Finite Group Databasebeta

← D23×#1270 #19212135

Subgroups — D23×#1270 #19212135

Order 5888.

994 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D23×#390C211yesmaximal, characteristic
2D23×#336C212yesmaximal, complemented
3#15671218C211yesmaximal, characteristic, complemented
4#15671217C212yesmaximal, complemented
5C23×#1270C211yesmaximal, characteristic, complemented
6C22×D4×D23C411yescharacteristic
7C22×C8×D23—22no—
8C2×C4⋊C4×D23C411yescharacteristic
9C2×C22⋊C4×D23—22no—
10C2×C42×D23C2211yescharacteristic
11#13092320C411yescharacteristic
12#13092319C2211yescharacteristic
13#13092318C411yescharacteristic
14#13092317—22no—
15#13092172C2212yescomplemented
16#13092171—22no—
17C23×#390C2211yescharacteristic
18C23×#336C2212yescomplemented
19C24×D23—21no—
20C22×D4×C23C2×C411yescharacteristic
21C22×C22⋊D23—21no—
22C2×D4×D23—22no—
23C2×D4×D23—42no—
24C2×D4×D23—42no—
25C2×D4×D23C814yescomplemented
26C22×D92C2×C411yescharacteristic
27C22×C4×D23—21no—
28C22×C4×D23C2×C411yescharacteristic
29C22×C4×D23D412yes—
30C22×C8×C23—22no—
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C22⋊C4×C23—22no—
33C2×C42×C23C2311yescharacteristic
34C2×C22⋊Dic23—22no—
35C2×(C2×D23)⋊C4—22no—
36C2×(C2×D23)⋊C4—22no—
37C2×C4⋊Dic23C2×C411yescharacteristic
38C2×Dic23⋊C4—21no—
39C2×C4×Dic23C2×C411yescharacteristic
40C2×C4×Dic23D412yes—
41C22×C23⋊C8—22no—
42C2×C8×D23—28no—
43C2×C8×D23—44no—
44C2×C8×D23—44no—
45C4⋊C4×D23—42no—
46C22⋊C4×D23—44no—
47C22⋊C4×D23—44no—
48C42×D23—42no—
49C24×C23—21no—
50C23×D23—21no—
51C23×D23—24no—
52C23×D23—42no—
53C23×D23—42no—
54C23×D23—44no—
55C23×D23—44no—
56C23×D23C22⋊C411yescharacteristic
57C2×D4×C23—21no—
58C2×D4×C23—41no—
59C2×D4×C23—41no—
60C2×D4×C23C2×C812yescomplemented
61C22×C4×C23—21no—
62C22×C4×C23C22×C411yescharacteristic
63C22×C4×C23C2×D412yes—
64C2×C22⋊D23—24no—
65C2×C22⋊D23—42no—
66C2×C22⋊D23—42no—
67C22×Dic23—21no—
68C22×Dic23C22⋊C411yescharacteristic
69C22×Dic23C22⋊C412yes—
70D4×D23—48no—
71D4×D23—48no—
72C2×D92—21no—
73C2×D92—41no—
74C2×D92—41no—
75C2×D92C2×C812yescomplemented
76C2×C4×D23—24no—
77C2×C4×D23—22no—
78C2×C4×D23—22no—
79C2×C4×D23—24no—
80C2×C4×D23—42no—
81C2×C4×D23—44no—
82C2×C4×D23C2×C812yes—
83C2×C4×D23M1612yes—
84C2×C4×D23D814yes—
85C2×C4×D23SD1614yes—
86C2×C8×C23—24no—
87C2×C8×C23—42no—
88C2×C8×C23—42no—
89C4⋊C4×C23—41no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C42×C23—41no—
93C22⋊Dic23—44no—
94(C2×D23)⋊C4—42no—
95(C2×D23)⋊C4—42no—
96(C2×D23)⋊C4—44no—
97C4⋊Dic23—41no—
98Dic23⋊C4—42no—
99C4×Dic23—41no—
100C4×Dic23—42no—
101C2×C23⋊C8—24no—
102C2×C23⋊C8—42no—
103C2×C23⋊C8—42no—
104C8×D23—416no—
105C2×#1270—231nomaximal, Sylow
106C23×C23—22no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—42no—
110C23×C23—42no—
111C23×C23C2×C22⋊C411yescharacteristic
112C22×D23—22no—
113C22×D23—41no—
