Finite Group Databasebeta

← C2×#11724311 #13181081

Subgroups — C2×#11724311 #13181081

Order 1728.

904 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 143 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×#89040C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic
3C2×#88988C211yesmaximal, characteristic
4C2×#88938C214yesmaximal, complemented
5#11724311C218yesmaximal, direct product factor
6C2×#64228—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#634—21no—
9C2×C32×Dic6—21no—
10C2×#6296C2212yes—
11C2×#6294—21no—
12#89040—22no—
13#89040C2214yes—
14C2×C3×#627C2212yes—
15C2×C3×#624—21no—
16C3×#3797—22no—
17C3×#3797C2214yes—
18C2×C3⋊#622C2212yes—
19#88988C2214yes—
20#88938C22116yescomplemented
21C2×C3×D3×Q8—61no—
22C2×#3797—31no—
23C2×#3797D311yescharacteristic
24C2×D3×Dic6—61no—
25C2×#3787—31no—
26C2×#3776—34no—
27#64228—38no—
28C22×C3×C3⋊D3D411yescharacteristic
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C3×C4×C3⋊D3D412yes—
33C3×C4×C3⋊D3C2312yes—
34C3×#634—24no—
35C2×C32×Dic3—22no—
36C32×Dic6—24no—
37#6296—22no—
38#6296C2314yes—
39#6294—24no—
40C3×#627—22no—
41C3×#627C2314yes—
42C3×#624—24no—
43C3⋊#622C2314yes—
44C22×#216—91nomaximal
45C2×C32×Q8—61no—
46C2×C4×C3⋊D3C2×D311yescharacteristic
47C2×#634—21no—
48C3×D3×Q8—64no—
49C3×D3×Q8—64no—
50C2×C3×C4×D3—62no—
51C2×C3×C4×D3—61no—
52C2×C3×C4×D3—61no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—41no—
56C2×C3×Dic6—62no—
57C2×C3×Dic6—61no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—121no—
60C2×#627—32no—
61C2×#627—62no—
62C2×#627C2×D312yes—
63C2×#624—21no—
64C2×#624—61no—
65C2×D3×Dic3—62no—
66#3797—22no—
67#3797—34no—
68#3797—62no—
69#3797C2×D314yes—
70D3×Dic6—64no—
71D3×Dic6—64no—
72C2×#622—32no—
73#3787—34no—
74#3776—316no—
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3Q1614yescomplemented
77C2×C3×C3⋊D3C2×D412yes—
78C33×C4C2×D412yes—
79C3×C3⋊Dic3—24no—
80C3×C3⋊Dic3C2×D411yescharacteristic
81C3×C3⋊Dic3C2411yescharacteristic
82C32×Dic3—24no—
83C22×C3×Q8—91no—
84C2×D3×Q8—61no—
85C2×D3×Q8—181no—
86C22×Dic6—91no—
87C2×#216—98no—
88C2×#216—94no—
89C22×C3⋊C8—91no—
90C22×C3⋊D3#6711yescharacteristic
91C22×C3×D3—61no—
92C22×C3×D3—61no—
93C32×Q8—64no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—62no—
99C2×C32×C4#6711yescharacteristic
100C2×C3⋊Dic3—22no—
101C2×C3⋊Dic3C22×D311yescharacteristic
102C4×C3⋊D3#6712yes—
103C4×C3⋊D3C22×D312yes—
104#634—24no—
105C2×C3×Dic3—22no—
106C2×C3×Dic3—22no—
107C2×C3×Dic3—42no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—122no—
113C3×C4×D3—62no—
114C3×C4×D3—62no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—68no—
118C3×Dic6—24no—
119C3×Dic6—24no—
120C3×Dic6—44no—
121C3×Dic6—64no—
122C3×Dic6—64no—
123C3×Dic6—64no—
124C3×Dic6—64no—
125C3×Dic6—124no—
126#627—22no—
127#627—34no—
128#627—64no—
129#627—64no—
130#627—62no—
131#627C22×D314yes—
132#624—24no—
133#624—64no—
134D3×Dic3—68no—
135#622—34no—
136C22×Q16—271noSylow
137C2×C33C2×Q1612yescomplemented
138C2×C33C22×D411yescharacteristic
139C3×C3⋊D3C2×Q1614yescomplemented
140C2×C3×Q8—61no—
141C2×C3×Q8—61no—
142C2×C3×Q8—92no—
143C2×C3×Q8—94no—
144C2×C3×Q8—121no—
145C2×C3×Q8—181no—
146C2×C3×Q8—182no—
147C22×C3×C4—91no—
148C22×C3×C4—181no—
149C22×Dic3—181no—
150D3×Q8—64no—
151D3×Q8—64no—
152D3×Q8—184no—
153D3×Q8—184no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—62no—
157C2×C4×D3—182no—
158C2×Dic6—21no—
159C2×Dic6—62no—
160C2×Dic6—61no—
161C2×Dic6—61no—
162C2×Dic6—94no—
163C2×Dic6—92no—
164C2×Dic6—121no—
165C2×Dic6—182no—
166C2×Dic6—182no—
167#216—916no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—92no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—41no—
173C22×C32—41no—
174C22×C32C2×#6711yescharacteristic
175C2×C3⋊D3#21614yescomplemented
