Finite Group Databasebeta

← D23×#1271 #19212138

Subgroups — D23×#1271 #19212138

Order 5888.

714 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D23×#389C211yesmaximal, characteristic
2D23×#336C211yesmaximal, characteristic
3D23×#336C211yesmaximal, characteristic
4#15671221C211yesmaximal, characteristic, complemented
5#15671220C211yesmaximal, characteristic, complemented
6#15671219C211yesmaximal, characteristic, complemented
7C23×#1271C211yesmaximal, characteristic, complemented
8C22×C8×D23—21no—
9C22×C8×D23—21no—
10C2×C4⋊C4×D23—21no—
11C2×C4⋊C4×D23C411yescharacteristic
12C2×C4⋊C4×D23C411yescharacteristic
13C2×C42×D23C2211yescharacteristic
14#13092316C411yescharacteristic
15#13092315C2211yescharacteristic
16#13092314—21no—
17#13092313C411yescharacteristic
18#13092312—21no—
19#95×D23—21no—
20#13092172C2211yescharacteristic
21#13092172C2211yescharacteristic
22#13092171—21no—
23#13092171—21no—
24C23×#389C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×C4×D23—21no—
28C22×C4×D23—21no—
29C22×C4×D23C2×C411yescharacteristic
30C22×C4×D23D411yescharacteristic
31C22×C4×D23D411yescharacteristic
32C22×C8×C23—21no—
33C22×C8×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23C2×C411yescharacteristic
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C42×C23C2311yescharacteristic
38C2×C4⋊Dic23—21no—
39C2×C4⋊Dic23C2×C411yescharacteristic
40C2×C4⋊Dic23C2×C411yescharacteristic
41C2×Dic23⋊C4—21no—
42C2×Dic23⋊C4—21no—
43C2×Dic23⋊C4—21no—
44C2×Dic23⋊C4—21no—
45C2×C4×Dic23C2×C411yescharacteristic
46C2×C4×Dic23D411yescharacteristic
47C2×C4×Dic23D411yescharacteristic
48C22×C23⋊C8—21no—
49C22×C23⋊C8—21no—
50C2×C8×D23—24no—
51C2×C8×D23—24no—
52C2×C8×D23—42no—
53C2×C8×D23—42no—
54C2×C8×D23—42no—
55C2×C8×D23—42no—
56C4⋊C4×D23—22no—
57C4⋊C4×D23—42no—
58C4⋊C4×D23—42no—
59C4⋊C4×D23—42no—
60C4⋊C4×D23C814yescomplemented
61C42×D23—42no—
62C23×#95—21no—
63C23⋊#95—21no—
64C23⋊#95—21no—
65C23⋊#95—21no—
66C23×D23C22⋊C411yescharacteristic
67C22×C4×C23—21no—
68C22×C4×C23—21no—
69C22×C4×C23C22×C411yescharacteristic
70C22×C4×C23C2×D411yescharacteristic
71C22×C4×C23C2×D411yescharacteristic
72C22×Dic23—21no—
73C22×Dic23—21no—
74C22×Dic23C22⋊C411yescharacteristic
75C22×Dic23C22⋊C411yescharacteristic
76C22×Dic23C22⋊C411yescharacteristic
77C2×C4×D23—22no—
78C2×C4×D23—22no—
79C2×C4×D23—22no—
80C2×C4×D23—22no—
81C2×C4×D23—22no—
82C2×C4×D23—22no—
83C2×C4×D23—24no—
84C2×C4×D23—42no—
85C2×C4×D23—42no—
86C2×C4×D23—42no—
87C2×C4×D23—42no—
88C2×C4×D23—42no—
89C2×C4×D23C2×C812yes—
90C2×C4×D23M1612yes—
91C2×C4×D23D812yes—
92C2×C4×D23SD1612yes—
93C2×C4×D23SD1612yes—
94C2×C4×D23Q1612yes—
95C2×C8×C23—22no—
96C2×C8×C23—22no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C2×C8×C23—41no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23C2×C812yescomplemented
106C42×C23—41no—
107C4⋊Dic23—21no—
108C4⋊Dic23—42no—
109C4⋊Dic23—41no—
110C4⋊Dic23C2×C812yescomplemented
111Dic23⋊C4—24no—
112Dic23⋊C4—42no—
113Dic23⋊C4—41no—
114Dic23⋊C4—41no—
115Dic23⋊C4—42no—
116C4×Dic23—41no—
117C4×Dic23—41no—
118C4×Dic23—41no—
119C2×C23⋊C8—22no—
120C2×C23⋊C8—22no—
121C2×C23⋊C8—41no—
122C2×C23⋊C8—41no—
123C2×C23⋊C8—41no—
124C2×C23⋊C8—41no—
