Finite Group Databasebeta

← #13167586

Subgroups — #13167586

Order 1728.

1028 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C23×C3×#157C211yesmaximal, characteristic, complemented
2C2×#89049C211yesmaximal, characteristic
3C2×#88939C212yesmaximal, complemented
4#11723723C211yesmaximal, characteristic, complemented
5#11722791C212yesmaximal, complemented
6#10593515—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×#1113C2211yescharacteristic
9C22×C3×#157—22no—
10C22×C3×#157—24no—
11C22×C3×#157C2212yes—
12C2×C3×#638—21no—
13#89049—22no—
14#89049C414yescomplemented
15C2×#6295—21no—
16C2×#6294C2212yes—
17C2×#6294C2211yescharacteristic
18#88939—24no—
19C3×#3757—21no—
20#88814—21no—
21C23×#157D311yescharacteristic, complemented
22C2×#3811—31no—
23C2×D3×#67—61no—
24C2×D32⋊C4—32no—
25#63974—31no—
26D3×#217—61no—
27#63737—32no—
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3C2×C412yes—
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C3×C3⋊D3D412yes—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×#1113—22no—
37C2×#1113—22no—
38C2×#1113—21no—
39C2×#1113C2×C412yescomplemented
40C2×C3×#157—28no—
41C2×C3×#157—22no—
42C2×C3×#157—22no—
43C2×C3×#157D414yescomplemented
44C3×#638—24no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—22no—
47#6295—24no—
48#6294—21no—
49#6294—22no—
50#6294—22no—
51#6294C2×C412yescomplemented
52#5213—91nomaximal
53C23×C3⋊D3C2×D311yescharacteristic
54C23×C3×D3—61no—
55C23×C3×D3—61no—
56C22×D32—31no—
57C22×D32—61no—
58C22×#157—22no—
59C22×#157—24no—
60C22×#157C2×D312yes—
61C2×#638—21no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×#67—121no—
65#3811—34no—
66#3811—62no—
67D3×#67—64no—
68D3×#67—64no—
69C2×#626—61no—
70C2×#626—61no—
71C2×#625—61no—
72C2×#624—31no—
73C2×#624—32no—
74C2×D3×Dic3—61no—
75C2×D3×Dic3—62no—
76D32⋊C4—64no—
77#3757—21no—
78C3×#217—61no—
79C3×#217—61no—
80C3×#217—121no—
81#3722—61no—
82(C2×D3)⋊Dic3—62no—
83C22×C33—22no—
84C22×C33C22⋊C412yescomplemented
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—24no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—42no—
92C2×C3×C3⋊D3C22⋊C412yescomplemented
93C2×C3×C3⋊D3C22×C411yescharacteristic
94C2×C3×C3⋊D3C2×D412yes—
95C2×C3×C3⋊D3C2×D411yescharacteristic
96C2×C3×C3⋊D3#4112yes—
97#1113—24no—
98#1113—41no—
99#1113—41no—
100C3×#157—24no—
101C3×C3⋊Dic3—22no—
102C3×C3⋊Dic3—42no—
103C23×C3×C4—91no—
104C22×#67—91no—
105C2×D3×D4—181no—
106C2×#217—91no—
107C2×(C2×D3)⋊C4—92no—
108D3×C22⋊C4—181no—
109C3⋊#95—92no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32—41no—
114C23×C32D1211yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3⋊D3—22no—
117C22×C3⋊D3—22no—
118C22×C3⋊D3C4×D312yes—
119C22×C3⋊D3D1211yescharacteristic
120C22×C3⋊D3#6712yes—
121C22×C3⋊D3C22×D311yescharacteristic
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—61no—
129C22×C3×D3—62no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—122no—
133C22×C3×D3—122no—
134C22×C3×D3—122no—
135C22×C3×D3—122no—
136C22×C3×D3—121no—
137C2×D32—32no—
138C2×D32—62no—
139C2×D32—62no—
140C2×D32—61no—
141C2×D32—62no—
142C2×D32—62no—
143C2×D32—122no—
144C2×D32—122no—
145C2×#157—28no—
146C2×#157—22no—
147C2×#157—22no—
148C2×#157#6714yescomplemented
149#638—24no—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—22no—
152C3×#67—64no—
153C3×#67—64no—
154C3×#67—124no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—62no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—121no—
160C2×C3×Dic3—122no—
161#626—64no—
162#626—64no—
163#625—64no—
