Finite Group Databasebeta

← C2×#11724114 #13180553

Subgroups — C2×#11724114 #13180553

Order 1728.

1432 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1432 conjugacy classes of subgroups — too many to draw. Only its 148 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×C3×GO+(4,𝔽2)C211yesmaximal, characteristic, complemented
2C22×D3×#157C211yesmaximal, characteristic, complemented
3C2×#88952C212yesmaximal, complemented
4C2×#88940C211yesmaximal, characteristic, complemented
5C2×D32⋊Dic3C212yesmaximal, complemented
6#11724114C218yesmaximal, direct product factor
7C2×#64242—31nomaximal
8C22×D3×C3⋊D3C2211yescharacteristic
9C22×C3×D32C2212yes—
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×C3×GO+(4,𝔽2)—24no—
13C2×C3×GO+(4,𝔽2)C2214yescomplemented
14C2×D3×#157—24no—
15C2×D3×#157C2214yescomplemented
16C2×#6294C2212yes—
17C2×#6291—22no—
18C2×C3×#625—22no—
19#88952—24no—
20#88952C2218yescomplemented
21#88940C2214yescomplemented
22D32⋊Dic3C2218yescomplemented
23C23×#157—31no—
24C22×GO+(4,𝔽2)D311yescharacteristic, complemented
25C2×#3811—32no—
26C2×D3×#67—62no—
27#64242—38no—
28C2×#3773—31no—
29C2×D32⋊C4—32no—
30C22×C3⋊(C3⋊D3)D411yescharacteristic, complemented
31C22×C3×C3⋊D3C2311yescharacteristic
32C22×C32×D3—22no—
33C22×C32×D3D411yescharacteristic, complemented
34C2×D3×C3⋊D3—22no—
35C2×D3×C3⋊D3—22no—
36C2×D3×C3⋊D3D412yescomplemented
37C2×D3×C3⋊D3C2312yes—
38C2×C3×D32—24no—
39C2×C3×D32—24no—
40C2×C3×D32—22no—
41C2×C3×D32C2314yes—
42C2×C3⋊#157—22no—
43C2×C3⋊#157C2312yes—
44C2×C3×#157D414yescomplemented
45C2×C3×#157C2312yes—
46C3×GO+(4,𝔽2)—28no—
47D3×#157—28no—
48C2×C3×C3⋊Dic3—22no—
49#6294—22no—
50#6294C2314yes—
51#6291—28no—
52C3×#625—28no—
53C2×#938—91nomaximal
54C23×C3⋊D3—31no—
55C23×C3×D3—62no—
56C22×D32—62no—
57C22×D32C2×D312yes—
58C22×#157—31no—
59C22×#157—34no—
60C22×#157—64no—
61C22×#157C2×D311yescharacteristic
62C2×GO+(4,𝔽2)—24no—
63C2×GO+(4,𝔽2)C2×D314yescomplemented
64C2×#638—62no—
65C2×C3×#67—62no—
66C2×C3×#67—62no—
67#3811—38no—
68#3811—64no—
69D3×#67—68no—
70D3×#67—68no—
71C2×#626—62no—
72C2×#626—62no—
73C2×#626—122no—
74C2×#625—22no—
75C2×#624—32no—
76C2×D3×Dic3—62no—
77#3773—34no—
78D32⋊C4—38no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—22no—
81C2×C3⋊(C3⋊D3)C2×D412yescomplemented
82C2×C3×C3⋊D3C2×D411yescharacteristic
83C2×C3×C3⋊D3C2×D412yes—
84C2×C3×C3⋊D3#4112yes—
85C2×C3×C3⋊D3C2411yescharacteristic
86C2×C32×D3—22no—
87C2×C32×D3—24no—
88C2×C32×D3—44no—
89C2×C32×D3C2×D412yescomplemented
90D3×C3⋊D3—24no—
91D3×C3⋊D3—22no—
92D3×C3⋊D3—22no—
93C3×D32—24no—
