Finite Group Databasebeta

← C2×#11724120 #13180559

Subgroups — C2×#11724120 #13180559

Order 1728.

1128 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 140 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×#88957C211yesmaximal, characteristic, complemented
2C2×#88949C211yesmaximal, characteristic
3C2×#88939C211yesmaximal, characteristic
4C2×#88934C211yesmaximal, characteristic, complemented
5C2×#88932C211yesmaximal, characteristic
6C2×C3×#3773C211yesmaximal, characteristic, complemented
7C2×Dic3⋊#157C211yesmaximal, characteristic, complemented
8#11724120C218yesmaximal, direct product factor
9C2×D32⋊Q8—31nomaximal
10C22×#1113C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3×#157C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×#6293—21no—
15C2×Dic3×C3⋊D3—21no—
16C2×Dic3×C3⋊D3C2211yescharacteristic
17C2×C3×#627—21no—
18C2×C3×#624—21no—
19C2×C3×#624C2211yescharacteristic
20C2×C3×#624C2211yescharacteristic
21#88957—22no—
22#88957C2214yescomplemented
23#88949—22no—
24#88949C2214yes—
25#88939C2214yes—
26#88934C2214yescomplemented
27#88932C2214yes—
28C3×#3773C2214yescomplemented
29Dic3⋊#157C2214yescomplemented
30C2×C4×D32—31no—
31C2×C4×D32—61no—
32C2×#3797—31no—
33C2×D3×Dic6—61no—
34C2×C4⋊#157—31no—
35C2×#3773—31no—
36C2×#3773D311yescharacteristic, complemented
37C2×D32⋊C4—31no—
38C2×D32⋊C4—31no—
39D32⋊Q8—38no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3C2311yescharacteristic
42C2×#1113—21no—
43C2×#1113—22no—
44C2×#1113Q814yescomplemented
45C2×#1113C2312yes—
46C2×C3⋊#157—22no—
47C2×C3⋊#157C2312yes—
48C2×C3×#157—22no—
49C2×C3×#157C2312yes—
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55#6294—21no—
56#6294C2312yes—
57#6293—24no—
58Dic3×C3⋊D3—24no—
59Dic3×C3⋊D3D412yes—
60Dic3×C3⋊D3C2312yes—
61C3×#627—24no—
62C3×#624—21no—
63C3×#624—21no—
64C3×#624—24no—
65C3×#624C2312yes—
66C3×#624C2312yes—
67C2×#896—91nomaximal
68C22×D32—31no—
69C22×D32—61no—
70C22×#157—31no—
71C22×#157C2×D311yescharacteristic
72C2×C4×C3⋊D3—31no—
73C2×C4×C3⋊D3—61no—
74C2×#634—61no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—61no—
80C2×C3×Dic6—61no—
81C2×#627—21no—
82C2×#627—61no—
83C2×#627—61no—
84C2×#627—121no—
85C2×#624—21no—
86C2×#624—31no—
87C2×#624C2×D311yescharacteristic
88C2×#624C2×D311yescharacteristic
89C2×D3×Dic3—61no—
90C2×D3×Dic3—61no—
91C2×D3×Dic3—61no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94C2×D3×Dic3—121no—
95C4×D32—34no—
96C4×D32—62no—
97C4×D32—64no—
98C4×D32—64no—
99#3797—34no—
100#3797—62no—
101D3×Dic6—64no—
102D3×Dic6—64no—
103C4⋊#157—34no—
104#3773—34no—
105#3773C2×D314yescomplemented
106D32⋊C4—34no—
107D32⋊C4—34no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—24no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3C2×D411yescharacteristic
112C2×C3×C3⋊D3C2×Q812yes—
113C2×C3×C3⋊D3#4112yes—
114C2×C3×C3⋊D3C2411yescharacteristic
115#1113—24no—
116#1113—22no—
117#1113C2×Q814yescomplemented
118C3⋊#157—22no—
119C3×#157—22no—
120C3⋊(C3⋊Dic3)C2×D412yes—
121C3×C3⋊Dic3—22no—
122C32×Dic3—22no—
123C32×Dic3—22no—
124C32×Dic3C2×D412yes—
125C2×D3×Q8—181no—
126C22×C4×D3—91no—
127C22×C4×D3—181no—
128C22×Dic6—91no—
129C2×C3×C4⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134#896—98no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3C22×D311yescharacteristic
137C22×C3×D3—61no—
138C22×C3×D3—61no—
139C22×C3×D3—61no—
140C22×C3×D3—61no—
141C22×C3×D3—121no—
142C2×D32—34no—
143C2×D32—32no—
144C2×D32—64no—
145C2×D32—62no—
146C2×D32—64no—
147C2×D32—62no—
148C2×D32—62no—
149C2×D32—61no—
150C2×#157—22no—
151C2×#157—32no—
152C2×#157—62no—
153C2×#157C22×D312yes—
154C2×C32×C4—31no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—121no—
163C4×C3⋊D3—32no—
164C4×C3⋊D3—32no—
165C4×C3⋊D3—64no—
