Finite Group Databasebeta

← #82402

Subgroups — #82402

Order 384.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic, complemented
2C2×C3×C4×D4C211yesmaximal, characteristic, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#924C211yesmaximal, characteristic, complemented
5#5223C212yesmaximal, complemented
6#5212C211yesmaximal, characteristic, complemented
7#4989C212yesmaximal, complemented
8#4959C212yesmaximal, complemented
9#4956C212yesmaximal, complemented
10#4939C211yesmaximal, characteristic, complemented
11#4929C211yesmaximal, characteristic, complemented
12#2407—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C2×#235—24no—
19C2×#235—24no—
20C2×#235—24no—
21C22×D12C2211yescharacteristic
22C22×C4×D3C2212yes—
23C22×Dic6C2211yescharacteristic
24C3×C4×D4—24no—
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C42C2211yescharacteristic
29C2×#217C2212yes—
30C2×#217C2212yes—
31#930—24no—
32C2×(C2×D3)⋊C4C2212yes—
33C2×(C2×D3)⋊C4C2212yes—
34C2×(C2×D3)⋊C4C2212yes—
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37#924—24no—
38C2×Dic3⋊C4C2212yes—
39C3⋊#95C2212yes—
40C3⋊#95C2212yes—
41C3⋊#95C2212yes—
42C22×#41—31no—
43C2×C22⋊Q8—31no—
44C2×#121—31no—
45C2×C4×D4D311yescharacteristic, complemented
46#394—32no—
47#392—32no—
48#391—31no—
49#390—31no—
50#386—32no—
51#385—32no—
52#380—31no—
53C24×C3C2311yescharacteristic
54C24×C3C2311yescharacteristic
55C23×D3C2312yes—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—22no—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—22no—
62C22×C3×C4—22no—
63C22×C3×C4—22no—
64C22×C3×C4D414yescomplemented
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2311yescharacteristic
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—24no—
73C2×#67—24no—
74C2×#67—24no—
75C2×#67—42no—
76C22×Dic3C2312yes—
77C22×Dic3C2312yes—
78C22×Dic3C2312yes—
79#235—232no—
80C2×D12—22no—
81C2×D12—22no—
82C2×D12D414yescomplemented
83C2×C4×D3—24no—
84C2×C4×D3—24no—
85C2×C4×D3—24no—
86C2×Dic6—22no—
87C2×Dic6—22no—
88C2×Dic6D414yescomplemented
89C3×C4⋊C4—22no—
90C3×C22⋊C4—22no—
91C3×C22⋊C4—22no—
92C3×C42—22no—
93#217—24no—
94#217—42no—
95(C2×D3)⋊C4—24no—
96(C2×D3)⋊C4—24no—
97(C2×D3)⋊C4—42no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100Dic3⋊C4—24no—
101C2×#41—64no—
102C2×#41—64no—
103C2×#41—64no—
104C22×Q8—31no—
105C22×D4—32no—
106C22×D4—32no—
107C22×D4—31no—
108C22×D4C2×D311yescharacteristic
109C23×C4—32no—
110C23×C4C2×D311yescharacteristic
111C23×C4C2×D311yescharacteristic
112C22⋊Q8—64no—
113#121—64no—
114C4×D4—24no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4C2×D311yescharacteristic
119C2×C22⋊C4—32no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—32no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4C2×D311yescharacteristic
125C2×C22⋊C4C2×D311yescharacteristic
126C2×C42C2×D311yescharacteristic
127#95—32no—
128#95—32no—
129#95—32no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3C2×D412yes—
143C23×C3#4112yescomplemented
144C23×C3C2411yescharacteristic
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—22no—
148C22×D3—22no—
149C22×D3—22no—
150C22×D3—22no—
151C22×D3C2×D414yes—
152C3×D4—28no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—28no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—44no—
165C2×C3×C4C2×D412yes—
166C2×C3×C4C2×D412yes—
167C2×C3×C4C2×D412yes—
168C2×C3×C4C2×D412yes—
169C2×C3×C4#4112yescomplemented
170#67—216no—
171#67—48no—
172C2×Dic3—22no—
173C2×Dic3—22no—
174C2×Dic3—22no—
175C2×Dic3—22no—
176C2×Dic3—22no—
177C2×Dic3—22no—
178C2×Dic3—22no—
179C2×Dic3—22no—
180C2×Dic3C2×D414yes—
181D12—28no—
182C4×D3—216no—
183Dic6—28no—
