Finite Group Databasebeta

← C2×#11724355 #13181172

Subgroups — C2×#11724355 #13181172

Order 1728.

1432 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×C3×#157C211yesmaximal, characteristic, complemented
2C22×#157⋊D3C211yesmaximal, characteristic, complemented
3C2×#89049C212yesmaximal, complemented
4#11724355C218yesmaximal, direct product factor
5C2×#88940C211yesmaximal, characteristic, complemented
6C2×#88939C212yesmaximal, complemented
7C2×#64242—31nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×#1113C2212yes—
10C22×C3×#157—24no—
11C22×C3×#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C22×C3×#157C2214yescomplemented
14C2×#157⋊D3—24no—
15C2×#157⋊D3C2214yescomplemented
16C2×C3×#638—22no—
17#89049—24no—
18#89049C2218yescomplemented
19C2×#6295—22no—
20C2×#6294C2212yes—
21#88940C2214yescomplemented
22#88939C2218yescomplemented
23C23×#157D311yescharacteristic, complemented
24C22×GO+(4,𝔽2)—31no—
25C2×#3811—32no—
26C2×D3×#67—62no—
27#64242—38no—
28C2×#3773—31no—
29C2×D32⋊C4—32no—
30C23×C33D411yescharacteristic, complemented
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3—22no—
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3D412yescomplemented
35C22×C3×C3⋊D3D411yescharacteristic, complemented
36C22×C3×C3⋊D3C2312yes—
37C22×C3×C3⋊D3C2311yescharacteristic
38C2×#1113—24no—
39C2×#1113—24no—
40C2×#1113—22no—
41C2×#1113C2314yes—
42C2×C3×#157—22no—
43C2×C3×#157—28no—
44C2×C3×#157D414yescomplemented
45C2×C3×#157C2312yes—
46C2×C3×#157C2312yes—
47#157⋊D3—28no—
48C3×#638—28no—
49C2×C3×C3⋊Dic3—22no—
50#6295—28no—
51#6294—22no—
52#6294C2314yes—
53C2×#930—91nomaximal
54C23×C3⋊D3C2×D311yescharacteristic
55C23×C3×D3—62no—
56C22×D32—32no—
57C22×D32—62no—
58C22×#157—24no—
59C22×#157C2×D311yescharacteristic
60C22×#157C2×D311yescharacteristic
61C22×#157C2×D314yescomplemented
62C2×GO+(4,𝔽2)—34no—
63C2×GO+(4,𝔽2)—64no—
64C2×#638—22no—
65C2×C3×#67—62no—
66C2×C3×#67—62no—
67C2×C3×#67—122no—
68#3811—38no—
69#3811—64no—
70D3×#67—68no—
71D3×#67—68no—
72C2×#626—62no—
73C2×#626—62no—
74C2×#625—62no—
75C2×#624—32no—
76C2×D3×Dic3—62no—
77#3773—34no—
78D32⋊C4—38no—
79C22×C33—22no—
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D412yescomplemented
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—24no—
87C2×C3×C3⋊D3—44no—
88C2×C3×C3⋊D3C2×D411yescharacteristic
89C2×C3×C3⋊D3C2×D412yes—
90C2×C3×C3⋊D3C2×D412yescomplemented
91C2×C3×C3⋊D3#4112yes—
92C2×C3×C3⋊D3C2411yescharacteristic
93#1113—24no—
94#1113—24no—
95#1113—44no—
96C3×#157—22no—
97C3×#157C2×D414yescomplemented
98C3×C3⋊Dic3—24no—
99C23×C3×C4—91no—
100C22×#67—92no—
101C2×D3×D4—182no—
102C22×D12—91no—
103#930—98no—
104C2×(C2×D3)⋊C4—92no—
105C2×C4⋊Dic3—91no—
106C23×C32—22no—
107C23×C32—42no—
108C23×C32D1211yescharacteristic, complemented
109C22×C3⋊D3—22no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3—22no—
112C22×C3⋊D3D1212yescomplemented
113C22×C3⋊D3D1211yescharacteristic, complemented
114C22×C3⋊D3C22×D312yes—
115C22×C3⋊D3C22×D311yescharacteristic
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—64no—
119C22×C3×D3—64no—
120C22×C3×D3—62no—
121C22×C3×D3—62no—
122C22×C3×D3—124no—
123C22×C3×D3—124no—
124C22×C3×D3—122no—
125C2×D32—34no—
126C2×D32—64no—
127C2×D32—64no—
128C2×D32—62no—
129C2×D32—64no—
130C2×D32—64no—
131C2×D32—124no—
132C2×D32—124no—
133C2×#157—22no—
134C2×#157—28no—
135C2×#157#6714yescomplemented
136C2×#157C22×D312yes—
137C2×#157C22×D312yes—
138GO+(4,𝔽2)—68no—
139#638—28no—
