Finite Group Databasebeta

← C2×#11724300 #13181061

Subgroups — C2×#11724300 #13181061

Order 1728.

1128 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1128 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
1C2×#89040C211yesmaximal, characteristic, complemented
2C2×C3×C4×D32C211yesmaximal, characteristic
3C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
4C2×#88934C212yesmaximal, complemented
5C2×D32⋊Dic3C212yesmaximal
6#11724300C218yesmaximal, direct product factor
7C2×D32⋊Q8—31nomaximal
8C22×C3×D32C2211yescharacteristic
9C22×C3⋊#157C2212yes—
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C3×#634—21no—
12C2×C32×C4×D3—21no—
13C2×#6296—21no—
14C2×#6294C2212yes—
15#89040—22no—
16#89040C2214yescomplemented
17C2×C3×#624C2211yescharacteristic
18C2×C3×D3×Dic3—21no—
19C3×C4×D32—22no—
20C3×C4×D32C2214yes—
21C4⋊(C3⋊#157)C2214yescomplemented
22#88934C2218yescomplemented
23D32⋊Dic3C2218yes—
24C22×C3×C4×D3—61no—
25C2×C4×D32D311yescharacteristic
26C2×#3797—31no—
27C2×D3×Dic6—61no—
28C2×C4⋊#157—31no—
29C2×#3773—32no—
30C2×D32⋊C4—32no—
31D32⋊Q8—38no—
32C22×C3×C3⋊D3C2311yescharacteristic
33C22×C32×D3—21no—
34C2×C3×D32—21no—
35C2×C3×D32—22no—
36C2×C3×D32Q814yescomplemented
37C2×C3×D32C2312yes—
38C2×C3⋊#157—24no—
39C2×C3⋊#157C2314yes—
40C2×C33×C4D411yescharacteristic
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4×C3⋊D3D412yes—
44C3×C4×C3⋊D3C2312yes—
45C3×#634—24no—
46C2×C32×Dic3—21no—
47C32×C4×D3—24no—
48#6296—24no—
49#6294—22no—
50#6294C2314yes—
51C3×#624—21no—
52C3×#624C2312yes—
53C3×D3×Dic3—24no—
54C2×#924—91nomaximal
55C23×C3×D3—61no—
56C22×D32C2×D311yescharacteristic
57C22×#157—32no—
58C22×C32×C4—61no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634—21no—
61C22×C3×Dic3—61no—
62C2×C3×C4×D3—21no—
63C2×C3×C4×D3—21no—
64C2×C3×C4×D3—41no—
65C2×C3×C4×D3—64no—
66C2×C3×C4×D3—64no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—61no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6—121no—
74C2×#627—62no—
75C2×#627—61no—
76C2×#624—32no—
77C2×#624C2×D311yescharacteristic
78C2×D3×Dic3—21no—
79C2×D3×Dic3—62no—
80C4×D32—22no—
81C4×D32C2×D314yes—
82#3797—34no—
83#3797—62no—
84D3×Dic6—64no—
85D3×Dic6—64no—
86C4⋊#157—34no—
87#3773—38no—
88D32⋊C4—38no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3C2×D411yescharacteristic
91C2×C3×C3⋊D3C2×Q812yes—
92C2×C3×C3⋊D3#4112yes—
93C2×C3×C3⋊D3C2411yescharacteristic
94C2×C32×D3—24no—
95C2×C32×D3—22no—
96C3×D32—24no—
97C3×D32—22no—
98C3×D32C2×Q814yescomplemented
99C3⋊#157—24no—
100C33×C4C2×D412yes—
101C3×C3⋊Dic3—24no—
102C3×C3⋊Dic3C2×D412yes—
103C32×Dic3—22no—
104C23×C3×C4—91no—
105C2×D3×Q8—181no—
106C22×C4×D3—61no—
107C22×Dic6—91no—
108C2×#217—92no—
109C2×C4⋊Dic3—91no—
110#924—98no—
111C2×Dic3⋊C4—92no—
112C23×C32—61no—
113C22×C3⋊D3C22×D311yescharacteristic
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—41no—
117C22×C3×D3—64no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—64no—
122C22×C3×D3—62no—
123C2×D32—22no—
124C2×D32—21no—
125C2×D32Dic614yescomplemented
126C2×D32C22×D312yes—
127C2×#157—34no—
128C2×#157—64no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4—64no—
134C2×C32×C4—61no—
135C2×C32×C4#6711yescharacteristic
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3#6711yescharacteristic
138C4×C3⋊D3#6712yes—
139C4×C3⋊D3C22×D312yes—
140#634—24no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—122no—
150C3×C4×D3—24no—
