Finite Group Databasebeta

← #84097

Subgroups — #84097

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×C3×#41C211yesmaximal, characteristic, complemented
2C2×#946C211yesmaximal, characteristic, complemented
3C2×#938C211yesmaximal, characteristic, complemented
4C2×C4×#67C211yesmaximal, characteristic, complemented
5#5229C211yesmaximal, characteristic, complemented
6#5223C212yesmaximal, complemented
7#5222C212yesmaximal, complemented
8#5221C211yesmaximal, characteristic, complemented
9#5213C212yesmaximal, complemented
10#5212C211yesmaximal, characteristic, complemented
11#5210C212yesmaximal, complemented
12#2407—31nomaximal, Sylow
13C2×C3×#41—24no—
14C2×C3×#41—24no—
15C2×C3×#41—24no—
16C22×C3×Q8C2211yescharacteristic
17C22×C3×D4C2212yes—
18C22×C3×D4C2211yescharacteristic
19C23×C3×C4C2212yes—
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2212yes—
23C22×C4×D3C2211yescharacteristic
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×#217C2211yescharacteristic
27C2×#217C2212yes—
28C2×#217C2212yes—
29#946—24no—
30#938—24no—
31C4×#67—24no—
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2212yes—
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2212yes—
38C2×C4×Dic3C2211yescharacteristic
39C3⋊#95C2212yes—
40C3⋊#95C2212yes—
41C3⋊#95C2212yes—
42C22×#41D311yescharacteristic, complemented
43C2×C22⋊Q8—31no—
44C2×#121—31no—
45C2×C4×D4—31no—
46#394—32no—
47#392—32no—
48#391—31no—
49#390—31no—
50#386—32no—
51#385—32no—
52#380—31no—
53C24×C3C2312yes—
54C24×C3C2311yescharacteristic
55C23×D3C2311yescharacteristic
56C3×#41—232no—
57C2×C3×Q8—22no—
58C2×C3×Q8—22no—
59C2×C3×Q8D414yescomplemented
60C2×C3×D4—24no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4—22no—
64C2×C3×D4—22no—
65C2×C3×D4D414yescomplemented
66C22×C3×C4—24no—
67C22×C3×C4—24no—
68C22×C3×C4—24no—
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4D414yescomplemented
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2312yes—
74C22×C3×C4C2311yescharacteristic
75C2×#67—22no—
76C2×#67—22no—
77C2×#67—24no—
78C2×#67—24no—
79C2×#67—22no—
80C2×#67—42no—
81C22×Dic3C2312yes—
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2312yes—
84C22×Dic3C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C2×C4×D3—22no—
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89#217—22no—
90#217—24no—
91#217—24no—
92#217—42no—
93#217—42no—
94(C2×D3)⋊C4—22no—
95(C2×D3)⋊C4—24no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98Dic3⋊C4—22no—
99Dic3⋊C4—24no—
100C4×Dic3—22no—
101C2×#41—24no—
102C2×#41—24no—
103C2×#41—24no—
104C22×Q8C2×D311yescharacteristic
105C22×D4—31no—
106C22×D4—32no—
107C22×D4C2×D312yes—
108C22×D4C2×D311yescharacteristic
109C23×C4—31no—
110C23×C4C2×D312yes—
111C23×C4C2×D311yescharacteristic
112C22⋊Q8—64no—
113#121—64no—
114C4×D4—64no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—32no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4—32no—
121C2×C22⋊C4—32no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—32no—
124C2×C22⋊C4—32no—
125C2×C22⋊C4—32no—
126C2×C42—31no—
127#95—32no—
128#95—32no—
129#95—32no—
130C23×C3—22no—
131C23×C3—21no—
132C23×C3—22no—
133C23×C3—21no—
134C23×C3—22no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—22no—
138C23×C3—21no—
139C23×C3—22no—
140C23×C3—21no—
141C23×C3—22no—
142C23×C3C2×D414yes—
143C23×C3C2×D412yes—
144C23×C3C2411yescharacteristic
145C22×D3—21no—
146C22×D3—21no—
147C22×D3—21no—
148C22×D3—21no—
149C22×D3—21no—
150C22×D3—21no—
151C22×D3#4112yescomplemented
152C3×Q8—28no—
153C3×D4—216no—
154C3×D4—28no—
155C2×C3×C4—216no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—28no—
161C2×C3×C4C2×D412yes—
162C2×C3×C4C2×D414yes—
163C2×C3×C4C2×D412yes—
164C2×C3×C4C2×D412yes—
165C2×C3×C4C2×D412yes—
166#67—28no—
167#67—48no—
168C2×Dic3—21no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—21no—
172C2×Dic3—22no—
173C2×Dic3—21no—
174C2×Dic3—22no—
175C2×Dic3—21no—
176C2×Dic3—22no—
177C2×Dic3—22no—
178C2×Dic3—21no—