94C3×D32—24no—
95C3×D32—44no—
96C3⋊#157—22no—
97C3×#157C2×D414yescomplemented
98C3×C3⋊Dic3—24no—
99C22×C3×D4—91no—
100C22×#67—92no—
101C2×D3×D4—182no—
102C22×C4×D3—91no—
103C2×#217—92no—
104#938—98no—
105C2×C4⋊Dic3—91no—
106C23×C32—31no—
107C23×C32—62no—
108C22×C3⋊D3—31no—
109C22×C3⋊D3—32no—
110C22×C3⋊D3—32no—
111C22×C3⋊D3—62no—
112C22×C3⋊D3—62no—
113C22×C3⋊D3—62no—
114C22×C3⋊D3—122no—
115C22×C3⋊D3C22×D311yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—22no—
118C22×C3×D3—22no—
119C22×C3×D3—42no—
120C22×C3×D3—64no—
121C22×C3×D3—62no—
122C22×C3×D3—64no—
123C22×C3×D3—124no—
124C22×C3×D3—124no—
125C2×D32—22no—
126C2×D32—24no—
127C2×D32—24no—
128C2×D32—64no—
129C2×D32—64no—
130C2×D32—124no—
131C2×D32—124no—
132C2×D32C22×D314yes—
133C2×#157—32no—
134C2×#157—62no—
135C2×#157—68no—
136C2×#157#6714yescomplemented
137C2×#157C22×D312yes—
138GO+(4,𝔽2)—28no—
139#638—68no—
140C2×C3⋊Dic3—22no—
141C3×#67—68no—
142C3×#67—68no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—122no—
146#626—68no—
147#626—68no—
148#626—128no—
149#625—28no—
150#624—34no—
151#624—62no—
152D3×Dic3—64no—
153D3×Dic3—64no—
154C2×#121—271noSylow
155C2×C33#12112yescomplemented
156C2×C33C22×D411yescharacteristic
157C3⋊(C3⋊D3)—22no—
158C3×C3⋊D3#12112yescomplemented
159C3×C3⋊D3C22×D411yescharacteristic
160C3×C3⋊D3C2×#4111yescharacteristic
161C32×D3—22no—
162C32×D3—44no—
163C24×C3—92no—
164C23×D3—62no—
165C23×D3—91no—
166C23×D3—182no—
167C2×C3×D4—94no—
168C2×C3×D4—184no—
169C2×C3×D4—182no—
170C22×C3×C4—91no—
171C2×#67—62no—
172C2×#67—62no—
173C2×#67—98no—
174C2×#67—182no—
175C2×#67—182no—
176C2×#67—184no—
177C2×#67—362no—
178C22×Dic3—92no—
179C22×Dic3—91no—
180D3×D4—188no—
181D3×D4—188no—
182C2×D12—182no—
183C2×C4×D3—94no—
184C2×C4×D3—182no—
185C2×C4×D3—184no—
186#217—98no—
187C4⋊Dic3—94no—
188C22×C32—22no—
189C22×C32—32no—
190C22×C32—42no—
191C22×C32—64no—
192C22×C32—62no—
193C22×C32—124no—
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—32no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—62no—
198C2×C3⋊D3—64no—
199C2×C3⋊D3—62no—
200C2×C3⋊D3—62no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3⋊D3D3×D411yescharacteristic
205C2×C3⋊D3#23712yes—
206C2×C3⋊D3C2×#6712yes—
207C2×C3⋊D3C23×D311yescharacteristic
208C2×C3×D3—24no—
209C2×C3×D3—24no—
210C2×C3×D3—24no—
211C2×C3×D3—44no—
212C2×C3×D3—44no—
213C2×C3×D3—44no—
214C2×C3×D3—44no—
215C2×C3×D3—62no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—62no—
219C2×C3×D3—84no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3—128no—
223D32—24no—
224D32—24no—