140C2×C3⋊Dic3—22no—
141C3×#67—68no—
142C3×#67—68no—
143C3×#67—128no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—122no—
147#626—68no—
148#626—68no—
149#625—68no—
150#624—34no—
151#624—62no—
152D3×Dic3—64no—
153D3×Dic3—64no—
154C2×#121—271noSylow
155C2×C33—22no—
156C2×C33#12112yescomplemented
157C2×C33C22×D411yescharacteristic
158C3×C3⋊D3—22no—
159C3×C3⋊D3—44no—
160C3×C3⋊D3#12112yescomplemented
161C3×C3⋊D3C22×D411yescharacteristic
162C3×C3⋊D3C2×#4111yescharacteristic
163C24×C3—91no—
164C23×D3—62no—
165C23×D3—92no—
166C23×D3—182no—
167C2×C3×D4—182no—
168C22×C3×C4—91no—
169C22×C3×C4—91no—
170C22×C3×C4—94no—
171C22×C3×C4—184no—
172C2×#67—62no—
173C2×#67—62no—
174C2×#67—98no—
175C2×#67—122no—
176C2×#67—182no—
177C2×#67—182no—
178C2×#67—184no—
179C22×Dic3—92no—
180D3×D4—188no—
181D3×D4—188no—
182C2×D12—94no—
183C2×D12—184no—
184C2×D12—182no—
185C2×C4×D3—182no—
186(C2×D3)⋊C4—98no—
187C4⋊Dic3—94no—
188C22×C32—22no—
189C22×C32—24no—
190C22×C32—22no—
191C22×C32—42no—
192C22×C32—44no—
193C22×C32—44no—
194C22×C32—84no—
195C22×C32C2×D1211yescharacteristic
196C22×C32C2×D1212yescomplemented
197C2×C3⋊D3—24no—
198C2×C3⋊D3—22no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3—44no—
203C2×C3⋊D3C2×D1211yescharacteristic
204C2×C3⋊D3C2×D1212yescomplemented
205C2×C3⋊D3#23512yes—
206C2×C3⋊D3C2×#6712yes—
207C2×C3⋊D3C23×D311yescharacteristic
208C2×C3×D3—62no—
209C2×C3×D3—64no—
210C2×C3×D3—64no—
211C2×C3×D3—62no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—124no—
216C2×C3×D3—128no—
217C2×C3×D3—124no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3—244no—
223D32—64no—
224D32—64no—
225D32—124no—
226D32—124no—
227D32—128no—
228D32—124no—
229#157—22no—
230#157C2×#6714yescomplemented
231C3⋊Dic3—24no—
232C3×Dic3—64no—
233C3×Dic3—64no—
234C3×Dic3—124no—
235C22×D4—271no—
236C22×D4—272no—
237C23×C4—91no—
238#121—278no—
239C2×C4⋊C4—271no—
240C2×C22⋊C4—272no—
241C33C2×#12111yescharacteristic, complemented, Sylow
242C23×C3—22no—
243C23×C3—42no—
244C23×C3—91no—
245C23×C3—91no—
246C23×C3—92no—
247C23×C3—92no—
248C23×C3—182no—
249C23×C3—182no—
250C23×C3—182no—
251C23×C3GO+(4,𝔽2)11yescharacteristic, complemented
252C22×D3—62no—
253C22×D3—62no—
254C22×D3—64no—
255C22×D3—64no—
256C22×D3—62no—
257C22×D3—62no—
258C22×D3—94no—
259C22×D3—124no—
260C22×D3—124no—
261C22×D3—122no—
262C22×D3—184no—
263C22×D3—182no—
264C22×D3—184no—
265C22×D3—184no—
266C22×D3—184no—
267C22×D3—364no—
268C22×D3—364no—
269C3×D4—188no—
270C2×C3×C4—94no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—188no—
274C2×C3×C4—182no—
275C2×C3×C4—182no—
276#67—68no—
277#67—68no—
278#67—128no—
279#67—188no—
280#67—188no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—94no—
284C2×Dic3—122no—
285C2×Dic3—182no—
286D12—188no—
287D12—188no—
288C4×D3—184no—
289C4×D3—184no—
290C2×C32—22no—
291C2×C32—24no—
292C2×C32—22no—
293C2×C32—44no—
294C2×C32—42no—
295C2×C32—44no—
296C2×C32—84no—
297C2×C32#93012yescomplemented
298C2×C32C22×D1211yescharacteristic
299C3⋊D3—22no—
300C3⋊D3—44no—
301C3⋊D3#93012yescomplemented
302C3⋊D3C2×#23511yescharacteristic
303C3⋊D3C22×#6711yescharacteristic
304C3×D3—64no—
305C3×D3—62no—
306C3×D3—62no—
307C3×D3—124no—
308C3×D3—124no—
309C3×D3—124no—
310C3×D3—244no—
311C24—91no—
312C24—272no—
313C2×D4—182no—
314C2×D4—274no—
315C2×D4—278no—
316C2×D4—544no—
317C2×D4—542no—
318C2×D4—544no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—94no—
322C22×C4—184no—
323C22×C4—272no—
324C4⋊C4—274no—