164#624—32no—
165#624—61no—
166#624—62no—
167#624—62no—
168D3×Dic3—62no—
169D3×Dic3—62no—
170D3×Dic3—122no—
171D3×Dic3—122no—
172#386—271noSylow
173C2×C33—22no—
174C2×C33C2×C22⋊C411yescharacteristic
175C2×C33#12112yes—
176C3×C3⋊D3—22no—
177C3×C3⋊D3—42no—
178C3×C3⋊D3C4×D412yes—
179C3×C3⋊D3#12011yescharacteristic
180C3×C3⋊D3#12311yescharacteristic
181C24×C3—91no—
182C23×D3—61no—
183C23×D3—61no—
184C23×D3—91no—
185C23×D3—181no—
186C2×C3×D4—181no—
187C22×C3×C4—92no—
188C22×C3×C4—184no—
189C22×C3×C4—182no—
190C2×#67—61no—
191C2×#67—61no—
192C2×#67—94no—
193C2×#67—121no—
194C2×#67—181no—
195C2×#67—181no—
196C2×#67—182no—
197C22×Dic3—91no—
198C22×Dic3—92no—
199D3×D4—184no—
200D3×D4—184no—
201C2×D12—181no—
202C2×C4×D3—181no—
203C2×C4×D3—182no—
204C3×C22⋊C4—181no—
205#217—61no—
206#217—61no—
207#217—121no—
208#217—181no—
209(C2×D3)⋊C4—182no—
210(C2×D3)⋊C4—184no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—42no—
222C22×C32—82no—
223C22×C32—82no—
224C22×C32(C2×D3)⋊C412yescomplemented
225C22×C32C2×D1211yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3—22no—
228C2×C3⋊D3—24no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—22no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3(C2×D3)⋊C412yescomplemented
233C2×C3⋊D3C2×C4×D311yescharacteristic
234C2×C3⋊D3#23512yes—
235C2×C3⋊D3C2×#6712yes—
236C2×C3⋊D3C2×#6711yescharacteristic
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—61no—
240C2×C3×D3—62no—
241C2×C3×D3—62no—
242C2×C3×D3—61no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—122no—
250C2×C3×D3—122no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C2×C3×D3—122no—
254C2×C3×D3—122no—
255C2×C3×D3—122no—
256C2×C3×D3—122no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—242no—
261D32—64no—
262D32—124no—
263D32—122no—
264D32—122no—
265D32—121no—
266D32—121no—
267#157—24no—
268C3⋊Dic3—22no—
269C3⋊Dic3—42no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—122no—
273C3×Dic3—122no—
274C3×Dic3—122no—
275C3×Dic3—242no—
276C22×D4—271no—
277C23×C4—91no—
278C2×C22⋊C4—271no—
279C2×C22⋊C4—272no—
280#95—272no—
281C33#38611yescharacteristic, complemented, Sylow
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—91no—
287C23×C3—91no—
288C23×C3—92no—
289C23×C3—92no—
290C23×C3—182no—
291C23×C3—182no—
292C23×C3—181no—
293C23×C3GO+(4,𝔽2)11yescharacteristic
294C22×D3—61no—
295C22×D3—61no—
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—62no—
301C22×D3—62no—
302C22×D3—61no—
303C22×D3—61no—
304C22×D3—92no—
305C22×D3—122no—
306C22×D3—122no—
307C22×D3—122no—
308C22×D3—122no—
309C22×D3—121no—
310C22×D3—182no—
311C22×D3—181no—
312C22×D3—182no—
313C22×D3—182no—
314C22×D3—182no—
315C22×D3—362no—
316C22×D3—362no—
317C3×D4—184no—
318C2×C3×C4—94no—
319C2×C3×C4—181no—
320C2×C3×C4—182no—
321C2×C3×C4—182no—
322C2×C3×C4—188no—
323C2×C3×C4—182no—
324#67—64no—
325#67—64no—
326#67—124no—
327#67—184no—
328#67—184no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—92no—
334C2×Dic3—121no—
335C2×Dic3—122no—
336C2×Dic3—181no—
337C2×Dic3—182no—
338C2×Dic3—182no—
339D12—184no—
340C4×D3—182no—
341C4×D3—182no—
342C4×D3—362no—
343C4×D3—362no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—42no—
352C2×C32—42no—
353C2×C32—42no—
354C2×C32—42no—
355C2×C32—82no—
356C2×C32—82no—
357C2×C32C2×(C2×D3)⋊C411yescharacteristic
358C2×C32#93012yes—
359C3⋊D3—22no—
360C3⋊D3—42no—
361C3⋊D3C4×#6712yes—
362C3⋊D3#92911yescharacteristic
363C3⋊D3#95311yescharacteristic
364C3×D3—62no—
365C3×D3—62no—
366C3×D3—61no—
367C3×D3—61no—
368C3×D3—61no—
