Finite Group Databasebeta

← C2×D3×#3773 #13180533

Subgroups — C2×D3×#3773 #13180533

Order 1728.

1416 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 228 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×D3×#157C211yesmaximal, characteristic
2C2×D3×#624C212yesmaximal
3C2×#88940C211yesmaximal, characteristic, complemented
4C2×#88934C212yesmaximal, complemented
5C2×C3×#3773C211yesmaximal, characteristic, complemented
6D3×#3773C218yesmaximal, direct product factor
7C22×#3773—31nomaximal
8C22×D3×C3⋊D3C2211yescharacteristic
9C22×C3⋊#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×D3×#157C418yescomplemented
12C2×D3×#157C2214yes—
13C2×#6294C2212yes—
14C2×#6289—22no—
15C2×C3×#624C2212yes—
16C2×C3×D3×Dic3—22no—
17D3×#624—24no—
18D3×#624C2218yes—
19#88940C2214yescomplemented
20#88934C2218yescomplemented
21C3×#3773C2214yescomplemented
22C23×#157—31no—
23C22×#624—32no—
24C2×C4×D32—62no—
25C2×#3773—32no—
26C2×#3773—38no—
27C2×#3773—31no—
28C2×#3773D311yescharacteristic, direct product factor
29C22×C3⋊(C3⋊D3)D411yescharacteristic
30C22×C3×C3⋊D3C2311yescharacteristic
31C22×C32×D3D411yescharacteristic
32C2×D3×C3⋊D3C2×C414yes—
33C2×D3×C3⋊D3D412yes—
34C2×D3×C3⋊D3Q814yes—
35C2×D3×C3⋊D3C2312yes—
36C2×C3⋊#157C2×C414yescomplemented
37C2×C3⋊#157C2312yes—
38C2×C3×#157C2×C414yescomplemented
39C2×C3×#157C2312yes—
40D3×#157C2×C4116yescomplemented
41C2×C3×C3⋊Dic3—22no—
42C2×C32×Dic3—22no—
43#6294—22no—
44#6294C2314yes—
45#6289—28no—
46C3×#624—22no—
47C3×#624C2314yes—
48C3×D3×Dic3—28no—
49C2×D3×C4⋊C4—91nomaximal
50C23×C3⋊D3—31no—
51C22×D32—62no—
52C22×#157—31no—
53C22×#157—38no—
54C22×#157—34no—
55C22×#157C2×D311yescharacteristic
56C2×C4×C3⋊D3—62no—
57C22×C3×Dic3—62no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C2×#624—32no—
61C2×#624—38no—
62C2×#624—64no—
63C2×#624—62no—
64C2×#624—62no—
65C2×#624—122no—
66C2×#624C2×D312yes—
67C2×D3×Dic3—22no—
68C2×D3×Dic3—62no—
69C4×D32—68no—
70C4×D32—68no—
71#3773—34no—
72#3773—38no—
73#3773C2×D314yesdirect product factor
74C22×C33C2×D411yescharacteristic
75C2×C3⋊(C3⋊D3)C4⋊C414yescomplemented
76C2×C3⋊(C3⋊D3)C2×D412yes—
77C2×C3×C3⋊D3C22×C412yes—
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C2×C3×C3⋊D3C2×Q812yes—
80C2×C3×C3⋊D3C2411yescharacteristic
81C2×C32×D3C4⋊C414yescomplemented
82C2×C32×D3C2×D412yes—
83D3×C3⋊D3C4⋊C418yescomplemented
84D3×C3⋊D3C22×C414yes—
85D3×C3⋊D3C2×Q814yes—
86C3⋊#157C22×C414yescomplemented
87C3×#157C22×C414yescomplemented
88C3×C3⋊Dic3—24no—
89C32×Dic3—24no—
90C22×C4×D3—92no—
91C22×C4×D3—91no—
92C22×C4×D3—182no—
93C2×C3×C4⋊C4—91no—
94C2×C4⋊Dic3—91no—
95C2×Dic3⋊C4—92no—
96D3×C4⋊C4—98no—
97C23×C32—31no—
98C22×C3⋊D3—31no—
99C22×C3⋊D3—34no—
100C22×C3⋊D3—32no—
101C22×C3⋊D3—34no—
102C22×C3⋊D3—32no—
103C22×C3⋊D3—62no—
104C22×C3⋊D3—122no—
105C22×C3⋊D3C22×D311yescharacteristic
106C22×C3×D3—22no—
107C22×C3×D3—62no—
108C2×D32—68no—
109C2×D32—64no—
110C2×D32—68no—
111C2×D32—64no—
112C2×#157—34no—
113C2×#157—32no—
114C2×#157—316no—
115C2×#157C4×D314yescomplemented
116C2×#157C22×D312yes—
117C2×C32×C4—62no—
118C2×C3⋊Dic3—22no—
119C4×C3⋊D3—68no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—68no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—122no—