176C2×C3⋊D3C2×#6712yes—
177C2×C3×D3—64no—
178C2×C3×D3—64no—
179C2×C3×D3—62no—
180C2×C3×D3—62no—
181C32×C4—22no—
182C32×C4—22no—
183C32×C4—42no—
184C32×C4—42no—
185C32×C4—64no—
186C32×C4C2×#6712yes—
187C3⋊Dic3—24no—
188C3⋊Dic3C2×#6711yescharacteristic
189C3⋊Dic3C23×D311yescharacteristic
190C3×Dic3—24no—
191C3×Dic3—24no—
192C3×Dic3—44no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C3×Dic3—124no—
200C22×Q8—91no—
201C22×Q8—271no—
202C2×Q16—278no—
203C2×Q16—274no—
204C22×C8—271no—
205C33C22×Q1611yescharacteristic, complemented, Sylow
206C23×C3—91no—
207C22×D3—61no—
208C22×D3—61no—
209C3×Q8—64no—
210C3×Q8—64no—
211C3×Q8—94no—
212C3×Q8—124no—
213C3×Q8—184no—
214C3×Q8—182no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4—91no—
222C2×C3×C4—92no—
223C2×C3×C4—92no—
224C2×C3×C4—122no—
225C2×C3×C4—184no—
226C2×C3×C4—182no—
227C2×C3×C4GO+(4,𝔽2)11yescharacteristic
228C2×Dic3—22no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—62no—
232C2×Dic3—62no—
233C2×Dic3—122no—
234C2×Dic3—184no—
235C4×D3—62no—
236C4×D3—62no—
237C4×D3—62no—
238C4×D3—62no—
239C4×D3—68no—
240C4×D3—188no—
241Dic6—24no—
242Dic6—64no—
243Dic6—64no—
244Dic6—64no—
245Dic6—64no—
246Dic6—94no—
247Dic6—124no—
248Dic6—184no—
249Dic6—182no—
250Dic6—184no—
251C3⋊C8—94no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—42no—
257C2×C32—42no—
258C2×C32—41no—
259C2×C32—41no—
260C2×C32C2×#21612yescomplemented
261C2×C32C22×#6711yescharacteristic
262C3⋊D3C2×#21614yescomplemented
263C3×D3—64no—
264C3×D3—64no—
265C2×Q8—61no—
266C2×Q8—92no—
267C2×Q8—94no—
268C2×Q8—182no—
269C2×Q8—181no—
270C2×Q8—272no—
271C2×Q8—274no—
272C2×Q8—542no—
273C22×C4—91no—
274C22×C4—181no—
275C22×C4—541no—
276Q16—2716no—
277C2×C8—274no—
278C2×C8—272no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—92no—
284C22×C3—94no—
285C22×C3C2×GO+(4,𝔽2)11yescharacteristic
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—64no—
289C2×D3—64no—
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—64no—
295C3×C4—64no—
296C3×C4—91no—
297C3×C4—91no—
298C3×C4—124no—
299C3×C4—184no—
300C3×C4C2×GO+(4,𝔽2)12yes—
301Dic3—24no—
302Dic3—61no—
303Dic3—61no—
304Dic3—61no—
305Dic3—61no—
306Dic3—64no—
307Dic3—64no—
308Dic3—124no—
309C32—21no—
310C32—21no—
311C32—41no—
312C32—41no—
313C32C22×#21611yescharacteristic, complemented
314C23—91no—
315Q8—64no—
316Q8—94no—
317Q8—182no—
318Q8—184no—
319Q8—274no—
320Q8—542no—
321C2×C4—62no—
322C2×C4—91no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—184no—
326C2×C4—182no—
327C2×C4—544no—
328C2×C4D32⋊D311yescharacteristic
329C8—274no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—42no—
335C2×C3—42no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—94no—
339C2×C3#6422812yescomplemented
340C2×C3C22×GO+(4,𝔽2)11yescharacteristic
341D3—64no—
342D3—64no—
343C22—94no—
344C22—92no—
345C22C2×D32⋊D311yescharacteristic
346C4—64no—
347C4—91no—
348C4—91no—
349C4—184no—
350C4C2×D32⋊D312yes—
351C3—21no—
352C3—21no—
353C3—41no—
354C3—41no—
355C3C2×#6422811yescharacteristic, complemented
356C2—94no—
357C2#1172431112yesdirect product factor
358C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC3×C3⋊Dic3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4D32⋊D3
hypocenterC33C22×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88938C2semidirect
5. #11724311C2direct (N × H)
20. #88938C22semidirect
76. C2×C3×C3⋊D3Q16semidirect
137. C2×C33C2×Q16semidirect
139. C3×C3⋊D3C2×Q16semidirect
175. C2×C3⋊D3#216semidirect
205. C33C22×Q16semidirect
260. C2×C32C2×#216semidirect
262. C3⋊D3C2×#216semidirect
313. C32C22×#216semidirect
339. C2×C3#64228semidirect
355. C3C2×#64228semidirect
357. C2#11724311direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×#37972
C2×#889882
C2×#889382
#117243112
C2×#642283
C22×#2169