Finite Group Databasebeta

← #19212225

Subgroups — #19212225

Order 5888.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15675859C211yesmaximal, characteristic
2#15673539C211yesmaximal, characteristic, complemented
3#15673141C211yesmaximal, characteristic, complemented
4#15671221C211yesmaximal, characteristic, complemented
5#15671219C211yesmaximal, characteristic, complemented
6#15671218C211yesmaximal, characteristic
7C23×#1270C211yesmaximal, characteristic
8C2×D4×Dic23C411yescharacteristic
9C2×#10625048—21no—
10C2×C8×Dic23—21no—
11#13092320C411yescharacteristic
12#13092315C2211yescharacteristic
13#13092313C411yescharacteristic
14#13092311—21no—
15#13092305—21no—
16C4⋊C4×Dic23C411yescharacteristic
17C22⋊C4×Dic23—21no—
18C42×Dic23C2211yescharacteristic
19#13092172C2211yescharacteristic
20#13092172C2211yescharacteristic
21#13092171—21no—
22#13092171—21no—
23#13092171—21no—
24C23×#390C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×D4×C23C2×C411yescharacteristic
28C23×Dic23—21no—
29C22×C8×C23—21no—
30C22×C8×C23—21no—
31C2×C4⋊C4×C23C2×C411yescharacteristic
32C2×C22⋊C4×C23—21no—
33C2×C22⋊C4×C23—21no—
34C2×C42×C23C2311yescharacteristic
35C2×C22⋊Dic23—21no—
36C2×C22⋊Dic23—21no—
37C2×C22⋊Dic23—21no—
38D4×Dic23—22no—
39D4×Dic23C814yescomplemented
40C2×C4⋊Dic23C2×C411yescharacteristic
41C2×C4⋊Dic23C2×C411yescharacteristic
42C2×Dic23⋊C4—21no—
43C2×Dic23⋊C4—21no—
44C2×C4×Dic23—21no—
45C2×C4×Dic23C2×C411yescharacteristic
46C2×C4×Dic23C2×C411yescharacteristic
47C2×C4×Dic23D411yescharacteristic
48C2×C4×Dic23D411yescharacteristic
49C2×C4×Dic23D411yescharacteristic
50C2×C4×Dic23D411yescharacteristic
51C22×C23⋊C8—21no—
52C22×C23⋊C8—21no—
53#10625048—42no—
54#10625048—42no—
55C23⋊#95—21no—
56C23⋊#95—21no—
57C23⋊#95—21no—
58C8×Dic23—24no—
59C24×C23—21no—
60C2×D4×C23—21no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23C2×C812yes—
64C22×C4×C23—21no—
65C22×C4×C23C22×C411yescharacteristic
66C22×C4×C23C2×D411yescharacteristic
67C22×C4×C23C2×D411yescharacteristic
68C22×Dic23—21no—
69C22×Dic23—24no—
70C22×Dic23—21no—
71C22×Dic23—42no—
72C22×Dic23—42no—
73C22×Dic23C22⋊C411yescharacteristic
74C22×Dic23C22⋊C411yescharacteristic
75C22×Dic23C22⋊C411yescharacteristic
76C22×Dic23C22⋊C411yescharacteristic
77C2×C8×C23—22no—
78C2×C8×C23—41no—
79C2×C8×C23—41no—
80C2×C8×C23—41no—
81C2×C8×C23—41no—
82C2×C8×C23—41no—
83C4⋊C4×C23—41no—
84C22⋊C4×C23—41no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—41no—
87C22⋊C4×C23—41no—
88C42×C23—41no—
89C22⋊Dic23—24no—
90C22⋊Dic23—42no—
91C22⋊Dic23—42no—
92C4⋊Dic23—21no—
93C4⋊Dic23—41no—
94C4⋊Dic23C2×C812yes—
95Dic23⋊C4—42no—
96Dic23⋊C4—41no—
97Dic23⋊C4—41no—
98C4×Dic23—21no—
99C4×Dic23—21no—
100C4×Dic23—21no—
101C4×Dic23—21no—
102C4×Dic23—22no—
103C4×Dic23—41no—
104C4×Dic23—41no—
105C4×Dic23—42no—
106C4×Dic23C2×C812yes—
