Finite Group Databasebeta

← C23×#8969 #19156595

Subgroups — C23×#8969 #19156595

Order 5888.

516 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3106C211yesmaximal, characteristic
2C23×#2990C211yesmaximal, characteristic, complemented
3C23×#1962C211yesmaximal, characteristic, complemented
4C4×C23×#104C211yesmaximal, characteristic, complemented
5C23×#1484C211yesmaximal, characteristic, complemented
6C23×#1299C211yesmaximal, characteristic, complemented
7C23×#1276C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C23×#482—21no—
11C23×#477C411yescharacteristic, complemented
12C23×#476C411yescharacteristic, complemented
13C23×#460C411yescharacteristic, complemented
14C23×#459C411yescharacteristic, complemented
15C23×#449—21no—
16C23×#446—21no—
17C4×SD16×C23—21no—
18C4×D8×C23—21no—
19C23×#435—22no—
20C8×D4×C23—21no—
21C23×#432—21no—
22C23×#431C2211yescharacteristic
23C23×#431C2211yescharacteristic
24C23×#423C2211yescharacteristic
25C2×C23×C4⋊C8—21no—
26C2×C23×#104—21no—
27C2×C23×#104—22no—
28C4×M16×C23C2211yescharacteristic
29C43×C23C2211yescharacteristic
30C23×#339—21no—
31C23×D4⋊C8—21no—
32C23×D4⋊C8—21no—
33C2×#41×C23—21no—
34C2×#41×C23D411yescharacteristic
35C2×SD16×C23—21no—
36C2×D8×C23—21no—
37C2×M16×C23—21no—
38C2×M16×C23D412yes—
39C22×C8×C23—22no—
40C23×C4⋊Q8C2×C411yescharacteristic
41C23×#127C2×C411yescharacteristic
42C23×#124—21no—
43C23×#121—21no—
44C23×#121—21no—
45C4×Q8×C23C2×C411yescharacteristic
46C4×Q8×C23D411yescharacteristic
47C4×D4×C23—21no—
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23—21no—
51C4×D4×C23C2×C411yescharacteristic
52C4×D4×C23D411yescharacteristic
53C23×#117—21no—
54C23×#117D411yescharacteristic
55C2×C42×C23—21no—
56C2×C42×C23—21no—
57C2×C42×C23—21no—
58C2×C42×C23—21no—
59C2×C42×C23C2311yescharacteristic
60C23×#107—21no—
61C23×#106—21no—
62C23×C4⋊C8—22no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×C4⋊C8C2×C411yescharacteristic
66C23×C4⋊C8C2×C411yescharacteristic
67C23×C4⋊C8D411yescharacteristic
68C23×C4⋊C8D411yescharacteristic
69C23×#104—24no—
70C23×#104—44no—
71C23×#103—21no—
72C23×#102—21no—
73C23×#102—21no—
74C23×#102—21no—
75C23×C22⋊C8—22no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×#97—22no—
79C23×#97—21no—
80C4×C8×C23—21no—
81C4×C8×C23—21no—
82C4×C8×C23—21no—
83#41×C23—22no—
84#41×C23—41no—
85#41×C23—42no—
86#41×C23—42no—
87C2×Q8×C23—21no—
88C2×Q8×C23C22⋊C411yescharacteristic
89C2×D4×C23—21no—
90C2×D4×C23—21no—
91C2×D4×C23—21no—
92C2×D4×C23—21no—
93C2×D4×C23—41no—
94C2×D4×C23C22⋊C411yescharacteristic
95C22×C4×C23—21no—
96C22×C4×C23—21no—
97C22×C4×C23—21no—
98C22×C4×C23—41no—
99C22×C4×C23—41no—
100C22×C4×C23C2×D411yescharacteristic
101C22×C4×C23C2×D411yescharacteristic
102C22×C4×C23#4111yescharacteristic
103SD16×C23—42no—
104D8×C23—42no—
105M16×C23—22no—
106M16×C23—41no—
107M16×C23—42no—
108C2×C8×C23—21no—
109C2×C8×C23—21no—
110C2×C8×C23—22no—
111C2×C8×C23—21no—
112C2×C8×C23—21no—
113C2×C8×C23—41no—
114C2×C8×C23—82no—
115C4⋊C4×C23—21no—
116C4⋊C4×C23—21no—
117C4⋊C4×C23—21no—
118C4⋊C4×C23—21no—
119C4⋊C4×C23C22⋊C411yescharacteristic
120C4⋊C4×C23C22⋊C411yescharacteristic
121C22⋊C4×C23—21no—
122C22⋊C4×C23—21no—
123C22⋊C4×C23—21no—
124C22⋊C4×C23—41no—
125C22⋊C4×C23—41no—
126C42×C23—21no—
127C42×C23—21no—
128C42×C23—21no—
129C42×C23—22no—
130C42×C23—22no—
131C42×C23—22no—
132C42×C23—41no—