127C3×C4×D3—64no—
128C3×C4×D3—64no—
129C3×C4×D3—68no—
130#624—22no—
131#624—34no—
132#624—68no—
133#624—68no—
134#624—62no—
135#624—128no—
136#624C22×D314yes—
137D3×Dic3—28no—
138D3×Dic3—64no—
139D3×Dic3—64no—
140C22×C4⋊C4—271noSylow
141C2×C33C2×C4⋊C412yescomplemented
142C2×C33C22×D411yescharacteristic
143C3⋊(C3⋊D3)C2×C4⋊C414yescomplemented
144C3×C3⋊D3C2×C4⋊C412yescomplemented
145C3×C3⋊D3C23×C411yescharacteristic
146C3×C3⋊D3C22×Q811yescharacteristic
147C32×D3C2×C4⋊C414yescomplemented
148C23×D3—91no—
149C23×D3—182no—
150C22×C3×C4—92no—
151C22×C3×C4—91no—
152C22×C3×C4—182no—
153C22×Dic3—62no—
154C22×Dic3—92no—
155C22×Dic3—91no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—98no—
159C2×C4×D3—98no—
160C2×C4×D3—94no—
161C2×C4×D3—188no—
162C2×C4×D3—182no—
163C2×C4×D3—184no—
164C2×C4×D3—188no—
165C2×C4×D3—182no—
166C2×C4×D3—182no—
167C2×C4×D3—182no—
168C2×C4×D3—362no—
169C3×C4⋊C4—94no—
170C4⋊Dic3—94no—
171Dic3⋊C4—98no—
172C22×C32—22no—
173C22×C32—34no—
174C22×C32—32no—
175C22×C32—42no—
176C22×C32D3×D411yescharacteristic
177C2×C3⋊D3—34no—
178C2×C3⋊D3—32no—
179C2×C3⋊D3—38no—
180C2×C3⋊D3—34no—
181C2×C3⋊D3—34no—
182C2×C3⋊D3—68no—
183C2×C3⋊D3—64no—
184C2×C3⋊D3—128no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3C2×C4×D312yes—
187C2×C3⋊D3D3×D411yescharacteristic
188C2×C3⋊D3D3×Q812yes—
189C2×C3⋊D3C23×D311yescharacteristic
190C2×C3×D3—24no—
191C2×C3×D3—28no—
192C2×C3×D3—64no—
193C2×C3×D3—62no—
194C2×C3×D3—64no—
195C2×C3×D3—62no—
196D32—616no—
197D32—68no—
198D32—68no—
199#157—34no—
200#157C2×C4×D314yescomplemented
201C32×C4—64no—
202C3⋊Dic3—24no—
203C3×Dic3—24no—
204C3×Dic3—24no—
205C3×Dic3—44no—
206C3×Dic3—64no—
207C3×Dic3—64no—
208C3×Dic3—124no—
209C23×C4—272no—
210C23×C4—271no—
211C2×C4⋊C4—91no—
212C2×C4⋊C4—278no—
213C2×C4⋊C4—272no—
214C2×C4⋊C4—271no—
215C33C22×C4⋊C411yescharacteristic, complemented, Sylow
216C23×C3—62no—
217C23×C3—91no—
218C22×D3—62no—
219C22×D3—91no—
220C22×D3—92no—
221C22×D3—92no—
222C22×D3—94no—
223C22×D3—94no—
224C22×D3—184no—
225C22×D3—184no—
226C22×D3—188no—
227C22×D3—188no—
228C22×D3—182no—
229C22×D3—362no—
230C22×D3GO+(4,𝔽2)11yescharacteristic
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—94no—
234C2×C3×C4—94no—
235C2×C3×C4—92no—
236C2×C3×C4—122no—
237C2×C3×C4—182no—
238C2×C3×C4—188no—
239C2×C3×C4—182no—
240C2×Dic3—22no—
241C2×Dic3—68no—
242C2×Dic3—62no—
243C2×Dic3—62no—
244C2×Dic3—94no—
245C2×Dic3—94no—
246C2×Dic3—92no—
247C2×Dic3—122no—
248C2×Dic3—182no—
249C4×D3—64no—
250C4×D3—64no—
251C4×D3—68no—
252C4×D3—916no—
253C4×D3—184no—
254C4×D3—188no—
255C4×D3—188no—
256C4×D3—188no—
257C4×D3—184no—
258C4×D3—368no—
259C2×C32—22no—
260C2×C32—24no—
261C2×C32—34no—
262C2×C32—44no—
263C2×C32—42no—
264C2×C32D3×C4⋊C412yescomplemented
265C2×C32C2×D3×D411yescharacteristic
266C3⋊D3—34no—
267C3⋊D3—68no—
268C3⋊D3—128no—
269C3⋊D3D3×C4⋊C412yescomplemented
270C3⋊D3C22×C4×D311yescharacteristic
271C3⋊D3C2×D3×Q811yescharacteristic
272C3×D3—28no—
273C3×D3—64no—
274C3×D3—62no—
275C3×D3—62no—
276C24—271no—
277C22×C4—92no—
278C22×C4—91no—
279C22×C4—182no—
280C22×C4—278no—