107C4×Dic23M1612yes—
108C2×C23⋊C8—22no—
109C2×C23⋊C8—41no—
110C2×C23⋊C8—41no—
111C2×C23⋊C8—41no—
112C2×C23⋊C8—41no—
113C2×C23⋊C8—41no—
114#7580—231nomaximal, Sylow
115C23×C23—22no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—41no—
119C23×C23—41no—
120C23×C23—41no—
121C23×C23—41no—
122C23×C23C2×C22⋊C411yescharacteristic
123D4×C23—42no—
124D4×C23—42no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23C22×C811yescharacteristic
135C2×C4×C23C2×M1611yescharacteristic
136C2×C4×C23#13511yescharacteristic
137C2×C4×C23#13511yescharacteristic
138C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
140C2×Dic23—21no—
141C2×Dic23—21no—
142C2×Dic23—21no—
143C2×Dic23—21no—
144C2×Dic23—21no—
145C2×Dic23—21no—
146C2×Dic23—22no—
147C2×Dic23—21no—
148C2×Dic23—21no—
149C2×Dic23—21no—
150C2×Dic23—48no—
151C2×Dic23—41no—
152C2×Dic23—41no—
153C2×Dic23—41no—
154C2×Dic23—41no—
155C2×Dic23—41no—
156C2×Dic23C22⋊C812yes—
157C2×Dic23C22⋊C812yes—
158C2×Dic23#10412yes—
159C8×C23—42no—
160C8×C23—81no—
161C23⋊C8—42no—
162C23⋊C8—81no—
163#2293—231no—
164#1839—231no—
165#1744—231no—
166#1271—231no—
167#1271—231no—
168#1270—231no—
169#1270D2311yescharacteristic
170C22×C23—42no—
171C22×C23—42no—
172C22×C23—42no—
173C22×C23—81no—
174C22×C23—81no—
175C22×C23—81no—
176C22×C23—81no—
177C22×C23C2×C22⋊C811yescharacteristic
178C22×C23#41011yescharacteristic
179C22×C23#41311yescharacteristic
180C22×C23#41611yescharacteristic
181C22×C23#41811yescharacteristic
182C22×C23C2×#10411yescharacteristic
183C22×C23#42111yescharacteristic
184C4×C23—21no—
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—81no—
191Dic23—22no—
192Dic23—22no—
193Dic23—22no—
194Dic23—41no—
195Dic23—42no—
196Dic23—81no—
197C2×C4×D4—231no—
198C2×#117—461no—
199C2×C4×C8—461no—
200#390—231no—
201#390C2×D2311yescharacteristic
202#389—231no—
203#389—231no—
204#388—461no—
205#388—461no—
206C4×C4⋊C4—231no—
207C4×C22⋊C4—461no—
208C43—231no—
209#336—231no—
210#336—231no—
211#336—461no—
212#336—461no—
213#336—461no—
214#336C2×D2311yescharacteristic
215#336C2×D2311yescharacteristic
216C2×C23—82no—
217C2×C23#145811yescharacteristic
218C2×C23#147611yescharacteristic
219C2×C23#147811yescharacteristic
220C2×C23#150511yescharacteristic
221C2×C23#152111yescharacteristic
222C2×C23#153211yescharacteristic
223C2×C23#155311yescharacteristic
224C22×D4C4×D2311yescharacteristic
225C23×C4—461no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—461no—
229C22×C8—461no—
230C4×D4—234no—
231C4×D4—462no—
232#117—922no—
233#117—922no—
234C2×C4⋊C4—231no—
235C2×C4⋊C4—231no—
236C2×C4⋊C4—461no—
237C2×C4⋊C4—461no—
238C2×C4⋊C4C4×D2311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—461no—
242C2×C22⋊C4—461no—
243C2×C22⋊C4—461no—
244C2×C42—231no—
245C2×C42—231no—
246C2×C42—231no—
247C2×C42—231no—
248C2×C42—231no—