225D32—44no—
226D32—124no—
227D32—128no—
228D32—124no—
229#157—62no—
230#157C2×#6714yescomplemented
231C3⋊Dic3—24no—
232C3×Dic3—64no—
233C3×Dic3—64no—
234C3×Dic3—124no—
235C22×D4—91no—
236C22×D4—272no—
237C23×C4—271no—
238#121—278no—
239C2×C4⋊C4—271no—
240C2×C22⋊C4—272no—
241C33C2×#12111yescharacteristic, complemented, Sylow
242C23×C3—62no—
243C23×C3—62no—
244C23×C3—62no—
245C23×C3—94no—
246C23×C3—91no—
247C23×C3—122no—
248C23×C3—182no—
249C23×C3—184no—
250C23×C3—184no—
251C22×D3—22no—
252C22×D3—64no—
253C22×D3—62no—
254C22×D3—64no—
255C22×D3—91no—
256C22×D3—92no—
257C22×D3—92no—
258C22×D3—124no—
259C22×D3—124no—
260C22×D3—184no—
261C22×D3—184no—
262C22×D3—182no—
263C22×D3—182no—
264C22×D3—182no—
265C22×D3—362no—
266C22×D3—364no—
267C22×D3—364no—
268C22×D3GO+(4,𝔽2)11yescharacteristic, complemented
269C3×D4—188no—
270C3×D4—188no—
271C2×C3×C4—94no—
272C2×C3×C4—92no—
273C2×C3×C4—182no—
274#67—68no—
275#67—68no—
276#67—188no—
277#67—188no—
278#67—368no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—94no—
282C2×Dic3—92no—
283C2×Dic3—122no—
284C2×Dic3—182no—
285C2×Dic3—182no—
286D12—188no—
287C4×D3—184no—
288C4×D3—184no—
289C4×D3—188no—
290C2×C32—22no—
291C2×C32—24no—
292C2×C32—44no—
293C2×C32—42no—
294C2×C32—62no—
295C2×C32—124no—
296C2×C32#93812yescomplemented
297C2×C32C2×D3×D411yescharacteristic
298C3⋊D3—62no—
299C3⋊D3—124no—
300C3⋊D3—244no—
301C3⋊D3#93812yescomplemented
302C3⋊D3C2×#23711yescharacteristic
303C3⋊D3C22×#6711yescharacteristic
304C3×D3—44no—
305C3×D3—44no—
306C3×D3—44no—
307C3×D3—64no—
308C3×D3—62no—
309C3×D3—62no—
310C3×D3—84no—
311C24—92no—
312C24—271no—
313C2×D4—94no—
314C2×D4—184no—
315C2×D4—182no—
316C2×D4—278no—
317C2×D4—542no—
318C2×D4—544no—
319C22×C4—91no—
320C22×C4—274no—
321C22×C4—272no—
322C22×C4—271no—
323C22×C4—544no—
324C4⋊C4—274no—
325C22⋊C4—278no—
326C22×C3—22no—
327C22×C3—42no—
328C22×C3—64no—
329C22×C3—64no—
330C22×C3—64no—
331C22×C3—91no—
332C22×C3—91no—
333C22×C3—92no—
334C22×C3—92no—
335C22×C3—124no—
336C22×C3—124no—
337C22×C3—124no—
338C22×C3—124no—
339C22×C3—184no—
340C22×C3—184no—
341C22×C3—244no—
342C22×C3—364no—
343C22×C3C2×GO+(4,𝔽2)11yescharacteristic
344C2×D3—22no—
345C2×D3—24no—
346C2×D3—44no—
347C2×D3—62no—
348C2×D3—62no—
349C2×D3—64no—
350C2×D3—64no—
351C2×D3—92no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—128no—
355C2×D3—184no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3—184no—
359C2×D3—182no—
360C2×D3—364no—
361C2×D3—364no—
362C2×D3—368no—
363C2×D3—364no—
364C2×D3—364no—
365C2×D3—724no—
366C2×D3C2×GO+(4,𝔽2)12yescomplemented