369C3×D3—61no—
370C3×D3—122no—
371C3×D3—122no—
372C3×D3—122no—
373C3×D3—122no—
374C3×D3—242no—
375C24—91no—
376C24—271no—
377C2×D4—181no—
378C2×D4—274no—
379C2×D4—541no—
380C2×D4—542no—
381C22×C4—92no—
382C22×C4—184no—
383C22×C4—182no—
384C22×C4—271no—
385C22×C4—272no—
386C22⋊C4—181no—
387C22⋊C4—544no—
388C22⋊C4—541no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—91no—
403C22×C3—91no—
404C22×C3—92no—
405C22×C3—92no—
406C22×C3—92no—
407C22×C3—182no—
408C22×C3—184no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3—362no—
413C22×C3D32⋊C412yescomplemented
414C22×C3C2×GO+(4,𝔽2)11yescharacteristic
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—61no—
420C2×D3—61no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—122no—
428C2×D3—122no—
429C2×D3—124no—
430C2×D3—122no—
431C2×D3—124no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—122no—
435C2×D3—122no—
436C2×D3—122no—
437C2×D3—122no—
438C2×D3—184no—
439C2×D3—242no—
440C2×D3—364no—
441C2×D3—362no—
442C2×D3—362no—
443C2×D3—361no—
444C2×D3—361no—
445C3×C4—182no—
446C3×C4—184no—
447C3×C4—362no—
448Dic3—62no—
449Dic3—62no—
450Dic3—122no—
451Dic3—122no—
452Dic3—122no—
453Dic3—242no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32—41no—
458C32#521311yescharacteristic, complemented
459C23—91no—
460C23—91no—
461C23—92no—
462C23—92no—
463C23—182no—
464C23—182no—
465C23—181no—
466C23—272no—
467C23—541no—
468C23—542no—
469C23—542no—
470C23#157⋊D311yescharacteristic
471D4—184no—
472D4—544no—
473C2×C4—94no—
474C2×C4—182no—
475C2×C4—182no—
476C2×C4—188no—
477C2×C4—181no—
478C2×C4—182no—
479C2×C4—272no—
480C2×C4—541no—
481C2×C4—542no—
482C2×C4—542no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—22no—
487C2×C3—22no—
488C2×C3—42no—
489C2×C3—42no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—42no—
493C2×C3—42no—
494C2×C3—82no—
495C2×C3—82no—
496C2×C3—91no—
497C2×C3—92no—
498C2×C3—91no—
499C2×C3—182no—
500C2×C3—362no—
501C2×C3C2×D32⋊C411yescharacteristic
502C2×C3#6424212yes—
503D3—62no—
504D3—61no—
505D3—61no—
506D3—62no—
507D3—61no—
508D3—61no—
509D3—122no—
510D3—122no—
511D3—122no—
512D3—122no—
513D3—242no—
514C22—22no—
515C22—91no—
516C22—91no—
517C22—92no—
518C22—92no—
519C22—92no—
520C22—182no—
521C22—184no—
522C22—182no—
523C22—182no—
524C22—182no—
525C22—362no—
526C22—544no—
527C22—1081no—
528C22—1081no—
529C22#8893912yescomplemented
530C22C2×#157⋊D311yescharacteristic
531C4—184no—
532C4—182no—
533C4—362no—
534C3—21no—
535C3—21no—
536C3—41no—
537C3—41no—
538C3#1059351511yescharacteristic, complemented
539C2—22no—
540C2—91no—
541C2—92no—
542C2—91no—
543C2—182no—
544C2—362no—
545C2C2×#8893911yescharacteristic
546C2#1172435512yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89049C2
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#386
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3×#157C2semidirect
3. C2×#88939C2semidirect
4. #11723723C2semidirect
5. #11722791C2semidirect
14. #89049C4semidirect
21. C23×#157D3semidirect
39. C2×#1113C2×C4semidirect
43. C2×C3×#157D4semidirect
51. #6294C2×C4semidirect
84. C22×C33C22⋊C4semidirect
92. C2×C3×C3⋊D3C22⋊C4semidirect
148. C2×#157#67semidirect
224. C22×C32(C2×D3)⋊C4semidirect
232. C2×C3⋊D3(C2×D3)⋊C4semidirect
281. C33#386semidirect
413. C22×C3D32⋊C4semidirect
458. C32#5213semidirect
529. C22#88939semidirect
538. C3#10593515semidirect

Maximal subgroups

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

subgroupindex
C23×C3×#1572
C2×#890492
C2×#889392
#117237232
#117227912
#105935153
#52139