Finite Group Databasebeta

← C4×#88932 #13167103

Subgroups — C4×#88932 #13167103

Order 1728.

944 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 123 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×C4×#1113C211yesmaximal, characteristic
2C2×C4×C3⋊#157C211yesmaximal, characteristic, complemented
3C2×#88932C212yesmaximal
4C3×C4×#624C211yesmaximal, characteristic, complemented
5#11722769C212yesmaximal, complemented
6C4×D32⋊C4—31nomaximal
7C22×#1113C2211yescharacteristic
8C22×C3⋊#157C2212yes—
9C2×C3×C4×C3⋊D3—21no—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×#6294—21no—
12C2×#6294C2211yescharacteristic
13C4×#1113—22no—
14C4×#1113C414yescomplemented
15C2×C3×#624C2212yes—
16C4×C3⋊#157—22no—
17#88932C418yesdirect product factor
18C32×C4×Dic3—21no—
19C3×Dic32—21no—
20C2×C4×D32—31no—
21C2×C4×D32—61no—
22C2×C4×#157—31no—
23C2×D32⋊C4—32no—
24C3×C42×D3—61no—
25C4×#624D311yescharacteristic, complemented
26#63737—32no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×#1113—21no—
30C2×#1113—22no—
31C2×#1113C2×C414yes—
32C2×#1113C2×C412yes—
33C2×C3⋊#157—22no—
34C2×C3⋊#157—22no—
35C2×C3⋊#157C2×C414yescomplemented
36C2×C33×C4D411yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3—24no—
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3D412yes—
42C2×C32×Dic3—22no—
43#6294—21no—
44#6294—24no—
45#6294C2×C412yes—
46C3×#624—22no—
47C3×#624C2×C414yescomplemented
48C4×(C2×D3)⋊C4—91nomaximal
49C22×D32—31no—
50C22×D32—61no—
51C22×#157—32no—
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—121no—
60C2×#624—31no—
61C2×#624—61no—
62C2×#624C2×D312yes—
63C2×D3×Dic3—61no—
64C2×D3×Dic3—61no—
65C2×D3×Dic3—61no—
66C4×D32—34no—
67C4×D32—64no—
68C4×D32—64no—
69C4×D32—62no—
70C4×#157—62no—
71D32⋊C4—38no—
72C32×C42—61no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3—41no—
76C3×C4×Dic3—62no—
77Dic32—21no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3—24no—
81C2×C3×C3⋊D3C22×C411yescharacteristic
82C2×C3×C3⋊D3C22×C412yes—
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C3×C3⋊D3#4112yes—
85#1113—22no—
86#1113—24no—
87#1113C4214yescomplemented
88C3⋊#157—24no—
89C33×C4C22⋊C412yescomplemented
90C3×C3⋊Dic3—22no—
91C3×C3⋊Dic3C22⋊C412yescomplemented
92C32×Dic3—24no—
93C22×C4×D3—91no—
94C22×C4×D3—181no—
95C2×C3×C42—91no—
96C2×(C2×D3)⋊C4—92no—
97C2×C4×Dic3—91no—
98C42×D3—61no—
99C3⋊#95—92no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3C22×D311yescharacteristic
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C22×C3×D3—121no—
107C2×D32—34no—
108C2×D32—32no—
109C2×D32—64no—
110C2×D32—62no—
111C2×D32—64no—
112C2×D32—62no—
113C2×D32—61no—
114C2×D32—62no—
115C2×#157—34no—
116C2×#157—62no—
117C2×#157—62no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4—62no—
123C2×C32×C4#6711yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C4×C3⋊D3—24no—
127C4×C3⋊D3C4×D312yes—
128C4×C3⋊D3D1212yes—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—121no—
137C3×C4×D3—64no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×C4×D3—64no—
143C3×C4×D3—64no—