184C24—32no—
185C24C22×D311yescharacteristic
186C24C22×D311yescharacteristic
187#41—632no—
188C2×Q8—34no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—34no—
195C2×D4—64no—
196C2×D4—64no—
197C2×D4—64no—
198C2×D4—64no—
199C2×D4—64no—
200C2×D4—62no—
201C2×D4—62no—
202C2×D4—122no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—32no—
209C22×C4—32no—
210C22×C4—32no—
211C22×C4—64no—
212C22×C4—64no—
213C22×C4—64no—
214C22×C4#6714yescomplemented
215C22×C4C22×D311yescharacteristic
216C22×C4C22×D311yescharacteristic
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D311yescharacteristic
219C22×C4C22×D311yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—64no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—64no—
227C22⋊C4—64no—
228C22⋊C4—64no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C42—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3C22×D411yescharacteristic
247C22×C3C22×D411yescharacteristic
248C22×C3C22×D411yescharacteristic
249C22×C3C2×#4111yescharacteristic
250C22×C3C2×#4111yescharacteristic
251C22×C3H(5,𝔽2)11yescharacteristic
252C22×C32−1+411yescharacteristic
253C2×D3—24no—
254C2×D3—24no—
255C2×D3—24no—
256C2×D3—42no—
257C2×D3—42no—
258C2×D3—42no—
259C2×D3—42no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4#12014yes—
265Dic3—24no—
266Dic3—42no—
267Dic3—42no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—34no—
281C23—62no—
282C23—62no—
283C23—62no—
284C23—62no—
285C23—62no—
286C23—62no—
287C23#23512yescomplemented
288C23C2×#6712yes—
289C23C23×D311yescharacteristic
290Q8—68no—
291D4—28no—
292D4—616no—
293D4—68no—
294D4—128no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—28no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—34no—
307C2×C4—44no—
308C2×C4—616no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—62no—
317C2×C4#23512yescomplemented
318C2×C4D3×D412yes—
319C2×C4D3×D412yes—
320C2×C4C2×#6712yes—
321C2×C4C2×#6712yes—
322C2×C3—42no—
323C2×C3—42no—
324C2×C3C2×#12011yescharacteristic
325C2×C3#52511yescharacteristic
326C2×C3#53711yescharacteristic
327C2×C3#54611yescharacteristic
328C2×C3#54711yescharacteristic
329C2×C3#54811yescharacteristic
330C2×C3#55011yescharacteristic
331D3—44no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—64no—
347C22—64no—
348C22—64no—
349C22—122no—
350C22—122no—
351C22—122no—
352C22—122no—
353C22C2×#23511yescharacteristic
354C22C2×#23511yescharacteristic
355C22C2×D3×D411yescharacteristic
356C22C2×D3×D411yescharacteristic
357C22#100911yescharacteristic
358C22#101011yescharacteristic
359C22C22×#6711yescharacteristic
360C4—22no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—64no—
365C4—122no—
366C4—122no—
367C4#93714yes—
368C3#240711yescharacteristic, complemented, Sylow
369C2—42no—
370C2—42no—
371C2—124no—
372C2#555011yescharacteristic
373C2#555111yescharacteristic
374C2#555211yescharacteristic
375C2#555311yescharacteristic
376C2#579111yescharacteristic
377C2C2×#93711yescharacteristic
378C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C2×C3×C4×D4C2semidirect
3. C2×#930C2semidirect
4. C2×#924C2semidirect
5. #5223C2semidirect
6. #5212C2semidirect
7. #4989C2semidirect
8. #4959C2semidirect
9. #4956C2semidirect
10. #4939C2semidirect
11. #4929C2semidirect
45. C2×C4×D4D3semidirect
64. C22×C3×C4D4semidirect
82. C2×D12D4semidirect
88. C2×Dic6D4semidirect
143. C23×C3#41semidirect
169. C2×C3×C4#41semidirect
214. C22×C4#67semidirect
287. C23#235semidirect
317. C2×C4#235semidirect
368. C3#2407semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×C3×C4×D42
C2×#9302
C2×#9242
#52232
#52122
#49892
#49592
#49562
#49392
#49292
#24073