133C42×C23—41no—
134C42×C23—41no—
135C42×C23C22×C411yescharacteristic
136C42×C23C2×D411yescharacteristic
137C42×C23C2×D411yescharacteristic
138C42×C23#4111yescharacteristic
139#8969C2311yesmaximal, characteristic, direct product factor, Sylow
140C23×C23—21no—
141C23×C23—41no—
142C23×C23#12111yescharacteristic
143Q8×C23—21no—
144Q8×C23—21no—
145Q8×C23—42no—
146D4×C23—21no—
147D4×C23—42no—
148D4×C23—42no—
149D4×C23—41no—
150D4×C23—41no—
151D4×C23—41no—
152D4×C23—41no—
153D4×C23—82no—
154D4×C23#10412yescomplemented
155C2×C4×C23—21no—
156C2×C4×C23—21no—
157C2×C4×C23—21no—
158C2×C4×C23—21no—
159C2×C4×C23—21no—
160C2×C4×C23—21no—
161C2×C4×C23—21no—
162C2×C4×C23—21no—
163C2×C4×C23—21no—
164C2×C4×C23—42no—
165C2×C4×C23—42no—
166C2×C4×C23—42no—
167C2×C4×C23—42no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—81no—
174C2×C4×C23—81no—
175C2×C4×C23C2×C22⋊C411yescharacteristic
176C2×C4×C23C4×D411yescharacteristic
177C2×C4×C23C4×D411yescharacteristic
178C2×C4×C23#12011yescharacteristic
179C2×C4×C23#12111yescharacteristic
180C2×C4×C23#12311yescharacteristic
181C8×C23—42no—
182C8×C23—41no—
183C8×C23—41no—
184C8×C23—41no—
185C8×C23—42no—
186C8×C23—41no—
187#3106C2×C2311yescharacteristic
188#2990C2×C2311yescharacteristic, complemented
189#1962C2×C2311yescharacteristic, complemented
190C4×#104C2×C2311yescharacteristic, complemented
191#1484C2×C2311yescharacteristic, complemented
192#1299C2×C2311yescharacteristic, complemented
193#1276C2×C2311yescharacteristic, complemented
194C22×C23—21no—
195C22×C23—22no—
196C22×C23—21no—
197C22×C23—21no—
198C22×C23—41no—
199C22×C23—41no—
200C22×C23—81no—
201C22×C23—81no—
202C22×C23—81no—
203C22×C23#38611yescharacteristic
204C4×C23—21no—
205C4×C23—21no—
206C4×C23—21no—
207C4×C23—44no—
208C4×C23—41no—
209C4×C23—41no—
210C4×C23—41no—
211C4×C23—81no—
212C4×C23C2×#10411yescharacteristic
213C4×C23#42111yescharacteristic
214C4×C23#44311yescharacteristic
215C4×C23#44411yescharacteristic
216C4×C23#45311yescharacteristic
217C4×C23#45411yescharacteristic
218#532C22×C2311yescharacteristic
219C4×#41C22×C2311yescharacteristic
220#482—21no—
221#477C4×C2311yescharacteristic, complemented
222#476C4×C2311yescharacteristic, complemented
223#460C4×C2311yescharacteristic, complemented
224#459C4×C2311yescharacteristic, complemented
225#449—21no—
226#446—21no—
227C4×SD16—21no—
228C4×D8—21no—
229#435—22no—
230C8×D4—21no—
231#432—21no—
232#431C22×C2311yescharacteristic
233#431C22×C2311yescharacteristic
234#423C22×C2311yescharacteristic
235C2×C4⋊C8—21no—
236C2×#104—21no—
237C2×#104—22no—
238C4×M16C22×C2311yescharacteristic
239C43C22×C2311yescharacteristic
240#339—21no—
241D4⋊C8—21no—
242D4⋊C8—21no—
243C2×C23—41no—
244C2×C23—42no—
245C2×C23—161no—
246C2×C23#187111yescharacteristic
247C2×C23#187611yescharacteristic
248C2×C23#188311yescharacteristic
249C2×#41—21no—
250C2×#41D4×C2311yescharacteristic
251C2×SD16—21no—
252C2×D8—21no—
253C2×M16—21no—
254C2×M16D4×C2312yes—
255C22×C8—22no—
256C4⋊Q8C2×C4×C2311yescharacteristic
257#127C2×C4×C2311yescharacteristic
258#124—21no—
259#121—21no—
260#121—21no—
261C4×Q8C2×C4×C2311yescharacteristic
262C4×Q8D4×C2311yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4C2×C4×C2311yescharacteristic
268C4×D4D4×C2311yescharacteristic
269#117—21no—
270#117D4×C2311yescharacteristic
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C23×C2311yescharacteristic