151C3×C4×D3—24no—
152C3×C4×D3—44no—
153C3×C4×D3—62no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—64no—
160C3×Dic6—64no—
161C3×Dic6—64no—
162C3×Dic6—124no—
163#627—68no—
164#627—64no—
165#624—21no—
166#624—34no—
167#624—62no—
168#624C22×D312yes—
169D3×Dic3—24no—
170D3×Dic3—64no—
171D3×Dic3—64no—
172C2×C22⋊Q8—271noSylow
173C2×C33C22⋊Q812yescomplemented
174C2×C33C22×D411yescharacteristic
175C3×C3⋊D3C22⋊Q812yescomplemented
176C3×C3⋊D3C22×Q811yescharacteristic
177C3×C3⋊D3C2×#4111yescharacteristic
178C32×D3—24no—
179C24×C3—91no—
180C23×D3—61no—
181C2×C3×Q8—181no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4—91no—
185C22×C3×C4—94no—
186C22×C3×C4—91no—
187C22×C3×C4—121no—
188C22×C3×C4—182no—
189C22×C3×C4—181no—
190C22×Dic3—61no—
191C22×Dic3—92no—
192C22×Dic3—92no—
193D3×Q8—184no—
194D3×Q8—184no—
195C2×C4×D3—21no—
196C2×C4×D3—64no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×C4×D3—61no—
200C2×C4×D3—64no—
201C2×C4×D3—61no—
202C2×C4×D3—182no—
203C2×Dic6—61no—
204C2×Dic6—61no—
205C2×Dic6—94no—
206C2×Dic6—121no—
207C2×Dic6—182no—
208C2×Dic6—181no—
209C2×Dic6—182no—
210#217—98no—
211C4⋊Dic3—94no—
212Dic3⋊C4—98no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—62no—
218C22×C32—64no—
219C22×C32C2×#6711yescharacteristic
220C2×C3⋊D3C2×Dic612yes—
221C2×C3⋊D3#23512yes—
222C2×C3⋊D3C2×#6711yescharacteristic
223C2×C3⋊D3C23×D311yescharacteristic
224C2×C3×D3—22no—
225C2×C3×D3—24no—
226C2×C3×D3—24no—
227C2×C3×D3—22no—
228C2×C3×D3—42no—
229C2×C3×D3—44no—
230C2×C3×D3—64no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—68no—
240C2×C3×D3—64no—
241D32—24no—
242D32—22no—
243D32C2×Dic614yescomplemented
244#157—64no—
245C32×C4—22no—
246C32×C4—22no—
247C32×C4—42no—
248C32×C4—42no—
249C32×C4—62no—
250C32×C4C2×#6712yes—
251C3⋊Dic3—24no—
252C3⋊Dic3C2×#6712yes—
253C3×Dic3—22no—
254C3×Dic3—22no—
255C3×Dic3—42no—
256C3×Dic3—62no—
257C3×Dic3—62no—
258C3×Dic3—64no—
259C3×Dic3—64no—
260C3×Dic3—124no—
261C22×Q8—271no—
262C23×C4—91no—
263C22⋊Q8—278no—
264C2×C4⋊C4—271no—
265C2×C4⋊C4—272no—
266C2×C22⋊C4—272no—
267C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
268C23×C3—61no—
269C23×C3—61no—
270C23×C3—92no—
271C23×C3—94no—
272C23×C3—91no—
273C23×C3—121no—
274C23×C3—181no—
275C23×C3—182no—
276C22×D3—21no—
277C22×D3—64no—
278C22×D3—62no—
279C22×D3—61no—
280C22×D3—61no—
281C22×D3—62no—
282C22×D3—64no—
283C3×Q8—184no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—64no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—64no—
292C2×C3×C4—92no—
293C2×C3×C4—91no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—124no—
297C2×C3×C4—121no—
298C2×C3×C4—181no—
299C2×C3×C4—184no—
300C2×C3×C4—182no—
301C2×C3×C4GO+(4,𝔽2)11yescharacteristic
302C2×Dic3—21no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—64no—
306C2×Dic3—62no—
307C2×Dic3—62no—
308C2×Dic3—94no—
309C2×Dic3—94no—
310C2×Dic3—122no—
311C2×Dic3—182no—
312C2×Dic3—184no—
313C4×D3—24no—
314C4×D3—62no—
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—62no—
318C4×D3—62no—
319C4×D3—62no—
320C4×D3—64no—
321C4×D3—184no—
322C4×D3—184no—
323Dic6—64no—
324Dic6—64no—
325Dic6—124no—
326Dic6—184no—
327Dic6—188no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—22no—
331C2×C32—22no—
332C2×C32—42no—
333C2×C32—42no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—64no—
337C2×C32#92412yescomplemented
338C2×C32C22×#6711yescharacteristic