179C2×Dic3—22no—
180C2×Dic3—22no—
181C2×Dic3—21no—
182C2×Dic3#4112yescomplemented
183C4×D3—44no—
184C24—31no—
185C24C22×D312yes—
186C24C22×D311yescharacteristic
187#41—232no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8#6714yescomplemented
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—64no—
200C2×D4—64no—
201C2×D4—122no—
202C2×D4#6714yescomplemented
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—31no—
209C22×C4—32no—
210C22×C4—31no—
211C22×C4—32no—
212C22×C4—31no—
213C22×C4—62no—
214C22×C4—62no—
215C22×C4—62no—
216C22×C4#6714yescomplemented
217C22×C4C22×D311yescharacteristic
218C22×C4C22×D312yes—
219C22×C4C22×D311yescharacteristic
220C4⋊C4—62no—
221C4⋊C4—62no—
222C4⋊C4—64no—
223C4⋊C4—62no—
224C22⋊C4—64no—
225C22⋊C4—62no—
226C22⋊C4—64no—
227C22⋊C4—62no—
228C22⋊C4—64no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C42—62no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—24no—
237C22×C3—24no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—42no—
244C22×C3—42no—
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—22no—
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—41no—
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—22no—
261C3×C4—24no—
262C3×C4#12014yes—
263Dic3—22no—
264Dic3—41no—
265Dic3—41no—
266Dic3—42no—
267Dic3—42no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—21no—
273C23—22no—
274C23—22no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—22no—
279C23—21no—
280C23—32no—
281C23—61no—
282C23—61no—
283C23—61no—
284C23—61no—
285C23—61no—
286C23—61no—
287C23C2×#6714yes—
288C23C2×#6712yes—
289C23C23×D311yescharacteristic
290Q8—28no—
291D4—216no—
292D4—28no—
293D4—68no—
294D4—128no—
295C2×C4—216no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—28no—
301C2×C4—32no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—61no—
307C2×C4—62no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—62no—
313C2×C4—61no—
314C2×C4—62no—
315C2×C4—61no—
316C2×C4—124no—
317C2×C4C2×#6712yes—
318C2×C4C2×#6714yes—
319C2×C4C2×#6712yes—
320C2×C4C2×#6712yes—
321C2×C4C2×#6712yes—
322C2×C3—42no—
323C2×C3—44no—
324C2×C3C2×#12011yescharacteristic
325C2×C3#52511yescharacteristic
326C2×C3#53711yescharacteristic
327C2×C3#54611yescharacteristic
328C2×C3#54711yescharacteristic
329C2×C3#54811yescharacteristic
330C2×C3#55011yescharacteristic
331D3—42no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—22no—
336C22—24no—
337C22—22no—
338C22—41no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—62no—
347C22—62no—
348C22—62no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22D3×#4111yescharacteristic
354C22D3×#4111yescharacteristic
355C22#100911yescharacteristic
356C22#101011yescharacteristic
357C22C22×#6711yescharacteristic
358C22C22×#6711yescharacteristic
359C22C22×#6711yescharacteristic
360C4—22no—
361C4—24no—
362C4—62no—
363C4—122no—
364C4—121no—
365C4—122no—
366C4—121no—
367C4#95314yes—
368C3#240711yescharacteristic, complemented, Sylow
369C2—44no—
370C2—42no—
371C2—122no—
372C2#582411yescharacteristic
373C2#582811yescharacteristic
374C2#582911yescharacteristic
375C2#583111yescharacteristic
376C2#583311yescharacteristic
377C2#583411yescharacteristic
378C2C2×#95311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×#41C2
hypercenterC22×#41D3
hypocenterC3#2407
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C2×#946C2semidirect
3. C2×#938C2semidirect
4. C2×C4×#67C2semidirect
5. #5229C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5221C2semidirect
9. #5213C2semidirect
10. #5212C2semidirect
11. #5210C2semidirect
42. C22×#41D3semidirect
59. C2×C3×Q8D4semidirect
65. C2×C3×D4D4semidirect
71. C22×C3×C4D4semidirect
151. C22×D3#41semidirect
182. C2×Dic3#41semidirect
190. C2×Q8#67semidirect
202. C2×D4#67semidirect
216. C22×C4#67semidirect
368. C3#2407semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×#9462
C2×#9382
C2×C4×#672
#52292
#52232
#52222
#52212
#52132
#52122
#52102
#24073