249C2×C42—231no—
250C2×C42—461no—
251C2×C42C22×D2311yescharacteristic
252C4×C8—464no—
253#95—461no—
254#95—461no—
255#95—461no—
256C23#758011yescharacteristic, complemented, Sylow
257C24—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4C8×D2312yes—
262C22×C4—21no—
263C22×C4—231no—
264C22×C4—231no—
265C22×C4—231no—
266C22×C4—231no—
267C22×C4—464no—
268C22×C4—461no—
269C22×C4—461no—
270C22×C4—922no—
271C22×C4—922no—
272C22×C4C2×C4×D2311yescharacteristic
273C22×C4D4×D2311yescharacteristic
274C22×C4D4×D2311yescharacteristic
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—462no—
282C2×C8—921no—
283C2×C8—921no—
284C2×C8—921no—
285C2×C8—921no—
286C2×C8—921no—
287C4⋊C4—41no—
288C4⋊C4—232no—
289C4⋊C4—461no—
290C4⋊C4—921no—
291C4⋊C4—922no—
292C4⋊C4—921no—
293C4⋊C4—921no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—464no—
299C22⋊C4—922no—
300C22⋊C4—922no—
301C42—41no—
302C42—232no—
303C42—232no—
304C42—462no—
305C42—461no—
306C42—461no—
307C42—461no—
308C42—461no—
309C42—921no—
310C42—921no—
311C42—922no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23C22⋊C4×D2311yescharacteristic
320D4—42no—
321D4—42no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—232no—
332C2×C4—232no—
333C2×C4—232no—
334C2×C4—461no—
335C2×C4—461no—
336C2×C4—461no—
337C2×C4—461no—
338C2×C4—461no—
339C2×C4—461no—
340C2×C4—462no—
341C2×C4—461no—
342C2×C4—461no—
343C2×C4—461no—
344C2×C4—928no—
345C2×C4—921no—
346C2×C4—921no—
347C2×C4—921no—
348C2×C4—921no—
349C2×C4—921no—
350C2×C4C2×C8×D2311yescharacteristic
351C2×C4M16×D2311yescharacteristic
352C2×C4#1062508211yescharacteristic
353C2×C4#1062508311yescharacteristic
354C2×C4#1062508511yescharacteristic
355C2×C4#1062508711yescharacteristic
356C8—42no—
357C8—81no—
358C8—922no—
359C8—1841no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C22⋊C8×D2311yescharacteristic
368C22#1309238911yescharacteristic
369C22#1309239811yescharacteristic
370C22#1309241011yescharacteristic
371C22#1309242011yescharacteristic
372C22#104×D2311yescharacteristic
373C22#1309243211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—81no—
381C4—462no—
382C4—462no—
383C4—462no—
384C4—921no—
385C4—922no—
386C4—1841no—
387C2—82no—
388C2#1567170811yescharacteristic
389C2#1567177611yescharacteristic
390C2#1567178611yescharacteristic
391C2#1567190311yescharacteristic
392C2#1567197111yescharacteristic
393C2#1567201411yescharacteristic
394C2#1567209911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D23
commutatorC2×C4×C23C22×C8
FrattiniC2×C42C22×D23
FittingC23×#1270C2
socleC23×C23C2×C22⋊C4
hypercenter#1270D23
hypocenterC23#7580
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. #15673539C2semidirect
3. #15673141C2semidirect
4. #15671221C2semidirect
5. #15671219C2semidirect
39. D4×Dic23C8semidirect
256. C23#7580semidirect