125C8×D23—48no—
126C8×D23—48no—
127C2×#1271—231nomaximal, Sylow
128C23×C23C2×C22⋊C411yescharacteristic
129C22×D23C22⋊C812yes—
130C22×D23#9912yes—
131C22×D23#10112yes—
132C22×D23#10212yes—
133C22×D23#10312yes—
134C22×D23#10412yes—
135C22×D23#10412yes—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—22no—
143C2×C4×C23—41no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23C22×C811yescharacteristic
149C2×C4×C23C2×M1611yescharacteristic
150C2×C4×C23C2×D811yescharacteristic
151C2×C4×C23C2×SD1611yescharacteristic
152C2×C4×C23C2×SD1611yescharacteristic
153C2×C4×C23C2×Q1611yescharacteristic
154C2×Dic23—21no—
155C2×Dic23—21no—
156C2×Dic23—21no—
157C2×Dic23—21no—
158C2×Dic23—21no—
159C2×Dic23—21no—
160C2×Dic23—22no—
161C2×Dic23—41no—
162C2×Dic23—41no—
163C2×Dic23—41no—
164C2×Dic23—41no—
165C2×Dic23—41no—
166C2×Dic23C22⋊C812yes—
167C2×Dic23#10212yes—
168C2×Dic23#10312yes—
169C4×D23—24no—
170C4×D23—24no—
171C4×D23—24no—
172C4×D23—24no—
173C4×D23—24no—
174C4×D23—24no—
175C4×D23—48no—
176C4×D23—84no—
177C8×C23—42no—
178C8×C23—42no—
179C23⋊C8—42no—
180C23⋊C8—42no—
181C2×#389—231no—
182C2×#336—231no—
183C2×#336—231no—
184#1271—231no—
185#1271—231no—
186#1271—231no—
187#1271D2311yescharacteristic, direct product factor
188C22×C23C2×C22⋊C811yescharacteristic
189C22×C23C2×#9911yescharacteristic
190C22×C23C2×#10111yescharacteristic
191C22×C23C2×#10211yescharacteristic
192C22×C23C2×#10311yescharacteristic
193C22×C23C2×#10411yescharacteristic
194C22×C23C2×#10411yescharacteristic
195C2×D23#32414yes—
196C2×D23D4⋊C814yes—
197C2×D23Q8⋊C814yes—
198C2×D23#32714yes—
199C2×D23#32814yes—
200C2×D23#33014yes—
201C2×D23#33214yes—
202C4×C23—21no—
203C4×C23—21no—
204C4×C23—21no—
205C4×C23—21no—
206C4×C23—21no—
207C4×C23—21no—
208C4×C23—42no—
209C4×C23—81no—
210Dic23—22no—
211Dic23—22no—
212Dic23—22no—
213Dic23—42no—
214Dic23—81no—
215C23×C8—461no—
216C23×C8—461no—
217C22×C4⋊C4—231no—
218C22×C4⋊C4—231no—
219C22×C4⋊C4—461no—
220C22×C42—231no—
221#389—231no—
222#389—231no—
223#389—231no—
224#389—461no—
225#389—461no—
226#389C2×D2311yescharacteristic
227C2×#95—461no—
228#336—231no—
229#336—231no—
230#336—461no—
231#336—461no—
232#336C2×D2311yescharacteristic
233#336C2×D2311yescharacteristic
234C2×C23C2×#32411yescharacteristic
235C2×C23C2×D4⋊C811yescharacteristic
236C2×C23C2×Q8⋊C811yescharacteristic
237C2×C23C2×#32711yescharacteristic
238C2×C23C2×#32811yescharacteristic
239C2×C23C2×#33011yescharacteristic
240C2×C23C2×#33211yescharacteristic
241D23#127118yesdirect product factor
242C23×C4—231no—
243C23×C4—231no—
244C23×C4—231no—
245C23×C4—461no—
246C23×C4—461no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—464no—
250C22×C8—464no—
251C22×C8—461no—
252C22×C8—461no—
253C22×C8—922no—
254C22×C8—922no—
255C22×C8—922no—
256C22×C8—922no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—231no—
259C2×C4⋊C4—234no—
260C2×C4⋊C4—231no—
261C2×C4⋊C4—462no—
262C2×C4⋊C4—461no—
263C2×C4⋊C4—461no—
264C2×C4⋊C4—461no—
265C2×C4⋊C4—461no—
266C2×C4⋊C4—461no—
267C2×C4⋊C4—922no—
268C2×C4⋊C4—922no—
269C2×C4⋊C4—922no—
270C2×C4⋊C4C4×D2311yescharacteristic
271C2×C4⋊C4C4×D2311yescharacteristic
272C2×C42—231no—
273C2×C42—231no—