281C22×C4—274no—
282C22×C4—278no—
283C22×C4—272no—
284C22×C4—271no—
285C22×C4—542no—
286C22×C4—544no—
287C4⋊C4—94no—
288C4⋊C4—278no—
289C4⋊C4—274no—
290C22×C3—22no—
291C22×C3—42no—
292C22×C3—64no—
293C22×C3—68no—
294C22×C3—91no—
295C22×C3—91no—
296C22×C3—92no—
297C22×C3—92no—
298C22×C3C2×GO+(4,𝔽2)11yescharacteristic
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—92no—
304C2×D3—94no—
305C2×D3—94no—
306C2×D3—94no—
307C2×D3—98no—
308C2×D3—188no—
309C2×D3—1816no—
310C2×D3—188no—
311C2×D3—184no—
312C2×D3—188no—
313C2×D3—364no—
314C2×D3—368no—
315C2×D3#377314yesdirect product factor
316C2×D3C2×GO+(4,𝔽2)12yes—
317C3×C4—64no—
318C3×C4—64no—
319C3×C4—94no—
320C3×C4—124no—
321C3×C4—184no—
322Dic3—24no—
323Dic3—64no—
324Dic3—64no—
325Dic3—94no—
326Dic3—124no—
327C32—22no—
328C32—42no—
329C32C2×D3×C4⋊C411yescharacteristic, complemented
330C23—31no—
331C23—91no—
332C23—272no—
333C23—274no—
334C23—272no—
335C23—274no—
336C23—271no—
337C2×C4—62no—
338C2×C4—94no—
339C2×C4—92no—
340C2×C4—94no—
341C2×C4—182no—
342C2×C4—188no—
343C2×C4—182no—
344C2×C4—2716no—
345C2×C4—274no—
346C2×C4—272no—
347C2×C4—274no—
348C2×C4—548no—
349C2×C4—542no—
350C2×C3—24no—
351C2×C3—22no—
352C2×C3—44no—
353C2×C3—42no—
354C2×C3—68no—
355C2×C3—91no—
356C2×C3—91no—
357C2×C3—92no—
358C2×C3C2×#377312yescomplemented
359C2×C3C22×GO+(4,𝔽2)11yescharacteristic
360D3—64no—
361D3—62no—
362D3—62no—
363D3—94no—
364D3—188no—
365D3—368no—
366D3C2×#377314yesdirect product factor
367C22—34no—
368C22—32no—
369C22—92no—
370C22—92no—
371C22—91no—
372C22—91no—
373C22—274no—
374C22—274no—
375C22—278no—
376C22—274no—
377C22—272no—
378C22D3×GO+(4,𝔽2)11yescharacteristic
379C4—64no—
380C4—94no—
381C4—184no—
382C4—274no—
383C3—22no—
384C3—42no—
385C3C22×#377311yescharacteristic, complemented
386C2—34no—
387C2—91no—
388C2—92no—
389C2—91no—
390C2—274no—
391C2D3×#377312yesdirect product factor
392C2C2×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×D3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C4⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
3. C2×#88940C2semidirect
4. C2×#88934C2semidirect
5. C2×C3×#3773C2semidirect
6. D3×#3773C2direct (N × H)
11. C2×D3×#157C4semidirect
19. #88940C22semidirect
20. #88934C22semidirect
21. C3×#3773C22semidirect
28. C2×#3773D3direct (N × H)
36. C2×C3⋊#157C2×C4semidirect
38. C2×C3×#157C2×C4semidirect
40. D3×#157C2×C4semidirect
73. #3773C2×D3direct (N × H)
75. C2×C3⋊(C3⋊D3)C4⋊C4semidirect
81. C2×C32×D3C4⋊C4semidirect
83. D3×C3⋊D3C4⋊C4semidirect
86. C3⋊#157C22×C4semidirect
87. C3×#157C22×C4semidirect
115. C2×#157C4×D3semidirect
141. C2×C33C2×C4⋊C4semidirect
143. C3⋊(C3⋊D3)C2×C4⋊C4semidirect
144. C3×C3⋊D3C2×C4⋊C4semidirect
147. C32×D3C2×C4⋊C4semidirect
200. #157C2×C4×D3semidirect
215. C33C22×C4⋊C4semidirect
264. C2×C32D3×C4⋊C4semidirect
269. C3⋊D3D3×C4⋊C4semidirect
315. C2×D3#3773direct (N × H)
329. C32C2×D3×C4⋊C4semidirect
358. C2×C3C2×#3773semidirect
366. D3C2×#3773direct (N × H)
385. C3C22×#3773semidirect
391. C2D3×#3773direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×D3×#6242
C2×#889402
C2×#889342
C2×C3×#37732
D3×#37732
C22×#37733
C2×D3×C4⋊C49