166#634—64no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C3×C4×D3—62no—
179C3×C4×D3—62no—
180C3×C4×D3—64no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×C4×D3—64no—
184C3×Dic6—64no—
185C3×Dic6—64no—
186#627—24no—
187#627—64no—
188#627—64no—
189#627—124no—
190#624—24no—
191#624—21no—
192#624—21no—
193#624—32no—
194#624—61no—
195#624C22×D312yes—
196#624C22×D312yes—
197D3×Dic3—62no—
198D3×Dic3—62no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201D3×Dic3—62no—
202D3×Dic3—64no—
203D3×Dic3—64no—
204D3×Dic3—62no—
205D3×Dic3—124no—
206C2×C22⋊Q8—271noSylow
207C2×C33C22⋊Q812yescomplemented
208C2×C33C22×D411yescharacteristic
209C3×C3⋊D3—24no—
210C3×C3⋊D3C22⋊Q812yescomplemented
211C3×C3⋊D3C22×Q811yescharacteristic
212C3×C3⋊D3C2×#4111yescharacteristic
213C23×D3—91no—
214C23×D3—181no—
215C2×C3×Q8—181no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—181no—
220C22×Dic3—91no—
221C22×Dic3—91no—
222C22×Dic3—91no—
223C22×Dic3—181no—
224D3×Q8—184no—
225D3×Q8—184no—
226C2×C4×D3—61no—
227C2×C4×D3—61no—
228C2×C4×D3—61no—
229C2×C4×D3—61no—
230C2×C4×D3—94no—
231C2×C4×D3—184no—
232C2×C4×D3—181no—
233C2×C4×D3—181no—
234C2×C4×D3—181no—
235C2×C4×D3—182no—
236C2×C4×D3—184no—
237C2×C4×D3—181no—
238C2×C4×D3—181no—
239C2×C4×D3—361no—
240C2×Dic6—61no—
241C2×Dic6—61no—
242C2×Dic6—94no—
243C2×Dic6—182no—
244C2×Dic6—181no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247C2×Dic6—361no—
248C3×C4⋊C4—94no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251Dic3⋊C4—94no—
252Dic3⋊C4—94no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32D3×D411yescharacteristic
258C2×C3⋊D3—22no—
259C2×C3⋊D3—24no—
260C2×C3⋊D3#23512yes—
261C2×C3⋊D3D3×D411yescharacteristic
262C2×C3⋊D3D3×Q812yes—
263C2×C3⋊D3C23×D311yescharacteristic
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—62no—
273C2×C3×D3—64no—
274C2×C3×D3—64no—
275C2×C3×D3—62no—
276C2×C3×D3—122no—
277C2×C3×D3—124no—
278D32—34no—
279D32—64no—
280D32—68no—
281D32—64no—
282D32—64no—
283D32—62no—
284#157—22no—
285#157—62no—
286C32×C4—32no—
287C32×C4—62no—
288C32×C4—62no—
289C3⋊Dic3—22no—
290C3⋊Dic3—32no—
291C3⋊Dic3—62no—
292C3⋊Dic3—62no—
293C3⋊Dic3—122no—
294C3⋊Dic3—122no—
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—22no—
300C3×Dic3—22no—
301C3×Dic3—42no—
302C3×Dic3—42no—
303C3×Dic3—62no—
304C3×Dic3—62no—
305C3×Dic3—122no—
306C22×Q8—271no—
307C23×C4—271no—
308C22⋊Q8—278no—
309C2×C4⋊C4—91no—
310C2×C4⋊C4—271no—
311C2×C4⋊C4—271no—
312C2×C22⋊C4—271no—
313C2×C22⋊C4—271no—
314C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C23×C3—181no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—92no—
322C22×D3—94no—
323C22×D3—121no—
324C22×D3—182no—
325C22×D3—184no—
326C22×D3—181no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—184no—
330C3×Q8—184no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4—121no—
341C2×C3×C4—121no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—184no—
345C2×C3×C4—181no—
346C2×C3×C4—182no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—92no—
352C2×Dic3—91no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—92no—
356C2×Dic3—121no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—184no—
360C2×Dic3—181no—
361C2×Dic3—182no—
362C2×Dic3—361no—
363C2×Dic3—361no—
364C2×Dic3GO+(4,𝔽2)11yescharacteristic
365C4×D3—62no—
366C4×D3—62no—
367C4×D3—64no—
368C4×D3—62no—
369C4×D3—62no—
370C4×D3—64no—
371C4×D3—182no—
372C4×D3—182no—
373C4×D3—182no—
374C4×D3—182no—
375C4×D3—182no—
376C4×D3—182no—
377C4×D3—184no—
378C4×D3—184no—
379C4×D3—364no—
380Dic6—64no—
381Dic6—64no—
382Dic6—184no—
383Dic6—184no—
384Dic6—184no—
385Dic6—364no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—22no—
389C2×C32—22no—
390C2×C32—42no—
