Finite Group Databasebeta

← C2×#11724333 #13181127

Subgroups — C2×#11724333 #13181127

Order 1728.

1128 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 151 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×#89040C211yesmaximal, characteristic, complemented
3C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
4C2×#88940C212yesmaximal, complemented
5C2×#88939C212yesmaximal
6#11724333C218yesmaximal, direct product factor
7C2×D32⋊Q8—31nomaximal
8C22×#1113C2211yescharacteristic
9C22×C3×#157C2212yes—
10C2×C3×C4×C3⋊D3—21no—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634—21no—
13C2×#6296—21no—
14C2×#6294—21no—
15C2×#6294C2212yes—
16C2×#6294C2211yescharacteristic
17C4×#1113—22no—
18C4×#1113C2214yes—
19#89040—22no—
20#89040C2214yescomplemented
21C3×C4⋊#157C2214yescomplemented
22#88940C2218yescomplemented
23#88939C2218yes—
24C2×C4×D32—31no—
25C2×C4×D32—61no—
26C2×#3797—31no—
27C2×D3×Dic6—61no—
28C2×C4⋊#157D311yescharacteristic, complemented
29C2×#3773—32no—
30C2×D32⋊C4—32no—
31D32⋊Q8—38no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3C2311yescharacteristic
34C2×#1113—21no—
35C2×#1113—22no—
36C2×#1113Q814yescomplemented
37C2×#1113C2312yes—
38C2×C3×#157—24no—
39C2×C3×#157C2314yes—
40C2×C33×C4D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3—24no—
45C3×C4×C3⋊D3D412yes—
46C3×C4×C3⋊D3C2312yes—
47C3×#634—24no—
48#6296—24no—
49#6294—22no—
50#6294—21no—
51#6294—24no—
52#6294C2314yes—
53#6294C2312yes—
54C2×#897—91nomaximal
55C22×D32—31no—
56C22×D32—61no—
57C22×#157C2×D312yes—
58C2×C4×C3⋊D3—21no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634—21no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—121no—
66C2×C3×Dic6—61no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6—121no—
69C2×#627—62no—
70C2×#627—61no—
71C2×#624—32no—
72C2×#624—31no—
73C2×#624—61no—
74C2×D3×Dic3—62no—
75C2×D3×Dic3—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C4×D32—34no—
79C4×D32—64no—
80C4×D32—64no—
81C4×D32—62no—
82#3797—34no—
83#3797—62no—
84D3×Dic6—64no—
85D3×Dic6—64no—
86C4⋊#157C2×D314yescomplemented
87#3773—38no—
88D32⋊C4—38no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—24no—
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3C2×D411yescharacteristic
93C2×C3×C3⋊D3C2×Q812yes—
94C2×C3×C3⋊D3#4112yes—
95C2×C3×C3⋊D3C2411yescharacteristic
96#1113—24no—
97#1113—22no—
98#1113C2×Q814yescomplemented
99C3×#157—24no—
100C33×C4C2×D412yes—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3—24no—
103C3×C3⋊Dic3C2×D412yes—
104C2×D3×Q8—181no—
105C22×C4×D3—91no—
106C22×C4×D3—181no—
107C22×Dic6—91no—
108C2×C3×C4⋊C4—91no—
109C2×(C2×D3)⋊C4—92no—
110C2×C4⋊Dic3—92no—
111#897—98no—
112C22×C3⋊D3—21no—
113C22×C3⋊D3C22×D311yescharacteristic
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—121no—
119C2×D32—34no—
120C2×D32—32no—
121C2×D32—64no—
122C2×D32—62no—
123C2×D32—64no—
124C2×D32—62no—
125C2×D32—61no—
126C2×D32—62no—
127C2×#157—24no—
128C2×#157C22×D314yes—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4D1211yescharacteristic
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3D1211yescharacteristic
137C4×C3⋊D3—24no—
138C4×C3⋊D3D1212yes—
139C4×C3⋊D3C22×D312yes—
140#634—24no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—122no—
148C2×C3×Dic3—121no—