114C22×D23—41no—
115C22×D23—42no—
116C22×D23—42no—
117C22×D23—48no—
118C22×D23—416no—
119C22×D23—88no—
120C22×D23—84no—
121C22×D23—84no—
122C22×D23C22⋊C812yes—
123C22×D23#9912yes—
124C22×D23#10012yes—
125C22×D23#10214yes—
126C22×D23#10414yes—
127D4×C23—42no—
128D4×C23—42no—
129C2×C4×C23—22no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—22no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23C22×C811yescharacteristic
136C2×C4×C23C2×M1611yescharacteristic
137C2×C4×C23C2×D812yes—
138C2×C4×C23C2×SD1612yes—
139C22⋊D23—48no—
140C2×Dic23—22no—
141C2×Dic23—21no—
142C2×Dic23—21no—
143C2×Dic23—22no—
144C2×Dic23—41no—
145C2×Dic23—42no—
146C2×Dic23C22⋊C812yes—
147C2×Dic23#10214yes—
148D92—42no—
149D92—42no—
150C4×D23—24no—
151C4×D23—24no—
152C4×D23—28no—
153C4×D23—28no—
154C4×D23—84no—
155C8×C23—44no—
156C23⋊C8—44no—
157C2×#390—231no—
158C2×#336—232no—
159#1270—231no—
160#1270—232no—
161#1270D2311yescharacteristic, direct product factor
162C22×C23—42no—
163C22×C23—44no—
164C22×C23—81no—
165C22×C23—82no—
166C22×C23—81no—
167C22×C23C2×C22⋊C811yescharacteristic
168C22×C23C2×#9911yescharacteristic
169C22×C23C2×#10011yescharacteristic
170C22×C23C2×#10212yes—
171C22×C23C2×#10412yes—
172C2×D23—42no—
173C2×D23—44no—
174C2×D23—81no—
175C2×D23—82no—
176C2×D23—81no—
177C2×D23—816no—
178C2×D23#32314yes—
179C2×D23D4⋊C818yes—
180C2×D23#32718yes—
181C2×D23#32914yes—
182C2×D23#33114yes—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—22no—
186C4×C23—22no—
187C4×C23—81no—
188Dic23—22no—
189Dic23—24no—
190Dic23—81no—
191C23×D4—231no—
192C23×C8—462no—
193C22×C4⋊C4—231no—
194C22×C22⋊C4—462no—
195C22×C42—231no—
196#390—231no—
197#390—231no—
198#390—231no—
199#390—462no—
200#390C2×D2311yescharacteristic
201#336—232no—
202#336—462no—
203#336C2×D2312yescomplemented
204C2×C23—82no—
205C2×C23C2×#32311yescharacteristic
206C2×C23C2×D4⋊C812yes—
207C2×C23C2×#32712yes—
208C2×C23C2×#32911yescharacteristic
209C2×C23C2×#33111yescharacteristic
210D23—82no—
211D23#127018yesdirect product factor
212C25—461no—
213C22×D4—234no—
214C22×D4—231no—
215C22×D4—462no—
216C22×D4—461no—
217C22×D4—922no—
218C22×D4—922no—
219C22×D4C4×D2311yescharacteristic
220C23×C4—231no—
221C23×C4—232no—
222C23×C4—461no—
223C22×C8—22no—
224C22×C8—468no—
225C22×C8—462no—
226C22×C8—924no—
227C22×C8—924no—
228C2×C4⋊C4—231no—
229C2×C4⋊C4—461no—
230C2×C4⋊C4—922no—
231C2×C4⋊C4C4×D2311yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—462no—
234C2×C22⋊C4—462no—
235C2×C22⋊C4—462no—
236C2×C22⋊C4—924no—
237C2×C22⋊C4—924no—
238C2×C42—231no—
239C2×C42—232no—
240C2×C42—922no—
241C2×C42C22×D2311yescharacteristic
242C23C2×#127011yescharacteristic, complemented, Sylow
243C24—21no—
244C24—231no—
245C24—464no—
246C24—461no—
247C24—924no—
248C24—922no—
249C24—922no—
