Finite Group Databasebeta

← C2×#11724111 #13180543

Subgroups — C2×#11724111 #13180543

Order 1728.

1432 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C22×#157⋊D3C211yesmaximal, characteristic
2C2×#88949C212yesmaximal, complemented
3C2×#88932C212yesmaximal, complemented
4C2×C3×#3773C211yesmaximal, characteristic, complemented
5C2×Dic3×#157C211yesmaximal, characteristic, complemented
6#11724111C218yesmaximal, direct product factor
7C2×C4×GO+(4,𝔽2)—31nomaximal
8C22×#1113C2212yes—
9C22×C3⋊#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×#157⋊D3C418yescomplemented
12C2×#157⋊D3C2214yes—
13C2×Dic3×C3⋊D3—22no—
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×C3×#624—22no—
16C2×C3×#624C2212yes—
17#88949—24no—
18#88949C2218yescomplemented
19#88932C2218yescomplemented
20C3×#3773C2214yescomplemented
21Dic3×#157C2214yescomplemented
22C22×GO+(4,𝔽2)—31no—
23C2×C4×D32—32no—
24C2×C4×D32—62no—
25C2×C4×#157—31no—
26C2×#3773D311yescharacteristic, complemented
27C2×D32⋊C4—32no—
28C4×GO+(4,𝔽2)—38no—
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3C2311yescharacteristic
31C2×#1113—22no—
32C2×#1113—24no—
33C2×#1113C2×C418yescomplemented
34C2×#1113C2314yes—
35C2×C3⋊#157—22no—
36C2×C3⋊#157C2312yes—
37C2×C3×#157C2×C414yescomplemented
38C2×C3×#157C2312yes—
39#157⋊D3C2×C4116yescomplemented
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic, complemented
41C2×C32×Dic3—22no—
42C2×C32×Dic3D411yescharacteristic, complemented
43Dic3×C3⋊D3—28no—
44Dic3×C3⋊D3D412yescomplemented
45Dic3×C3⋊D3C2312yes—
46C3×#624—22no—
47C3×#624—28no—
48C3×#624C2314yes—
49C2×#893—91nomaximal
50C22×D32—32no—
51C22×D32—62no—
52C22×#157—31no—
53C22×#157C2×D311yescharacteristic
54C2×GO+(4,𝔽2)—38no—
55C2×GO+(4,𝔽2)—34no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—62no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—62no—
61C2×#624—22no—
62C2×#624C2×D312yes—
63C2×D3×Dic3—62no—
64C2×D3×Dic3—62no—
65C2×D3×Dic3—62no—
66C2×D3×Dic3—122no—
67C4×D32—38no—
68C4×D32—64no—
69C4×D32—68no—
70C4×D32—68no—
71C4×#157—34no—
72#3773C2×D314yescomplemented
73D32⋊C4—38no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—28no—
76C2×C3×C3⋊D3—24no—
77C2×C3×C3⋊D3C22×C412yes—
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C2×C3×C3⋊D3#4112yes—
80C2×C3×C3⋊D3C2411yescharacteristic
81#1113—28no—
82#1113—24no—
83#1113C22×C418yescomplemented
84C3⋊#157—22no—
85C3×#157C22×C414yescomplemented
86C3⋊(C3⋊Dic3)C2×D412yescomplemented
87C32×Dic3—24no—
88C32×Dic3C2×D412yescomplemented
89C22×D12—91no—
90C22×C4×D3—92no—
91C22×C4×D3—182no—
92C2×C3×C4⋊C4—91no—
93C2×(C2×D3)⋊C4—92no—
94C2×C4×Dic3—91no—
95#893—98no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3C22×D311yescharacteristic
98C22×C3×D3—62no—
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—122no—
102C2×D32—38no—
103C2×D32—34no—
104C2×D32—68no—
105C2×D32—64no—
106C2×D32—68no—
107C2×D32—64no—
108C2×D32—62no—
109C2×D32—64no—
110C2×#157—32no—
111C2×#157—62no—
112C2×#157C4×D314yescomplemented
113C2×#157C22×D312yes—
114GO+(4,𝔽2)—316no—
115C2×C32×C4—31no—
116C2×C32×C4—62no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—122no—
