Finite Group Databasebeta

← #74003

Subgroups — #74003

Order 384.

880 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5#5212C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4957C212yesmaximal, complemented
9#4956C212yesmaximal, complemented
10#4665C212yesmaximal, complemented
11#4646C212yesmaximal, complemented
12#2311—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C2×#235—24no—
17C2×#235—24no—
18C2×#235C418yescomplemented
19C22×D12C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2212yes—
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2212yes—
27C2×#217C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2212yes—
35#892—24no—
36D3×C4⋊C4—24no—
37C3×#95C2212yes—
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C3⋊#95C2212yes—
41C22×#41—31no—
42C2×#117—31no—
43C22×C4⋊C4—31no—
44#391—32no—
45#390—32no—
46#386—32no—
47#386—32no—
48#380—31no—
49#380—31no—
50#380—31no—
51#380D311yescharacteristic, complemented
52C24×C3C2311yescharacteristic
53C23×D3C2311yescharacteristic
54C23×D3C2311yescharacteristic
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4C2×C414yescomplemented
58C22×C3×C4C2312yes—
59C22×C3×C4C2312yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67C2×C414yes—
67C2×#67C2×C414yescomplemented
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72#235—232no—
73C2×D12—22no—
74C2×D12—22no—
75C2×D12C2×C414yes—
76C2×C4×D3—22no—
77C2×C4×D3—22no—
78C2×C4×D3—24no—
79C2×C4×D3—42no—
80C2×C4×D3—42no—
81C2×C4×D3C2×C414yes—
82C2×C4×D3C2×C414yescomplemented
83C2×C4×D3D414yes—
84C2×C4×D3D414yes—
85C2×Dic6—22no—
86C2×Dic6—22no—
87C2×Dic6C2×C414yescomplemented
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C22⋊C4—42no—
91#217—42no—
92(C2×D3)⋊C4—24no—
93(C2×D3)⋊C4—42no—
94(C2×D3)⋊C4—42no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—24no—
98C4×Dic3—24no—
99C2×#41—38no—
100C2×#41—64no—
101C2×#41—64no—
102C22×Q8—31no—
103C22×D4—31no—
104C22×D4—31no—
105C22×D4—31no—
106C23×C4—31no—
107C23×C4—31no—
108C23×C4—32no—
109C23×C4C2×D311yescharacteristic
110#117—64no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—64no—
115C2×C4⋊C4C2×D311yescharacteristic
116C2×C4⋊C4C2×D311yescharacteristic
117C2×C22⋊C4—32no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4C2×D312yes—
122C2×C42—32no—
123#95—32no—
124#95—32no—
125#95—32no—
126#95C2×D312yes—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3C22×C412yes—
134C23×C3C2411yescharacteristic
135C22×D3—21no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3—21no—
139C22×D3—21no—
140C22×D3—21no—
141C22×D3—21no—
142C22×D3—21no—
143C22×D3—21no—
144C22×D3—21no—
145C22×D3C22×C412yes—
146C22×D3C22×C412yes—
147C22×D3C2×D412yes—
148C22×D3C2×D412yes—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—28no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4C22×C412yes—
159C2×C3×C4C22×C412yes—
160C2×C3×C4C2×D412yes—
161C2×C3×C4C2×D412yes—
162#67—28no—
163#67—28no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—22no—
171C2×Dic3—22no—
172C2×Dic3C22×C412yes—
173C2×Dic3C22×C412yes—
174C2×Dic3C2×D412yes—
175C2×Dic3C2×D412yes—
176D12—28no—
177C4×D3—28no—
178C4×D3—48no—
179C4×D3C22⋊C4116yescomplemented
180Dic6—28no—
181C24—31no—
182C24—31no—
183C24C22×D311yescharacteristic
184#41—632no—
185C2×Q8—34no—
186C2×Q8—62no—
187C2×Q8—62no—
188C2×D4—34no—
189C2×D4—34no—
190C2×D4—34no—
191C2×D4—62no—
192C2×D4—62no—
193C2×D4—62no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—62no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—31no—
200C22×C4—34no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—31no—