276#107—21no—
277#106—21no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C2×C4×C2311yescharacteristic
282C4⋊C8C2×C4×C2311yescharacteristic
283C4⋊C8D4×C2311yescharacteristic
284C4⋊C8D4×C2311yescharacteristic
285#104—24no—
286#104—44no—
287#103—21no—
288#102—21no—
289#102—21no—
290#102—21no—
291C22⋊C8—22no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294#97—22no—
295#97—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C23#896911yescharacteristic, direct product factor, Sylow
300#41—22no—
301#41—41no—
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×Q8C22⋊C4×C2311yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4C22⋊C4×C2311yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×D4×C2311yescharacteristic
318C22×C4C2×D4×C2311yescharacteristic
319C22×C4#41×C2311yescharacteristic
320SD16—42no—
321D8—42no—
322M16—22no—
323M16—41no—
324M16—42no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—82no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4C22⋊C4×C2311yescharacteristic
337C4⋊C4C22⋊C4×C2311yescharacteristic
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42C22×C4×C2311yescharacteristic
353C42C2×D4×C2311yescharacteristic
354C42C2×D4×C2311yescharacteristic
355C42#41×C2311yescharacteristic
356C23—21no—
357C23—41no—
358C23C23×#12111yescharacteristic
359Q8—21no—
360Q8—21no—
361Q8—42no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—82no—
370D4C23×#10412yescomplemented
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4C2×C22⋊C4×C2311yescharacteristic
392C2×C4C4×D4×C2311yescharacteristic
393C2×C4C4×D4×C2311yescharacteristic
394C2×C4C23×#12011yescharacteristic
395C2×C4C23×#12111yescharacteristic
396C2×C4C23×#12311yescharacteristic
397C8—42no—
398C8—41no—
399C8—41no—
400C8—41no—
401C8—42no—
402C8—41no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—41no—
408C22—41no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C23×#38611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—81no—
421C4C2×C23×#10411yescharacteristic
422C4C23×#42111yescharacteristic
423C4C23×#44311yescharacteristic
424C4C23×#44411yescharacteristic
425C4C23×#45311yescharacteristic
426C4C23×#45411yescharacteristic
427C2—41no—
428C2—42no—
429C2—161no—
430C2C23×#187111yescharacteristic
431C2C23×#187611yescharacteristic
432C2C23×#188311yescharacteristic
subgroupG/H
centerC2×C4×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
socleC22×C23#386
prime radicalC23×#3106C2
LCSG ▷ C42 ▷ 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×#2990C2semidirect
3. C23×#1962C2semidirect
4. C4×C23×#104C2semidirect
5. C23×#1484C2semidirect
6. C23×#1299C2semidirect
7. C23×#1276C2semidirect
11. C23×#477C4semidirect
12. C23×#476C4semidirect
13. C23×#460C4semidirect
14. C23×#459C4semidirect
139. #8969C23direct (N × H)
154. D4×C23#104semidirect
188. #2990C2×C23semidirect
189. #1962C2×C23semidirect
190. C4×#104C2×C23semidirect
191. #1484C2×C23semidirect
192. #1299C2×C23semidirect
193. #1276C2×C23semidirect
221. #477C4×C23semidirect
222. #476C4×C23semidirect
223. #460C4×C23semidirect
224. #459C4×C23semidirect
299. C23#8969direct (N × H)
370. D4C23×#104semidirect

Maximal subgroups

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

subgroupindex
C23×#31062
C23×#29902
C23×#19622
C4×C23×#1042
C23×#14842
C23×#12992
C23×#12762
#896923