391C2×C32—42no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32#89612yescomplemented
395C2×C32C2×D3×D411yescharacteristic
396C3⋊D3—24no—
397C3⋊D3#89612yescomplemented
398C3⋊D3C2×#23511yescharacteristic
399C3⋊D3C2×D3×Q811yescharacteristic
400C3×D3—62no—
401C3×D3—62no—
402C3×D3—61no—
403C3×D3—61no—
404C3×D3—61no—
405C3×D3—61no—
406C3×D3—64no—
407C3×D3—64no—
408C3×D3—124no—
409C24—271no—
410C2×Q8—181no—
411C2×Q8—274no—
412C2×Q8—541no—
413C2×Q8—542no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—181no—
418C22×C4—271no—
419C22×C4—274no—
420C22×C4—271no—
421C22×C4—271no—
422C22×C4—541no—
423C22×C4—542no—
424C4⋊C4—94no—
425C4⋊C4—274no—
426C4⋊C4—274no—
427C22⋊C4—274no—
428C22⋊C4—274no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—91no—
434C22×C3—91no—
435C22×C3—92no—
436C22×C3—92no—
437C22×C3—182no—
438C22×C3—184no—
439C22×C3C2×GO+(4,𝔽2)11yescharacteristic
440C2×D3—61no—
441C2×D3—61no—
442C2×D3—61no—
443C2×D3—61no—
444C2×D3—62no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—64no—
450C2×D3—62no—
451C2×D3—64no—
452C2×D3—94no—
453C2×D3—122no—
454C2×D3—124no—
455C2×D3—184no—
456C2×D3—182no—
457C2×D3—184no—
458C2×D3—188no—
459C2×D3—184no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—62no—
463C3×C4—62no—
464C3×C4—62no—
465C3×C4—62no—
466C3×C4—122no—
467C3×C4—122no—
468C3×C4—182no—
469C3×C4—182no—
470Dic3—22no—
471Dic3—22no—
472Dic3—62no—
473Dic3—62no—
474Dic3—92no—
475Dic3—122no—
476Dic3—182no—
477Dic3—182no—
478Dic3—182no—
479Dic3—362no—
480Dic3—362no—
481Dic3C2×GO+(4,𝔽2)12yes—
482C32—21no—
483C32—21no—
484C32—41no—
485C32—41no—
486C32C2×#89611yescharacteristic, complemented
487C23—91no—
488C23—181no—
489C23—272no—
490C23—274no—
491C23—541no—
492C23—542no—
493Q8—184no—
494Q8—544no—
495C2×C4—31no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—92no—
501C2×C4—184no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C4—182no—
505C2×C4—181no—
506C2×C4—272no—
507C2×C4—272no—
508C2×C4—272no—
509C2×C4—271no—
510C2×C4—272no—
511C2×C4—544no—
512C2×C4—541no—
513C2×C4—542no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—42no—
519C2×C3—42no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—91no—
523C2×C3—91no—
524C2×C3—92no—
525C2×C3—184no—
526C2×C3D32⋊Q812yescomplemented
527C2×C3C22×GO+(4,𝔽2)11yescharacteristic
528D3—62no—
529D3—61no—
530D3—61no—
531D3—62no—
532D3—61no—
533D3—61no—
534D3—64no—
535D3—64no—
536D3—124no—
537C22—92no—
538C22—92no—
539C22—91no—
540C22—91no—
541C22—184no—
542C22—182no—
543C22—274no—
544C22—542no—
545C22—544no—
546C22D3×GO+(4,𝔽2)11yescharacteristic
547C4—32no—
548C4—62no—
549C4—62no—
550C4—182no—
551C4—182no—
552C4—272no—
553C4—542no—
554C3—21no—
555C3—21no—
556C3—41no—
557C3—41no—
558C3C2×D32⋊Q811yescharacteristic, complemented
559C2—91no—
560C2—91no—
561C2—92no—
562C2—184no—
563C2#1172412012yesdirect product factor
564C2C2×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
prime radicalC22×#1113C22
hypocenterC33C2×C22⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#88957C2semidirect
4. C2×#88934C2semidirect
6. C2×C3×#3773C2semidirect
7. C2×Dic3⋊#157C2semidirect
8. #11724120C2direct (N × H)
22. #88957C22semidirect
26. #88934C22semidirect
28. C3×#3773C22semidirect
29. Dic3⋊#157C22semidirect
36. C2×#3773D3semidirect
44. C2×#1113Q8semidirect
105. #3773C2×D3semidirect
117. #1113C2×Q8semidirect
207. C2×C33C22⋊Q8semidirect
210. C3×C3⋊D3C22⋊Q8semidirect
314. C33C2×C22⋊Q8semidirect
394. C2×C32#896semidirect
397. C3⋊D3#896semidirect
486. C32C2×#896semidirect
526. C2×C3D32⋊Q8semidirect
558. C3C2×D32⋊Q8semidirect
563. C2#11724120direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889572
C2×#889492
C2×#889392
C2×#889342
C2×#889322
C2×C3×#37732
C2×Dic3⋊#1572
#117241202
C2×D32⋊Q83
C2×#8969