204C22×C4—34no—
205C22×C4—32no—
206C22×C4—32no—
207C22×C4—64no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—122no—
211C22×C4—122no—
212C22×C4C4×D314yescomplemented
213C22×C4C22×D312yes—
214C22×C4C22×D312yes—
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—64no—
220C4⋊C4—62no—
221C4⋊C4—62no—
222C22⋊C4—42no—
223C22⋊C4—64no—
224C22⋊C4—122no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C42—64no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3C23×C411yescharacteristic
236C22×C3C22×D411yescharacteristic
237C22×C3C22×D411yescharacteristic
238C22×C3H(5,𝔽2)11yescharacteristic
239C22×C3H(5,𝔽2)11yescharacteristic
240C22×C32−1+411yescharacteristic
241C22×C32−1+411yescharacteristic
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—22no—
249C2×D3—22no—
250C2×D3—22no—
251C2×D3—41no—
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3C2×C22⋊C414yes—
256C3×C4—22no—
257C3×C4—42no—
258C3×C4—42no—
259C3×C4C2×C22⋊C414yes—
260Dic3—22no—
261Dic3—42no—
262Dic3—42no—
263Dic3C2×C22⋊C414yes—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—32no—
271C23—32no—
272C23—32no—
273C23—32no—
274C23—61no—
275C23—61no—
276C23—61no—
277C23—61no—
278C23—61no—
279C23—61no—
280C23—61no—
281C23—61no—
282C23—61no—
283C23—61no—
284C23C2×C4×D312yes—
285C23C23×D311yescharacteristic
286Q8—68no—
287D4—68no—
288D4—68no—
289D4—68no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—316no—
303C2×C4—32no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—68no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—128no—
314C2×C4C2×C4×D312yes—
315C2×C4C2×C4×D312yes—
316C2×C4D3×D412yes—
317C2×C4D3×D412yes—
318C2×C3—42no—
319C2×C3C22×C22⋊C411yescharacteristic
320C2×C3#52011yescharacteristic
321C2×C3#52011yescharacteristic
322C2×C3#53511yescharacteristic
323C2×C3#53611yescharacteristic
324C2×C3#53711yescharacteristic
325C2×C3#53711yescharacteristic
326D3—24no—
327D3—42no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—34no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—62no—
337C22—62no—
338C22—62no—
339C22—62no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22—62no—
344C22—62no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22C22×C4×D311yescharacteristic
350C22C2×D3×D411yescharacteristic
351C22C2×D3×D411yescharacteristic
352C22#100411yescharacteristic
353C22#100711yescharacteristic
354C22#100911yescharacteristic
355C22#101011yescharacteristic
356C4—22no—
357C4—34no—
358C4—42no—
359C4—42no—
360C4—62no—
361C4—122no—
362C4—122no—
363C4D3×C22⋊C414yes—
364C3#231111yescharacteristic, complemented, Sylow
365C2—42no—
366C2—64no—
367C2—122no—
368C2C2×D3×C22⋊C411yescharacteristic
369C2#550411yescharacteristic
370C2#552311yescharacteristic
371C2#560711yescharacteristic
372C2#560811yescharacteristic
373C2#562811yescharacteristic
374C2#562911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#380C2
socle, cosocleC23×C3C24
prime radicalC22×#235C2
hypercenter#380D3
hypocenterC3#2311
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#892C2semidirect
3. C2×D3×C4⋊C4C2semidirect
4. C3×#380C2semidirect
5. #5212C2semidirect
6. #4987C2semidirect
7. #4987C2semidirect
8. #4957C2semidirect
9. #4956C2semidirect
10. #4665C2semidirect
11. #4646C2semidirect
18. C2×#235C4semidirect
51. #380D3semidirect
57. C22×C3×C4C2×C4semidirect
67. C2×#67C2×C4semidirect
82. C2×C4×D3C2×C4semidirect
87. C2×Dic6C2×C4semidirect
179. C4×D3C22⋊C4semidirect
212. C22×C4C4×D3semidirect
364. C3#2311semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#8922
C2×D3×C4⋊C42
C3×#3802
#52122
#49872
#49572
#49562
#46652
#46462
#23113