120C4×C3⋊D3—32no—
121C4×C3⋊D3—32no—
122C4×C3⋊D3—68no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—42no—
127C3×C4×D3—64no—
128C3×C4×D3—68no—
129C3×C4×D3—64no—
130C3×C4×D3—68no—
131#624—28no—
132#624—22no—
133#624C22×D314yes—
134D3×Dic3—64no—
135D3×Dic3—64no—
136D3×Dic3—68no—
137D3×Dic3—68no—
138D3×Dic3—128no—
139C2×C4×D4—271noSylow
140C2×C33C4×D412yescomplemented
141C2×C33C22×D411yescharacteristic
142C3×C3⋊D3—28no—
143C3×C3⋊D3C4×D412yescomplemented
144C3×C3⋊D3C23×C411yescharacteristic
145C3×C3⋊D3C2×#4111yescharacteristic
146C23×D3—92no—
147C23×D3—182no—
148C22×C3×C4—92no—
149C22×C3×C4—91no—
150C22×C3×C4—182no—
151C22×Dic3—91no—
152C22×Dic3—91no—
153C22×Dic3—182no—
154C2×D12—98no—
155C2×D12—94no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159C2×C4×D3—98no—
160C2×C4×D3—188no—
161C2×C4×D3—182no—
162C2×C4×D3—184no—
163C2×C4×D3—188no—
164C2×C4×D3—182no—
165C2×C4×D3—182no—
166C2×C4×D3—362no—
167C3×C4⋊C4—94no—
168(C2×D3)⋊C4—98no—
169C4×Dic3—94no—
170C22×C32—22no—
171C22×C32—42no—
172C22×C32D3×D411yescharacteristic
173C2×C3⋊D3—24no—
174C2×C3⋊D3—28no—
175C2×C3⋊D3C2×C4×D312yes—
176C2×C3⋊D3D3×D411yescharacteristic
177C2×C3⋊D3#23912yes—
178C2×C3⋊D3C23×D311yescharacteristic
179C2×C3×D3—64no—
180C2×C3×D3—62no—
181C2×C3×D3—64no—
182C2×C3×D3—62no—
183C2×C3×D3—64no—
184C2×C3×D3—68no—
185C2×C3×D3—68no—
186C2×C3×D3—64no—
187C2×C3×D3—124no—
188C2×C3×D3—128no—
189D32—38no—
190D32—68no—
191D32—616no—
192D32—68no—
193D32—68no—
194D32—64no—
195#157—62no—
196#157C2×C4×D314yescomplemented
197C32×C4—32no—
198C32×C4—64no—
199C3⋊Dic3—32no—
200C3⋊Dic3—64no—
201C3⋊Dic3—124no—
202C3×Dic3—24no—
203C3×Dic3—24no—
204C3×Dic3—24no—
205C3×Dic3—44no—
206C22×D4—271no—
207C23×C4—272no—
208C4×D4—278no—
209C2×C4⋊C4—91no—
210C2×C22⋊C4—272no—
211C2×C42—271no—
212C33C2×C4×D411yescharacteristic, complemented, Sylow
213C23×C3—91no—
214C23×C3—182no—
215C22×D3—62no—
216C22×D3—62no—
217C22×D3—62no—
218C22×D3—94no—
219C22×D3—98no—
220C22×D3—122no—
221C22×D3—184no—
222C22×D3—188no—
223C22×D3—182no—
224C22×D3—184no—
225C22×D3—184no—
226C22×D3—188no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—94no—
231C2×C3×C4—94no—
232C2×C3×C4—92no—
233C2×C3×C4—122no—
234C2×C3×C4—182no—
235C2×C3×C4—188no—
236C2×Dic3—22no—
237C2×Dic3—91no—
238C2×Dic3—92no—
239C2×Dic3—92no—
240C2×Dic3—92no—
241C2×Dic3—182no—
242C2×Dic3—188no—
243C2×Dic3—182no—
244C2×Dic3—362no—
245C2×Dic3GO+(4,𝔽2)11yescharacteristic, complemented
246D12—916no—
247C4×D3—64no—
248C4×D3—68no—
249C4×D3—64no—
250C4×D3—68no—
251C4×D3—184no—
252C4×D3—184no—
253C4×D3—188no—
254C4×D3—188no—
255C4×D3—368no—
256C2×C32—22no—
257C2×C32—24no—
258C2×C32—44no—
259C2×C32—42no—
260C2×C32#89312yescomplemented
261C2×C32C2×D3×D411yescharacteristic
262C3⋊D3—28no—
263C3⋊D3#89312yescomplemented
264C3⋊D3C22×C4×D311yescharacteristic
265C3⋊D3C2×#23911yescharacteristic
266C3×D3—64no—
267C3×D3—62no—
268C3×D3—62no—
269C3×D3—68no—
270C3×D3—68no—
271C3×D3—128no—
272C24—272no—
273C2×D4—274no—
274C2×D4—278no—
275C22×C4—92no—
276C22×C4—91no—
277C22×C4—182no—
278C22×C4—271no—
279C22×C4—278no—
280C22×C4—271no—