274C2×C42—231no—
275C2×C42—922no—
276C2×C42C22×D2311yescharacteristic
277#95—21no—
278#95—461no—
279#95—461no—
280#95—461no—
281C23C2×#127111yescharacteristic, complemented, Sylow
282C24—231no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—231no—
286C22×C4—231no—
287C22×C4—231no—
288C22×C4—232no—
289C22×C4—232no—
290C22×C4—232no—
291C22×C4—232no—
292C22×C4—232no—
293C22×C4—232no—
294C22×C4—462no—
295C22×C4—462no—
296C22×C4—462no—
297C22×C4—462no—
298C22×C4—462no—
299C22×C4—462no—
300C22×C4—464no—
301C22×C4—461no—
302C22×C4—461no—
303C22×C4—922no—
304C22×C4—922no—
305C22×C4—922no—
306C22×C4—922no—
307C22×C4—922no—
308C22×C4C2×C4×D2311yescharacteristic
309C22×C4D4×D2311yescharacteristic
310C22×C4D4×D2311yescharacteristic
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—462no—
318C2×C8—462no—
319C2×C8—928no—
320C2×C8—928no—
321C2×C8—921no—
322C2×C8—921no—
323C2×C8—921no—
324C2×C8—921no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—232no—
330C4⋊C4—464no—
331C4⋊C4—461no—
332C4⋊C4—921no—
333C4⋊C4—922no—
334C4⋊C4—922no—
335C4⋊C4—922no—
336C4⋊C4—921no—
337C4⋊C4—921no—
338C4⋊C4C8×D2312yescomplemented
339C42—41no—
340C42—921no—
341C42—921no—
342C42—921no—
343C23—232no—
344C23—232no—
345C23—232no—
346C23—232no—
347C23—232no—
348C23—232no—
349C23—232no—
350C23C22⋊C4×D2311yescharacteristic
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—232no—
364C2×C4—232no—
365C2×C4—232no—
366C2×C4—464no—
367C2×C4—464no—
368C2×C4—464no—
369C2×C4—464no—
370C2×C4—464no—
371C2×C4—464no—
372C2×C4—461no—
373C2×C4—461no—
374C2×C4—461no—
375C2×C4—461no—
376C2×C4—461no—
377C2×C4—461no—
378C2×C4—462no—
379C2×C4—928no—
380C2×C4—921no—
381C2×C4—921no—
382C2×C4—921no—
383C2×C4—921no—
384C2×C4—921no—
385C2×C4—1844no—
386C2×C4C2×C8×D2311yescharacteristic
387C2×C4M16×D2311yescharacteristic
388C2×C4D8×D2311yescharacteristic
389C2×C4SD16×D2311yescharacteristic
390C2×C4SD16×D2311yescharacteristic
391C2×C4Q16×D2311yescharacteristic
392C8—42no—
393C8—42no—
394C8—922no—
395C8—922no—
396C22—234no—
397C22—234no—
398C22—234no—
399C22—234no—
400C22—234no—
401C22—234no—
402C22—234no—
403C22C22⋊C8×D2311yescharacteristic
404C22#99×D2311yescharacteristic
405C22#101×D2311yescharacteristic
406C22#102×D2311yescharacteristic
407C22#103×D2311yescharacteristic
408C22#104×D2311yescharacteristic
409C22#104×D2311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—462no—
419C4—462no—
420C4—462no—
421C4—922no—
422C4—1841no—
423C2—238no—
424C2D23×#32411yescharacteristic
425C2D23×D4⋊C811yescharacteristic
426C2D23×Q8⋊C811yescharacteristic
427C2D23×#32711yescharacteristic
428C2D23×#32811yescharacteristic
429C2D23×#33011yescharacteristic
430C2D23×#33211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1271C2
socleC23×C23C2×C22⋊C4
hypercenter#1271D23
hypocenterC23C2×#1271
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #15671221C2semidirect
5. #15671220C2semidirect
6. #15671219C2semidirect
7. C23×#1271C2semidirect
60. C4⋊C4×D23C8semidirect
105. C4⋊C4×C23C2×C8semidirect
110. C4⋊Dic23C2×C8semidirect
187. #1271D23direct (N × H)
241. D23#1271direct (N × H)
281. C23C2×#1271semidirect
338. C4⋊C4C8×D23semidirect