Finite Group Databasebeta

← #70414

Subgroups — #70414

Order 384.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#1009C211yesmaximal, characteristic
2C3×#421C211yesmaximal, characteristic, complemented
3#5236C211yesmaximal, characteristic, complemented
4#5132C211yesmaximal, characteristic, complemented
5#5131C211yesmaximal, characteristic, complemented
6#5006C211yesmaximal, characteristic, complemented
7#5001C211yesmaximal, characteristic, complemented
8#1795—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10#1009—22no—
11#1009—22no—
12#1009—22no—
13#1009—22no—
14C2×#239C412yescomplemented
15C2×D3×D4—22no—
16C2×D3×D4C412yescomplemented
17C2×#235—21no—
18C2×#235C2211yescharacteristic
19C22×D12—21no—
20C22×D12C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×#117C2211yescharacteristic, complemented
23#926C2211yescharacteristic, complemented
24C2×#208C2211yescharacteristic, complemented
25C3×#104—22no—
26#837—22no—
27#824—21no—
28#823—21no—
29#821—22no—
30#809—22no—
31#806—22no—
32C2×H(5,𝔽2)—31no—
33#421—31no—
34#421D311yescharacteristic, complemented
35#418—31no—
36#418—31no—
37#413—31no—
38#410—31no—
39C23×D3—21no—
40C23×D3—22no—
41C3×#41—21no—
42C3×#41—21no—
43C3×#41D412yes—
44C3×#41D412yes—
45C2×C3×Q8C2×C411yescharacteristic
46C2×C3×D4—21no—
47C2×C3×D4C2×C411yescharacteristic
48C22×C3×C4—21no—
49C22×C3×C4C2311yescharacteristic
50C2×#67—22no—
51C2×#67D412yes—
52#239—24no—
53#239—24no—
54D3×D4—28no—
55D3×D4—28no—
56D3×D4—24no—
57D3×D4—24no—
58#235—24no—
59#235—24no—
60#235—21no—
61#235—21no—
62#235—21no—
63#235—21no—
64C2×D12—24no—
65C2×D12—24no—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12—21no—
70C2×D12C2×C412yes—
71C2×D12D411yescharacteristic
72C2×D12D412yes—
73C2×D12D412yes—
74C2×C4×D3—22no—
75C2×C4×D3C2×C412yes—
76C2×Dic6D411yescharacteristic
77C3×M16—21no—
78C3×M16—21no—
79C2×C3×C8—21no—
80C3×C4⋊C4—21no—
81C3×C22⋊C4—21no—
82C3×C42—21no—
83#217—21no—
84C4⋊Dic3—21no—
85C4×Dic3—21no—
86#208—21no—
87#208—21no—
88C2×C3⋊C8—21no—
89H(5,𝔽2)—62no—
90H(5,𝔽2)—62no—
91H(5,𝔽2)—62no—
92H(5,𝔽2)—62no—
93C2×#41—31no—
94C2×#41—32no—
95C2×#41—61no—
96C2×#41C2×D311yescharacteristic
97C22×D4—32no—
98C22×D4—31no—
99C22×D4—61no—
100C22×D4—62no—
101C2×M16—31no—
102C2×M16C2×D311yescharacteristic, complemented
103#117—31no—
104#117C2×D311yescharacteristic, complemented
105#104—22no—
106#104—62no—
107#102—62no—
108#102—62no—
109#101—61no—
110#100—61no—
111#99—62no—
112C23×C3—21no—
113C23×C3C2×D411yescharacteristic
114C22×D3—22no—
115C22×D3—28no—
116C22×D3—22no—
117C22×D3—22no—
118C22×D3—82no—
119C22×D3—84no—
120C22×D3C22⋊C412yes—
121C3×Q8—21no—
122C3×Q8C22⋊C412yes—
123C3×D4—22no—
124C3×D4—22no—
125C3×D4—21no—
126C3×D4C22⋊C412yes—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—41no—
131C2×C3×C4C22×C411yescharacteristic
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D411yescharacteristic
137#67—28no—
138#67—22no—
139#67—22no—
140C2×Dic3—41no—
141C2×Dic3C22⋊C412yes—
142D12—28no—
143D12—22no—
144D12—21no—
145D12—22no—
146D12—21no—
147D12—22no—
148D12C22⋊C414yes—
149C4×D3—28no—
150C4×D3—22no—
151C4×D3—22no—
152Dic6—21no—
153Dic6—21no—
154C3×C8—41no—
155C3⋊C8—41no—
156C24—62no—
157C24—61no—
158#41—21no—
159#41—21no—
160#41—64no—
161#41—64no—
162#41—64no—
163#41—64no—
164#41—61no—
165#41—61no—
