximal, characteristic, direct product factor, Sylow
157C23×C23—22no—
158C23×C23C2×C22⋊C411yescharacteristic
159Q8×C23—44no—
160Q8×C23—42no—
161Q8×C23—42no—
162Q8×C23—81no—
163D4×C23—42no—
164D4×C23—42no—
165D4×C23—42no—
166D4×C23—42no—
167D4×C23—81no—
168C2×C4×C23—21no—
169C2×C4×C23—21no—
170C2×C4×C23—21no—
171C2×C4×C23—21no—
172C2×C4×C23—21no—
173C2×C4×C23—21no—
174C2×C4×C23—21no—
175C2×C4×C23—22no—
176C2×C4×C23—22no—
177C2×C4×C23—21no—
178C2×C4×C23—21no—
179C2×C4×C23—22no—
180C2×C4×C23—21no—
181C2×C4×C23—21no—
182C2×C4×C23—41no—
183C2×C4×C23—44no—
184C2×C4×C23—42no—
185C2×C4×C23—42no—
186C2×C4×C23—41no—
187C2×C4×C23—41no—
188C2×C4×C23—82no—
189C2×C4×C23#10412yes—
190C2×C4×C23C2×C22⋊C412yes—
191C2×C4×C23#12011yescharacteristic
192C2×C4×C23#12012yes—
193C2×C4×C23#12011yescharacteristic
194C8×C23—42no—
195C8×C23—44no—
196C8×C23—81no—
197#3461C2×C2311yescharacteristic
198#1985C2×C2311yescharacteristic, complemented
199C4×#101C2×C2311yescharacteristic, complemented
200#1524C2×C2312yescomplemented
201#1483C2×C2312yescomplemented
202C22×C23—21no—
203C22×C23—21no—
204C22×C23—42no—
205C22×C23—42no—
206C22×C23—42no—
207C22×C23#37911yescharacteristic
208C22×C23C2×#10411yescharacteristic
209C22×C23#42111yescharacteristic
210C4×C23—21no—
211C4×C23—22no—
212C4×C23—22no—
213C4×C23—22no—
214C4×C23—42no—
215C4×C23—41no—
216C4×C23—42no—
217C4×C23—41no—
218C4×C23—41no—
219C4×C23—42no—
220C4×C23#45411yescharacteristic
221C4×C23#45611yescharacteristic
222#544C4×C2311yescharacteristic
223#542—22no—
224#541—22no—
225#539—21no—
226#539C4×C2312yes—
227#537C4×C2311yescharacteristic
228#536—21no—
229#533—22no—
230#529—22no—
231#529—21no—
232#529C22×C2312yes—
233C4×#41—21no—
234C2×C4×Q8C22×C2311yescharacteristic
235C2×C4⋊C8C22×C2311yescharacteristic
236C2×#101—21no—
237C4×M16C22×C2312yes—
238C2×#97C22×C2311yescharacteristic
239#343—21no—
240#336—21no—
241#336—21no—
242#330—22no—
243#329—22no—
244Q8⋊C8—22no—
245D4⋊C8—22no—
246C2×C23—21no—
247C2×C23—21no—
248C2×C23—82no—
249C2×C23#178911yescharacteristic
250C2×C23#179111yescharacteristic
251C2×C23#179311yescharacteristic
252C2×#41—21no—
253C22×Q8D4×C2311yescharacteristic
254C2×M16—22no—
255C22×C8—21no—
256C22×C8—21no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259#126—22no—
260#126—22no—
261#126—22no—
262#126—22no—
263#125—21no—
264#125—22no—
265#125—22no—
266#125—22no—
267#125D4×C2312yes—
268#124—22no—
269#124—21no—
270#124D4×C2312yes—
271#123—22no—
272#123—22no—
273#123—22no—
274#123—22no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279C22⋊Q8—21no—
280C22⋊Q8—22no—
281C22⋊Q8C2×C4×C2312yes—
282#121—22no—
283#121—21no—
284#121C2×C4×C2312yes—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8D4×C2312yes—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4D4×C2312yes—
297#117—22no—
298#117—22no—
299#117—21no—
300#117—22no—
301C2×C4⋊C4C2×C4×C2311yescharacteristic
302C2×C4⋊C4C2×C4×C2311yescharacteristic
303C2×C4⋊C4D4×C2311yescharacteristic
304C2×C42—22no—
305C2×C42D4×C2311yescharacteristic
306C2×C42D4×C2311yescharacteristic
307C2×C42C23×C2311yescharacteristic
308C4⋊C8—22no—
309#101—24no—
310#97—22no—
311#97—22no—
312C4×C8—22no—
313C23#840211yescharacteristic, direct product factor, Sylow
314#41—42no—
315#41—42no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×Q8C22⋊C4×C2312yes—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—21no—
325C2×D4C22⋊C4×C2312yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4C22⋊C4×C2311yescharacteristic
330C22×C4C22⋊C4×C2311yescharacteristic
331C22×C4C22⋊C4×C2311yescharacteristic
332C22×C4C22⋊C4×C2311yescharacteristic
333C22×C4C22×C4×C2311yescharacteristic
334C22×C4C2×D4×C2311yescharacteristic
335C22×C4C2×D4×C2311yescharacteristic
336M16—24no—
337M16—42no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4C22⋊C4×C2312yes—
360C4⋊C4C22⋊C4×C2312yes—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C42—22no—
371C42—22no—
372C42—21no—
373C42—22no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42C2×D4×C2312yes—
378C42C2×D4×C2312yes—
379C23—22no—
380C23C2×C22⋊C4×C2311yescharacteristic
381Q8—44no—
382Q8—42no—
383Q8—42no—
384Q8—81no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—81no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—82no—
411C2×C4C23×#10412yes—
412C2×C4C2×C22⋊C4×C2312yes—
413C2×C4C23×#12011yescharacteristic
414C2×C4C23×#12012yes—
415C2×C4C23×#12011yescharacteristic
416C8—42no—
417C8—44no—
418C8—81no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22C23×#37911yescharacteristic
425C22C2×C23×#10411yescharacteristic
426C22C23×#42111yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—41no—
432C4—42no—
433C4—41no—
434C4—41no—
435C4—42no—
436C4—42no—
437C4C23×#45411yescharacteristic
438C4C23×#45611yescharacteristic
439C2—21no—
440C2—21no—
441C2—82no—
442C2C23×#178911yescharacteristic
443C2C23×#179111yescharacteristic
444C2C23×#179311yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#379
prime radicalC23×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1985C2semidirect
3. C4×C23×#101C2semidirect
4. C23×#1524C2semidirect
5. C23×#1483C2semidirect
156. #8402C23direct (N × H)
198. #1985C2×C23semidirect
199. C4×#101C2×C23semidirect
200. #1524C2×C23semidirect
201. #1483C2×C23semidirect
313. C23#8402direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34612
C23×#19852
C4×C23×#1012
C23×#15242
C23×#14832
#840223