325C22⋊C4—278no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—24no—
329C22×C3—44no—
330C22×C3—42no—
331C22×C3—44no—
332C22×C3—84no—
333C22×C3—91no—
334C22×C3—91no—
335C22×C3—92no—
336C22×C3—92no—
337C22×C3—92no—
338C22×C3—182no—
339C22×C3—182no—
340C22×C3—184no—
341C22×C3—184no—
342C22×C3—182no—
343C22×C3—364no—
344C22×C3C2×GO+(4,𝔽2)11yescharacteristic
345C22×C3C2×GO+(4,𝔽2)12yescomplemented
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—64no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—64no—
352C2×D3—64no—
353C2×D3—124no—
354C2×D3—128no—
355C2×D3—124no—
356C2×D3—124no—
357C2×D3—124no—
358C2×D3—124no—
359C2×D3—124no—
360C2×D3—184no—
361C2×D3—184no—
362C2×D3—244no—
363C2×D3—364no—
364C2×D3—364no—
365C2×D3—368no—
366C2×D3—364no—
367C3×C4—94no—
368C3×C4—184no—
369C3×C4—182no—
370Dic3—64no—
371Dic3—64no—
372Dic3—124no—
373C32—22no—
374C32—42no—
375C32C2×#93011yescharacteristic, complemented
376C23—91no—
377C23—91no—
378C23—92no—
379C23—92no—
380C23—182no—
381C23—182no—
382C23—182no—
383C23—274no—
384C23—542no—
385C23—544no—
386C23—544no—
387C23#157⋊D311yescharacteristic, complemented
388D4—188no—
389D4—548no—
390D4—548no—
391C2×C4—92no—
392C2×C4—94no—
393C2×C4—92no—
394C2×C4—188no—
395C2×C4—182no—
396C2×C4—182no—
397C2×C4—274no—
398C2×C4—542no—
399C2×C3—22no—
400C2×C3—24no—
401C2×C3—22no—
402C2×C3—44no—
403C2×C3—44no—
404C2×C3—42no—
405C2×C3—84no—
406C2×C3—91no—
407C2×C3—91no—
408C2×C3—92no—
409C2×C3—182no—
410C2×C3—364no—
411C2×C3#6424212yescomplemented
412C2×C3C22×GO+(4,𝔽2)11yescharacteristic
413D3—64no—
414D3—62no—
415D3—62no—
416D3—124no—
417D3—124no—
418D3—124no—
419D3—244no—
420C22—22no—
421C22—92no—
422C22—92no—
423C22—91no—
424C22—91no—
425C22—92no—
426C22—182no—
427C22—182no—
428C22—184no—
429C22—182no—
430C22—184no—
431C22—364no—
432C22—544no—
433C22—544no—
434C22—1084no—
435C22C2×#157⋊D311yescharacteristic
436C22C2×#157⋊D312yescomplemented
437C4—94no—
438C4—182no—
439C4—184no—
440C3—22no—
441C3—42no—
442C3C2×#6424211yescharacteristic, complemented
443C2—22no—
444C2—91no—
445C2—91no—
446C2—92no—
447C2—182no—
448C2—364no—
449C2#1172435512yesdirect product factor
450C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33C2×#121
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×#157C2semidirect
2. C22×#157⋊D3C2semidirect
3. C2×#89049C2semidirect
4. #11724355C2direct (N × H)
5. C2×#88940C2semidirect
6. C2×#88939C2semidirect
13. C22×C3×#157C22semidirect
15. C2×#157⋊D3C22semidirect
18. #89049C22semidirect
21. #88940C22semidirect
22. #88939C22semidirect
23. C23×#157D3semidirect
30. C23×C33D4semidirect
34. C22×C3×C3⋊D3D4semidirect
35. C22×C3×C3⋊D3D4semidirect
44. C2×C3×#157D4semidirect
61. C22×#157C2×D3semidirect
81. C22×C33C2×D4semidirect
90. C2×C3×C3⋊D3C2×D4semidirect
97. C3×#157C2×D4semidirect
108. C23×C32D12semidirect
112. C22×C3⋊D3D12semidirect
113. C22×C3⋊D3D12semidirect
135. C2×#157#67semidirect
156. C2×C33#121semidirect
160. C3×C3⋊D3#121semidirect
196. C22×C32C2×D12semidirect
204. C2×C3⋊D3C2×D12semidirect
230. #157C2×#67semidirect
241. C33C2×#121semidirect
251. C23×C3GO+(4,𝔽2)semidirect
297. C2×C32#930semidirect
301. C3⋊D3#930semidirect
345. C22×C3C2×GO+(4,𝔽2)semidirect
375. C32C2×#930semidirect
387. C23#157⋊D3semidirect
411. C2×C3#64242semidirect
436. C22C2×#157⋊D3semidirect
442. C3C2×#64242semidirect
449. C2#11724355direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C22×#157⋊D32
C2×#890492
#117243552
C2×#889402
C2×#889392
C2×#642423
C2×#9309