149C3×C4×D3—62no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—64no—
154C3×C4×D3—64no—
155C3×C4×D3—124no—
156C3×Dic6—64no—
157C3×Dic6—64no—
158C3×Dic6—124no—
159#627—68no—
160#627—64no—
161#624—34no—
162#624—32no—
163#624—64no—
164#624—62no—
165#624—61no—
166D3×Dic3—64no—
167D3×Dic3—62no—
168D3×Dic3—64no—
169D3×Dic3—64no—
170D3×Dic3—62no—
171D3×Dic3—64no—
172C2×C22⋊Q8—271noSylow
173C2×C33C22⋊Q812yescomplemented
174C2×C33C22×D411yescharacteristic
175C3×C3⋊D3—24no—
176C3×C3⋊D3C22⋊Q812yescomplemented
177C3×C3⋊D3C22×Q811yescharacteristic
178C3×C3⋊D3C2×#4111yescharacteristic
179C23×D3—91no—
180C23×D3—181no—
181C2×C3×Q8—181no—
182C22×C3×C4—91no—
183C22×C3×C4—92no—
184C22×C3×C4—181no—
185C22×Dic3—91no—
186C22×Dic3—92no—
187C22×Dic3—181no—
188D3×Q8—184no—
189D3×Q8—184no—
190C2×C4×D3—61no—
191C2×C4×D3—61no—
192C2×C4×D3—61no—
193C2×C4×D3—61no—
194C2×C4×D3—94no—
195C2×C4×D3—121no—
196C2×C4×D3—184no—
197C2×C4×D3—181no—
198C2×C4×D3—182no—
199C2×C4×D3—182no—
200C2×C4×D3—184no—
201C2×C4×D3—181no—
202C2×C4×D3—181no—
203C2×Dic6—61no—
204C2×Dic6—61no—
205C2×Dic6—94no—
206C2×Dic6—121no—
207C2×Dic6—182no—
208C2×Dic6—181no—
209C2×Dic6—182no—
210C3×C4⋊C4—94no—
211(C2×D3)⋊C4—98no—
212C4⋊Dic3—98no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32C2×D1211yescharacteristic
218C2×C3⋊D3—22no—
219C2×C3⋊D3—24no—
220C2×C3⋊D3C2×D1211yescharacteristic
221C2×C3⋊D3#23712yes—
222C2×C3⋊D3D3×Q812yes—
223C2×C3⋊D3C23×D311yescharacteristic
224C2×C3×D3—61no—
225C2×C3×D3—61no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C2×C3×D3—62no—
236C2×C3×D3—122no—
237C2×C3×D3—124no—
238D32—34no—
239D32—64no—
240D32—68no—
241D32—64no—
242D32—64no—
243D32—62no—
244#157—24no—
245C32×C4—22no—
246C32×C4—22no—
247C32×C4—42no—
248C32×C4—42no—
249C32×C4C2×D1212yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—24no—
252C3⋊Dic3C2×D1212yes—
253C3×Dic3—62no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C3×Dic3—64no—
257C3×Dic3—62no—
258C3×Dic3—64no—
259C3×Dic3—124no—
260C3×Dic3—122no—
261C22×Q8—271no—
262C23×C4—271no—
263C22⋊Q8—278no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—272no—
266C2×C22⋊C4—272no—
267C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
268C23×C3—91no—
269C23×C3—181no—
270C22×D3—61no—
271C22×D3—61no—
272C22×D3—61no—
273C22×D3—61no—
274C22×D3—92no—
275C22×D3—94no—
276C22×D3—121no—
277C22×D3—182no—
278C22×D3—184no—
279C22×D3—181no—
280C22×D3—182no—
281C22×D3—182no—
282C22×D3—184no—
283C3×Q8—184no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—91no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4—94no—
292C2×C3×C4—184no—
293C2×C3×C4—182no—
294C2×C3×C4—181no—
295C2×C3×C4—184no—
296C2×C3×C4GO+(4,𝔽2)11yescharacteristic
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3—62no—
300C2×Dic3—61no—
301C2×Dic3—62no—
302C2×Dic3—61no—
303C2×Dic3—92no—
304C2×Dic3—94no—
305C2×Dic3—122no—
306C2×Dic3—121no—
307C2×Dic3—182no—
308C2×Dic3—181no—
309C2×Dic3—184no—
310C4×D3—62no—
311C4×D3—62no—
312C4×D3—62no—
313C4×D3—62no—
314C4×D3—64no—
315C4×D3—64no—
316C4×D3—124no—
317C4×D3—182no—
318C4×D3—184no—
319C4×D3—184no—
320C4×D3—184no—
321C4×D3—182no—
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#89712yescomplemented
337C2×C32C22×D1211yescharacteristic
338C3⋊D3—24no—
339C3⋊D3#89712yescomplemented