144C3×C4×D3—64no—
145C3×C4×D3—124no—
146#624—22no—
147#624—32no—
148#624—64no—
149#624—61no—
150#624C4×D314yescomplemented
151D3×Dic3—62no—
152D3×Dic3—64no—
153D3×Dic3—62no—
154D3×Dic3—64no—
155C4×C22⋊C4—271noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33C4×D412yes—
158C3×C3⋊D3—24no—
159C3×C3⋊D3C2×C4211yescharacteristic
160C3×C3⋊D3#11711yescharacteristic
161C3×C3⋊D3C4×D412yes—
162C23×D3—91no—
163C23×D3—181no—
164C22×C3×C4—92no—
165C22×C3×C4—91no—
166C22×C3×C4—181no—
167C22×Dic3—91no—
168C22×Dic3—92no—
169C22×Dic3—181no—
170C2×C4×D3—62no—
171C2×C4×D3—61no—
172C2×C4×D3—61no—
173C2×C4×D3—61no—
174C2×C4×D3—61no—
175C2×C4×D3—94no—
176C2×C4×D3—121no—
177C2×C4×D3—184no—
178C2×C4×D3—181no—
179C2×C4×D3—182no—
180C2×C4×D3—184no—
181C2×C4×D3—181no—
182C2×C4×D3—181no—
183C3×C42—61no—
184C3×C42—61no—
185C3×C42—121no—
186C3×C42—181no—
187(C2×D3)⋊C4—98no—
188C4×Dic3—21no—
189C4×Dic3—62no—
190C4×Dic3—182no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32C2×#6711yescharacteristic
196C2×C3⋊D3—22no—
197C2×C3⋊D3—24no—
198C2×C3⋊D3C2×C4×D311yescharacteristic
199C2×C3⋊D3C2×C4×D312yes—
200C2×C3⋊D3C2×D1211yescharacteristic
201C2×C3⋊D3#23512yes—
202C2×C3×D3—61no—
203C2×C3×D3—62no—
204C2×C3×D3—61no—
205C2×C3×D3—62no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—64no—
212C2×C3×D3—62no—
213C2×C3×D3—64no—
214C2×C3×D3—122no—
215C2×C3×D3—124no—
216D32—34no—
217D32—64no—
218D32—64no—
219D32—68no—
220D32—62no—
221D32—64no—
222#157—64no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—42no—
226C32×C4—42no—
227C32×C4—64no—
228C32×C4(C2×D3)⋊C412yescomplemented
229C3⋊Dic3—22no—
230C3⋊Dic3(C2×D3)⋊C412yescomplemented
231C3×Dic3—24no—
232C3×Dic3—24no—
233C3×Dic3—44no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—122no—
239C23×C4—271no—
240C2×C22⋊C4—272no—
241C2×C42—91no—
242C2×C42—271no—
243#95—272no—
244C33C4×C22⋊C411yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—94no—
252C22×D3—92no—
253C22×D3—121no—
254C22×D3—182no—
255C22×D3—184no—
256C22×D3—181no—
257C22×D3—182no—
258C22×D3—182no—
259C22×D3—184no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—94no—
267C2×C3×C4—91no—
268C2×C3×C4—92no—
269C2×C3×C4—92no—
270C2×C3×C4—122no—
271C2×C3×C4—184no—
272C2×C3×C4—182no—
273C2×C3×C4—181no—
274C2×C3×C4GO+(4,𝔽2)11yescharacteristic
275C2×Dic3—22no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—92no—
281C2×Dic3—94no—
282C2×Dic3—121no—
283C2×Dic3—181no—
284C2×Dic3—184no—
285C2×Dic3—182no—
286C2×Dic3—182no—
287C4×D3—64no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—62no—
291C4×D3—62no—
292C4×D3—64no—
293C4×D3—64no—
294C4×D3—64no—
295C4×D3—124no—
296C4×D3—182no—
297C4×D3—184no—
298C4×D3—184no—
299C4×D3—182no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—42no—
307C2×C32—42no—
308C2×C32C2×(C2×D3)⋊C411yescharacteristic
309C2×C32C4×#6712yes—
310C3⋊D3—24no—
311C3⋊D3C42×D311yescharacteristic
312C3⋊D3#87211yescharacteristic
313C3⋊D3C4×D1212yes—
314C3×D3—61no—
315C3×D3—61no—
316C3×D3—61no—
317C3×D3—61no—
318C3×D3—62no—
319C3×D3—62no—
320C3×D3—64no—
321C3×D3—64no—