367C3×C4—94no—
368C3×C4—184no—
369Dic3—64no—
370Dic3—64no—
371Dic3—124no—
372Dic3—182no—
373C32—22no—
374C32—42no—
375C32C2×#93811yescharacteristic, complemented
376C23—31no—
377C23—62no—
378C23—94no—
379C23—91no—
380C23—182no—
381C23—184no—
382C23—184no—
383C23—272no—
384C23—272no—
385C23—271no—
386C23—542no—
387C23—542no—
388D4—188no—
389D4—188no—
390D4—548no—
391C2×C4—92no—
392C2×C4—94no—
393C2×C4—182no—
394C2×C4—274no—
395C2×C4—272no—
396C2×C4—548no—
397C2×C4—542no—
398C2×C4—542no—
399C2×C3—24no—
400C2×C3—22no—
401C2×C3—44no—
402C2×C3—42no—
403C2×C3—91no—
404C2×C3—91no—
405C2×C3—92no—
406C2×C3—124no—
407C2×C3—124no—
408C2×C3—124no—
409C2×C3—244no—
410C2×C3#6424212yescomplemented
411C2×C3C22×GO+(4,𝔽2)11yescharacteristic
412D3—22no—
413D3—44no—
414D3—64no—
415D3—62no—
416D3—62no—
417D3—182no—
418D3—364no—
419D3—724no—
420C22—32no—
421C22—62no—
422C22—64no—
423C22—92no—
424C22—92no—
425C22—91no—
426C22—91no—
427C22—124no—
428C22—184no—
429C22—184no—
430C22—272no—
431C22—364no—
432C22—542no—
433C22—542no—
434C22—544no—
435C22—542no—
436C22D3×GO+(4,𝔽2)11yescharacteristic
437C4—94no—
438C4—184no—
439C4—542no—
440C3—22no—
441C3—42no—
442C3C2×#6424211yescharacteristic, complemented
443C2—62no—
444C2—91no—
445C2—91no—
446C2—92no—
447C2—124no—
448C2—542no—
449C2#1172411412yesdirect product factor
450C2C2×D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C2×D3×GO+(4,𝔽2)
Fitting, socleC22×C33C2×D4
hypocenterC33C2×#121
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3×GO+(4,𝔽2)C2semidirect
2. C22×D3×#157C2semidirect
3. C2×#88952C2semidirect
4. C2×#88940C2semidirect
5. C2×D32⋊Dic3C2semidirect
6. #11724114C2direct (N × H)
13. C2×C3×GO+(4,𝔽2)C22semidirect
15. C2×D3×#157C22semidirect
20. #88952C22semidirect
21. #88940C22semidirect
22. D32⋊Dic3C22semidirect
24. C22×GO+(4,𝔽2)D3semidirect
30. C22×C3⋊(C3⋊D3)D4semidirect
33. C22×C32×D3D4semidirect
36. C2×D3×C3⋊D3D4semidirect
44. C2×C3×#157D4semidirect
63. C2×GO+(4,𝔽2)C2×D3semidirect
81. C2×C3⋊(C3⋊D3)C2×D4semidirect
89. C2×C32×D3C2×D4semidirect
97. C3×#157C2×D4semidirect
136. C2×#157#67semidirect
155. C2×C33#121semidirect
158. C3×C3⋊D3#121semidirect
230. #157C2×#67semidirect
241. C33C2×#121semidirect
268. C22×D3GO+(4,𝔽2)semidirect
296. C2×C32#938semidirect
301. C3⋊D3#938semidirect
366. C2×D3C2×GO+(4,𝔽2)semidirect
375. C32C2×#938semidirect
410. C2×C3#64242semidirect
442. C3C2×#64242semidirect
449. C2#11724114direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C22×D3×#1572
C2×#889522
C2×#889402
C2×D32⋊Dic32
#117241142
C2×#642423
C2×#9389