340C3⋊D3C2×#23711yescharacteristic
341C3⋊D3C2×D3×Q811yescharacteristic
342C3×D3—62no—
343C3×D3—62no—
344C3×D3—61no—
345C3×D3—61no—
346C3×D3—61no—
347C3×D3—61no—
348C3×D3—64no—
349C3×D3—64no—
350C3×D3—124no—
351C24—271no—
352C2×Q8—181no—
353C2×Q8—274no—
354C2×Q8—541no—
355C2×Q8—542no—
356C22×C4—91no—
357C22×C4—92no—
358C22×C4—181no—
359C22×C4—274no—
360C22×C4—272no—
361C22×C4—271no—
362C22×C4—542no—
363C22×C4—541no—
364C4⋊C4—94no—
365C4⋊C4—278no—
366C22⋊C4—278no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—91no—
372C22×C3—91no—
373C22×C3—92no—
374C22×C3—92no—
375C22×C3—182no—
376C22×C3—184no—
377C22×C3C2×GO+(4,𝔽2)11yescharacteristic
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—61no—
381C2×D3—61no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—64no—
388C2×D3—62no—
389C2×D3—64no—
390C2×D3—94no—
391C2×D3—122no—
392C2×D3—124no—
393C2×D3—184no—
394C2×D3—182no—
395C2×D3—184no—
396C2×D3—188no—
397C2×D3—184no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—92no—
403C3×C4—184no—
404C3×C4—182no—
405C3×C4—184no—
406C3×C4C2×GO+(4,𝔽2)12yes—
407Dic3—62no—
408Dic3—62no—
409Dic3—64no—
410Dic3—62no—
411Dic3—64no—
412Dic3—62no—
413Dic3—124no—
414Dic3—122no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32—41no—
419C32C2×#89711yescharacteristic, complemented
420C23—91no—
421C23—181no—
422C23—272no—
423C23—274no—
424C23—541no—
425C23—542no—
426Q8—184no—
427Q8—544no—
428C2×C4—91no—
429C2×C4—92no—
430C2×C4—92no—
431C2×C4—94no—
432C2×C4—184no—
433C2×C4—181no—
434C2×C4—182no—
435C2×C4—184no—
436C2×C4—274no—
437C2×C4—272no—
438C2×C4—544no—
439C2×C4—541no—
440C2×C4—542no—
441C2×C4#157⋊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—91no—
451C2×C3—91no—
452C2×C3—92no—
453C2×C3—184no—
454C2×C3D32⋊Q812yescomplemented
455C2×C3C22×GO+(4,𝔽2)11yescharacteristic
456D3—62no—
457D3—61no—
458D3—61no—
459D3—62no—
460D3—61no—
461D3—61no—
462D3—64no—
463D3—64no—
464D3—124no—
465C22—92no—
466C22—92no—
467C22—91no—
468C22—91no—
469C22—184no—
470C22—182no—
471C22—274no—
472C22—542no—
473C22—544no—
474C22C2×#157⋊D311yescharacteristic
475C4—92no—
476C4—184no—
477C4—182no—
478C4—184no—
479C4C2×#157⋊D312yes—
480C3—21no—
481C3—21no—
482C3—41no—
483C3—41no—
484C3C2×D32⋊Q811yescharacteristic, complemented
485C2—91no—
486C2—91no—
487C2—92no—
488C2—184no—
489C2#1172433312yesdirect product factor
490C2C22×#157⋊D311yescharacteristic
subgroupG/H
centerC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FrattiniC2C22×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×#1113C22
hypercenterC2×C4#157⋊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
2. C2×#89040C2semidirect
3. C2×C3×C4⋊#157C2semidirect
4. C2×#88940C2semidirect
6. #11724333C2direct (N × H)
20. #89040C22semidirect
21. C3×C4⋊#157C22semidirect
22. #88940C22semidirect
28. C2×C4⋊#157D3semidirect
36. C2×#1113Q8semidirect
86. C4⋊#157C2×D3semidirect
98. #1113C2×Q8semidirect
173. C2×C33C22⋊Q8semidirect
176. C3×C3⋊D3C22⋊Q8semidirect
267. C33C2×C22⋊Q8semidirect
336. C2×C32#897semidirect
339. C3⋊D3#897semidirect
419. C32C2×#897semidirect
454. C2×C3D32⋊Q8semidirect
484. C3C2×D32⋊Q8semidirect
489. C2#11724333direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×#890402
C2×C3×C4⋊#1572
C2×#889402
C2×#889392
#117243332
C2×D32⋊Q83
C2×#8979