339C3⋊D3#92412yescomplemented
340C3⋊D3C22×Dic611yescharacteristic
341C3⋊D3C2×#23511yescharacteristic
342C3×D3—24no—
343C3×D3—24no—
344C3×D3—44no—
345C3×D3—62no—
346C3×D3—62no—
347C3×D3—61no—
348C3×D3—61no—
349C3×D3—61no—
350C3×D3—61no—
351C24—91no—
352C2×Q8—181no—
353C2×Q8—274no—
354C2×Q8—541no—
355C2×Q8—542no—
356C22×C4—61no—
357C22×C4—91no—
358C22×C4—94no—
359C22×C4—91no—
360C22×C4—182no—
361C22×C4—181no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—278no—
365C4⋊C4—274no—
366C22⋊C4—278no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—62no—
372C22×C3—64no—
373C22×C3—62no—
374C22×C3—64no—
375C22×C3—94no—
376C22×C3—91no—
377C22×C3—91no—
378C22×C3—92no—
379C22×C3—92no—
380C22×C3—122no—
381C22×C3—124no—
382C22×C3—182no—
383C22×C3—184no—
384C22×C3C2×GO+(4,𝔽2)11yescharacteristic
385C2×D3—22no—
386C2×D3—24no—
387C2×D3—64no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—62no—
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—62no—
396C2×D3—68no—
397C2×D3—64no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—92no—
405C3×C4—122no—
406C3×C4—184no—
407C3×C4C2×GO+(4,𝔽2)12yes—
408Dic3—22no—
409Dic3—62no—
410Dic3—62no—
411Dic3—64no—
412Dic3—64no—
413Dic3—124no—
414Dic3—184no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32—41no—
419C32C2×#92411yescharacteristic, complemented
420C23—61no—
421C23—92no—
422C23—94no—
423C23—91no—
424C23—181no—
425C23—182no—
426Q8—184no—
427Q8—544no—
428C2×C4—61no—
429C2×C4—64no—
430C2×C4—92no—
431C2×C4—91no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—181no—
435C2×C4—184no—
436C2×C4—182no—
437C2×C4—274no—
438C2×C4—274no—
439C2×C4—542no—
440C2×C4—544no—
441C2×C4D32⋊D311yescharacteristic
442C2×C3—22no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—42no—
447C2×C3—42no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—64no—
451C2×C3—64no—
452C2×C3—91no—
453C2×C3—91no—
454C2×C3—92no—
455C2×C3—124no—
456C2×C3D32⋊Q812yescomplemented
457C2×C3C22×GO+(4,𝔽2)11yescharacteristic
458D3—24no—
459D3—62no—
460D3—61no—
461D3—61no—
462D3—62no—
463D3—61no—
464D3—61no—
465C22—64no—
466C22—62no—
467C22—94no—
468C22—92no—
469C22—92no—
470C22—91no—
471C22—91no—
472C22—182no—
473C22—184no—
474C22C2×D32⋊D311yescharacteristic
475C4—62no—
476C4—92no—
477C4—184no—
478C4—544no—
479C4C2×D32⋊D312yes—
480C3—21no—
481C3—21no—
482C3—41no—
483C3—41no—
484C3C2×D32⋊Q811yescharacteristic, complemented
485C2—64no—
486C2—91no—
487C2—91no—
488C2—92no—
489C2#1172430012yesdirect product factor
490C2C22×D32⋊D311yescharacteristic
subgroupG/H
centerC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×D32⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
hypercenterC2×C4D32⋊D3
hypocenterC33C2×C22⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
3. C2×C4⋊(C3⋊#157)C2semidirect
4. C2×#88934C2semidirect
6. #11724300C2direct (N × H)
16. #89040C22semidirect
21. C4⋊(C3⋊#157)C22semidirect
22. #88934C22semidirect
36. C2×C3×D32Q8semidirect
98. C3×D32C2×Q8semidirect
125. C2×D32Dic6semidirect
173. C2×C33C22⋊Q8semidirect
175. C3×C3⋊D3C22⋊Q8semidirect
243. D32C2×Dic6semidirect
267. C33C2×C22⋊Q8semidirect
337. C2×C32#924semidirect
339. C3⋊D3#924semidirect
419. C32C2×#924semidirect
456. C2×C3D32⋊Q8semidirect
484. C3C2×D32⋊Q8semidirect
489. C2#11724300direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890402
C2×C3×C4×D322
C2×C4⋊(C3⋊#157)2
C2×#889342
C2×D32⋊Dic32
#117243002
C2×D32⋊Q83
C2×#9249