250C24—924no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—232no—
255C2×D4—464no—
256C2×D4—461no—
257C2×D4—921no—
258C2×D4—921no—
259C2×D4—928no—
260C2×D4—928no—
261C2×D4—922no—
262C2×D4—922no—
263C2×D4C8×D2312yescomplemented
264C22×C4—21no—
265C22×C4—232no—
266C22×C4—231no—
267C22×C4—232no—
268C22×C4—232no—
269C22×C4—234no—
270C22×C4—234no—
271C22×C4—464no—
272C22×C4—462no—
273C22×C4—462no—
274C22×C4—464no—
275C22×C4—461no—
276C22×C4—924no—
277C22×C4—922no—
278C22×C4C2×C4×D2311yescharacteristic
279C22×C4D4×D2312yes—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—464no—
284C2×C8—9216no—
285C2×C8—922no—
286C2×C8—922no—
287C4⋊C4—41no—
288C4⋊C4—921no—
289C4⋊C4—922no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—924no—
293C22⋊C4—924no—
294C22⋊C4—922no—
295C22⋊C4—922no—
296C42—41no—
297C42—922no—
298C42—921no—
299C23—22no—
300C23—41no—
301C23—41no—
302C23—42no—
303C23—42no—
304C23—232no—
305C23—232no—
306C23—232no—
307C23—234no—
308C23—234no—
309C23—462no—
310C23—928no—
311C23—9216no—
312C23—921no—
313C23—921no—
314C23—922no—
315C23—922no—
316C23—1848no—
317C23—1844no—
318C23—1844no—
319C23C22⋊C4×D2311yescharacteristic
320D4—42no—
321D4—42no—
322D4—922no—
323D4—922no—
324D4—928no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—232no—
332C2×C4—234no—
333C2×C4—468no—
334C2×C4—464no—
335C2×C4—464no—
336C2×C4—468no—
337C2×C4—461no—
338C2×C4—462no—
339C2×C4—461no—
340C2×C4—462no—
341C2×C4—921no—
342C2×C4—922no—
343C2×C4—1844no—
344C2×C4C2×C8×D2311yescharacteristic
345C2×C4M16×D2311yescharacteristic
346C2×C4D8×D2312yes—
347C2×C4SD16×D2312yes—
348C8—44no—
349C8—924no—
350C22—42no—
351C22—44no—
352C22—82no—
353C22—81no—
354C22—81no—
355C22—238no—
356C22—234no—
357C22—234no—
358C22—238no—
359C22—234no—
360C22—922no—
361C22—924no—
362C22—18416no—
363C22—1842no—
364C22—1841no—
365C22—1841no—
366C22C22⋊C8×D2311yescharacteristic
367C22#99×D2311yescharacteristic
368C22#100×D2311yescharacteristic
369C22#102×D2312yes—
370C22#104×D2312yes—
371C4—22no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—464no—
377C4—462no—
378C4—1841no—
379C2—82no—
380C2—238no—
381C2—1842no—
382C2D23×#32311yescharacteristic
383C2D23×D4⋊C812yes—
384C2D23×#32712yes—
385C2D23×#32911yescharacteristic
386C2D23×#33111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1270C2
socleC23×C23C2×C22⋊C4
hypercenter#1270D23
hypocenterC23C2×#1270
LCSG ▷ C2×C4×C23 ▷ C22×C23 ▷ C23
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. D23×#336C2semidirect
3. #15671218C2semidirect
4. #15671217C2semidirect
5. C23×#1270C2semidirect
15. #13092172C22semidirect
18. C23×#336C22semidirect
25. C2×D4×D23C8semidirect
60. C2×D4×C23C2×C8semidirect
75. C2×D92C2×C8semidirect
161. #1270D23direct (N × H)
203. #336C2×D23semidirect
211. D23#1270direct (N × H)
242. C23C2×#1270semidirect
263. C2×D4C8×D23semidirect