322C3×D3—124no—
323C24—271no—
324C22×C4—92no—
325C22×C4—91no—
326C22×C4—181no—
327C22×C4—274no—
328C22×C4—271no—
329C22×C4—272no—
330C22×C4—542no—
331C22×C4—541no—
332C22⋊C4—278no—
333C42—61no—
334C42—181no—
335C42—542no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—92no—
341C22×C3—92no—
342C22×C3—91no—
343C22×C3—91no—
344C22×C3—182no—
345C22×C3—184no—
346C22×C3C2×GO+(4,𝔽2)11yescharacteristic
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—62no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—64no—
357C2×D3—64no—
358C2×D3—62no—
359C2×D3—94no—
360C2×D3—122no—
361C2×D3—124no—
362C2×D3—184no—
363C2×D3—184no—
364C2×D3—182no—
365C2×D3—184no—
366C2×D3—188no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—92no—
374C3×C4—124no—
375C3×C4—182no—
376C3×C4D32⋊C412yescomplemented
377Dic3—24no—
378Dic3—62no—
379Dic3—62no—
380Dic3—62no—
381Dic3—62no—
382Dic3—122no—
383Dic3—184no—
384C32—21no—
385C32—21no—
386C32—41no—
387C32—41no—
388C32C4×(C2×D3)⋊C411yescharacteristic, complemented
389C23—91no—
390C23—181no—
391C23—274no—
392C23—272no—
393C23—541no—
394C23—542no—
395C2×C4—62no—
396C2×C4—94no—
397C2×C4—91no—
398C2×C4—92no—
399C2×C4—92no—
400C2×C4—182no—
401C2×C4—181no—
402C2×C4—184no—
403C2×C4—272no—
404C2×C4—274no—
405C2×C4—544no—
406C2×C4—541no—
407C2×C4—542no—
408C2×C4—542no—
409C2×C4D32⋊D311yescharacteristic
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—42no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—92no—
419C2×C3—91no—
420C2×C3—91no—
421C2×C3—184no—
422C2×C3C4×GO+(4,𝔽2)12yes—
423C2×C3C2×D32⋊C411yescharacteristic
424D3—61no—
425D3—61no—
426D3—62no—
427D3—61no—
428D3—61no—
429D3—62no—
430D3—64no—
431D3—64no—
432D3—124no—
433C22—92no—
434C22—92no—
435C22—91no—
436C22—91no—
437C22—184no—
438C22—182no—
439C22—274no—
440C22—544no—
441C22—542no—
442C22C2×D32⋊D311yescharacteristic
443C4—64no—
444C4—92no—
445C4—182no—
446C4—544no—
447C4#8893212yesdirect product factor
448C3—21no—
449C3—21no—
450C3—41no—
451C3—41no—
452C3C4×D32⋊C411yescharacteristic, complemented
453C2—92no—
454C2—91no—
455C2—91no—
456C2—184no—
457C2C4×D32⋊D312yes—
458C2C2×#8893211yescharacteristic
subgroupG/H
center, hypercenterC2×C4D32⋊D3
commutatorC3×C3⋊D3C2×C42
FrattiniC22C2×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×#1113C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33C4×C22⋊C4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊#157C2semidirect
4. C3×C4×#624C2semidirect
5. #11722769C2semidirect
14. C4×#1113C4semidirect
17. #88932C4direct (N × H)
25. C4×#624D3semidirect
35. C2×C3⋊#157C2×C4semidirect
47. C3×#624C2×C4semidirect
87. #1113C42semidirect
89. C33×C4C22⋊C4semidirect
91. C3×C3⋊Dic3C22⋊C4semidirect
150. #624C4×D3semidirect
228. C32×C4(C2×D3)⋊C4semidirect
230. C3⋊Dic3(C2×D3)⋊C4semidirect
244. C33C4×C22⋊C4semidirect
376. C3×C4D32⋊C4semidirect
388. C32C4×(C2×D3)⋊C4semidirect
447. C4#88932direct (N × H)
452. C3C4×D32⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×C4×C3⋊#1572
C2×#889322
C3×C4×#6242
#117227692
C4×D32⋊C43
C4×(C2×D3)⋊C49