281C22×C4—542no—
282C22×C4—544no—
283C4⋊C4—94no—
284C22⋊C4—278no—
285C42—274no—
286C22×C3—22no—
287C22×C3—42no—
288C22×C3—91no—
289C22×C3—91no—
290C22×C3—92no—
291C22×C3—92no—
292C22×C3—184no—
293C22×C3—188no—
294C22×C3C2×GO+(4,𝔽2)11yescharacteristic
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—64no—
300C2×D3—68no—
301C2×D3—64no—
302C2×D3—68no—
303C2×D3—98no—
304C2×D3—124no—
305C2×D3—128no—
306C2×D3—188no—
307C2×D3—184no—
308C2×D3—188no—
309C2×D3—1816no—
310C2×D3—188no—
311C3×C4—64no—
312C3×C4—64no—
313C3×C4—64no—
314C3×C4—94no—
315C3×C4—124no—
316Dic3—24no—
317Dic3—92no—
318Dic3—184no—
319Dic3—182no—
320Dic3—364no—
321Dic3C2×GO+(4,𝔽2)12yescomplemented
322C32—22no—
323C32—42no—
324C32C2×#89311yescharacteristic, complemented
325C23—91no—
326C23—182no—
327C23—274no—
328C23—278no—
329C23—542no—
330C23—544no—
331D4—2716no—
332C2×C4—31no—
333C2×C4—62no—
334C2×C4—94no—
335C2×C4—92no—
336C2×C4—94no—
337C2×C4—188no—
338C2×C4—182no—
339C2×C4—272no—
340C2×C4—272no—
341C2×C4—271no—
342C2×C4—272no—
343C2×C4—548no—
344C2×C4—542no—
345C2×C3—24no—
346C2×C3—22no—
347C2×C3—44no—
348C2×C3—42no—
349C2×C3—91no—
350C2×C3—91no—
351C2×C3—92no—
352C2×C3—188no—
353C2×C3C4×GO+(4,𝔽2)12yescomplemented
354C2×C3C22×GO+(4,𝔽2)11yescharacteristic
355D3—64no—
356D3—62no—
357D3—62no—
358D3—68no—
359D3—68no—
360D3—128no—
361C22—92no—
362C22—92no—
363C22—91no—
364C22—91no—
365C22—188no—
366C22—184no—
367C22—278no—
368C22—544no—
369C22—548no—
370C22D3×GO+(4,𝔽2)11yescharacteristic
371C4—32no—
372C4—64no—
373C4—94no—
374C4—272no—
375C4—542no—
376C3—22no—
377C3—42no—
378C3C2×C4×GO+(4,𝔽2)11yescharacteristic, complemented
379C2—91no—
380C2—91no—
381C2—92no—
382C2—188no—
383C2#1172411112yesdirect product factor
384C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊D3C23×C4
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
prime radicalC22×#157⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypocenterC33C2×C4×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#88949C2semidirect
3. C2×#88932C2semidirect
4. C2×C3×#3773C2semidirect
5. C2×Dic3×#157C2semidirect
6. #11724111C2direct (N × H)
11. C2×#157⋊D3C4semidirect
18. #88949C22semidirect
19. #88932C22semidirect
20. C3×#3773C22semidirect
21. Dic3×#157C22semidirect
26. C2×#3773D3semidirect
33. C2×#1113C2×C4semidirect
37. C2×C3×#157C2×C4semidirect
39. #157⋊D3C2×C4semidirect
40. C2×C3⋊(C3⋊Dic3)D4semidirect
42. C2×C32×Dic3D4semidirect
44. Dic3×C3⋊D3D4semidirect
72. #3773C2×D3semidirect
83. #1113C22×C4semidirect
85. C3×#157C22×C4semidirect
86. C3⋊(C3⋊Dic3)C2×D4semidirect
88. C32×Dic3C2×D4semidirect
112. C2×#157C4×D3semidirect
140. C2×C33C4×D4semidirect
143. C3×C3⋊D3C4×D4semidirect
196. #157C2×C4×D3semidirect
212. C33C2×C4×D4semidirect
245. C2×Dic3GO+(4,𝔽2)semidirect
260. C2×C32#893semidirect
263. C3⋊D3#893semidirect
321. Dic3C2×GO+(4,𝔽2)semidirect
324. C32C2×#893semidirect
353. C2×C3C4×GO+(4,𝔽2)semidirect
378. C3C2×C4×GO+(4,𝔽2)semidirect
383. C2#11724111direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#157⋊D32
C2×#889492
C2×#889322
C2×C3×#37732
C2×Dic3×#1572
#117241112
C2×C4×GO+(4,𝔽2)3
C2×#8939