166#41—61no—
167#41—61no—
168#41D1212yes—
169#41#6712yes—
170C2×Q8—31no—
171C2×Q8C4×D311yescharacteristic
172C2×D4—21no—
173C2×D4—32no—
174C2×D4—32no—
175C2×D4—32no—
176C2×D4—32no—
177C2×D4—31no—
178C2×D4—62no—
179C2×D4—64no—
180C2×D4—64no—
181C2×D4—62no—
182C2×D4—64no—
183C2×D4—68no—
184C2×D4—68no—
185C2×D4—64no—
186C2×D4—61no—
187C2×D4—61no—
188C2×D4—61no—
189C2×D4C4×D311yescharacteristic
190C22×C4—21no—
191C22×C4—32no—
192C22×C4—62no—
193C22×C4C22×D311yescharacteristic
194M16—21no—
195M16—21no—
196M16—61no—
197M16—61no—
198C2×C8—21no—
199C2×C8—61no—
200C4⋊C4—21no—
201C4⋊C4—61no—
202C22⋊C4—21no—
203C22⋊C4—61no—
204C42—21no—
205C42—61no—
206C22×C3—22no—
207C22×C3—41no—
208C22×C3—42no—
209C22×C3C2×C22⋊C411yescharacteristic
210C22×C3#12011yescharacteristic
211C22×C3#12011yescharacteristic
212C2×D3—22no—
213C2×D3—24no—
214C2×D3—44no—
215C2×D3—42no—
216C2×D3—82no—
217C2×D3—82no—
218C2×D3—88no—
219C3×C4—22no—
220C3×C4—81no—
221C3×C4C2×C22⋊C411yescharacteristic
222C3×C4C2×C22⋊C411yescharacteristic
223C3×C4#12011yescharacteristic
224C3×C4#12011yescharacteristic
225Dic3—22no—
226Dic3—81no—
227C23—21no—
228C23—32no—
229C23—68no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—242no—
234C23—244no—
235C23D3×D411yescharacteristic
236Q8—21no—
237Q8—61no—
238Q8—61no—
239Q8(C2×D3)⋊C412yes—
240D4—22no—
241D4—22no—
242D4—21no—
243D4—34no—
244D4—68no—
245D4—68no—
246D4—62no—
247D4—62no—
248D4—62no—
249D4—61no—
250D4—62no—
251D4—62no—
252D4—61no—
253D4(C2×D3)⋊C412yes—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—32no—
258C2×C4—41no—
259C2×C4—68no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—121no—
263C2×C4C2×C4×D311yescharacteristic
264C2×C4C2×D1211yescharacteristic
265C2×C4D3×D411yescharacteristic
266C2×C4D3×D411yescharacteristic
267C2×C4D3×D411yescharacteristic
268C2×C4C2×#6711yescharacteristic
269C8—41no—
270C8—121no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—42no—
275C2×C3#37911yescharacteristic
276D3—44no—
277D3—42no—
278C22—22no—
279C22—41no—
280C22—42no—
281C22—64no—
282C22—62no—
283C22—122no—
284C22—124no—
285C22—242no—
286C22—242no—
287C22—248no—
288C22D3×C22⋊C411yescharacteristic
289C22#88211yescharacteristic
290C22#93711yescharacteristic
291C4—22no—
292C4—62no—
293C4—81no—
294C4—241no—
295C4D3×C22⋊C411yescharacteristic
296C4#88211yescharacteristic
297C4C2×(C2×D3)⋊C411yescharacteristic
298C4#93711yescharacteristic
299C3#179511yescharacteristic, complemented, Sylow
300C2—21no—
301C2—21no—
302C2—21no—
303C2—42no—
304C2—124no—
305C2—122no—
306C2#495511yescharacteristic
subgroupG/H
centerC2#4955
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#379
prime radicalC2×#1009C2
cosocleC22×C3×C4C23
hypercenter#421D3
hypocenterC3#1795
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C3×#421C2semidirect
3. #5236C2semidirect
4. #5132C2semidirect
5. #5131C2semidirect
6. #5006C2semidirect
7. #5001C2semidirect
14. C2×#239C4semidirect
16. C2×D3×D4C4semidirect
21. C2×C3×M16C22semidirect
22. C3×#117C22semidirect
23. #926C22semidirect
24. C2×#208C22semidirect
34. #421D3semidirect
102. C2×M16C2×D3semidirect
104. #117C2×D3semidirect
299. C3#1795semidirect

Maximal subgroups

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

subgroupindex
C2×#10092
C3×#4212
#52362
#51322
#51312
#50062
#50012
#17953