Finite Group Databasebeta

← #13167249

Subgroups — #13167249

Order 1728.

616 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C3×#64319C211yesmaximal, characteristic, complemented
2#11724314C211yesmaximal, characteristic, complemented
3#11724306C211yesmaximal, characteristic, complemented
4#11723708C211yesmaximal, characteristic, complemented
5#11723109C211yesmaximal, characteristic, complemented
6#11722758C211yesmaximal, characteristic, complemented
7D32⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10593478—31nomaximal
9C2×D32⋊D3—21no—
10C3×Q8×C3⋊D3C2211yescharacteristic
11C32×#239—21no—
12C4⋊#1113C2211yescharacteristic
13C3×#3802—21no—
14C3×#3802C2211yescharacteristic
15C3×C4×D32—21no—
16C3×C4×D32C2211yescharacteristic
17C3×#3795—21no—
18C3×#3795—22no—
19C4⋊(C3⋊#157)C2211yescharacteristic
20#88988C2211yescharacteristic
21#88935—22no—
22#88932—21no—
23C3×#3755—21no—
24#88811—21no—
25#88808C2211yescharacteristic
26C3×D3×#41—61no—
27#64319D311yescharacteristic, complemented
28#64230—31no—
29#64225—31no—
30#63940—31no—
31D3×#215—61no—
32#63784—31no—
33#63729—31no—
34D32⋊C8—31no—
35C2×#1113—21no—
36C2×C3×D32—21no—
37C2×C3×D32D411yescharacteristic
38C2×C3⋊#157—21no—
39D32⋊D3—42no—
40C33×Q8D411yescharacteristic, complemented
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4×C3⋊D3—21no—
43C3×C4×C3⋊D3C2311yescharacteristic
44C3×#634—21no—
45C3×#634D411yescharacteristic, complemented
46C32×D12—22no—
47C32×D12—21no—
48C32×C4×D3—22no—
49C32×C4×D3—21no—
50#6295—21no—
51C3×#625—22no—
52C3×#625D412yescomplemented
53C3×#624D411yescharacteristic
54C3×D3×Dic3—22no—
55C3×D3×Dic3—21no—
56C3⋊#622—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5238—91nomaximal
59C2×GO+(4,𝔽2)—61no—
60C32×#41—61no—
61Q8×C3⋊D3C2×D311yescharacteristic
62C3×#239—21no—
63C3×#239—21no—
64C3×#239—41no—
65C3×D3×Q8—61no—
66C3×D3×Q8—61no—
67C3×#237—61no—
68C3×#237—62no—
69C3×D3×D4—61no—
70C3×D3×D4—62no—
71C3×#235—61no—
72C3×#235—62no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—62no—
75#3802—21no—
76#3802—31no—
77#3802C2×D311yescharacteristic
78D3×D12—61no—
79C4×D32—21no—
80C4×D32C2×D311yescharacteristic
81#3795—21no—
82#3795—22no—
83C4⋊#157—31no—
84#3787—31no—
85#3774—62no—
86D32⋊C4—61no—
87#3755—21no—
88C3×#215—61no—
89C3×#215—61no—
90C3×#215—121no—
91#3717—61no—
92#3716—61no—
93#3715—61no—
94#3710—31no—
95D3×C3⋊C8—61no—
96C2×C3×C3⋊D3—41no—
97C2×C3×C3⋊D3C2×D411yescharacteristic
98C2×C32×D3—21no—
99C2×C32×D3—22no—
100#1113—41no—
101#1113—41no—
102C3×D32—21no—
103C3×D32—21no—
104C3×D32—41no—
105C3×D32—41no—
106C3⋊#157—41no—
107C33×C4—21no—
108C33×C4C2×D411yescharacteristic
109C3×C3⋊Dic3—21no—
110C3×C3⋊Dic3C2×D411yescharacteristic
111C32×Dic3—21no—
112C2×C3×#41—91no—
113D3×#41—61no—
114C2×#215—91no—
115C2×#213—91no—
116#930—91no—
117D3×SD16—181no—
118#835—91no—
119#832—91no—
120#830—91no—
121C22×C3×D3—61no—
122C22×C3×D3—62no—
123C2×D32—21no—
124C2×D32—61no—
125C2×D32—121no—
126C2×D32#6711yescharacteristic
127C2×#157—61no—
128GO+(4,𝔽2)—122no—
129C32×Q8—21no—
130C32×Q8—21no—
131C32×Q8—41no—
132C32×Q8—41no—
133C32×Q8#6711yescharacteristic, complemented
134C32×D4—62no—
135C32×D4—61no—
136C2×C32×C4—62no—
137C2×C32×C4—61no—
138C4⋊(C3⋊D3)—21no—
139C4×C3⋊D3—21no—
140C4×C3⋊D3C22×D311yescharacteristic
141#634—21no—
142#634#6711yescharacteristic, complemented
143C3×#67—62no—
144C3×#67—62no—
145C3×#67—62no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C3×D12—22no—
149C3×D12—22no—
150C3×D12—21no—
151C3×D12—21no—
152C3×D12—41no—
153C3×D12—42no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—121no—
157C3×C4×D3—22no—
158C3×C4×D3—22no—
159C3×C4×D3—21no—
160C3×C4×D3—21no—
161C3×C4×D3—41no—
162C3×C4×D3—42no—
163C3×C4×D3—61no—
164C3×C4×D3—61no—
165C3×C4×D3—61no—
166C3×C4×D3—62no—
167C3×C4×D3—61no—
168C3×C4×D3—62no—
169C3×C4×D3—121no—
170C3×Dic6—61no—
171C3×Dic6—61no—
172C3×Dic6—62no—
173C3×Dic6—121no—
174#626—121no—
175#625—22no—
176#625—61no—
177#625#6712yescomplemented
178#624#6711yescharacteristic
179D3×Dic3—22no—
180D3×Dic3—21no—
181#622—61no—
182C3⋊(C3⋊C8)—21no—
183C3×C3⋊C8—61no—
184C3×C3⋊C8—61no—
185C3×C3⋊C8—121no—
186#449—271noSylow
187C2×C33#12011yescharacteristic
188C3×C3⋊D3—81no—
189C3×C3⋊D3#13511yescharacteristic
190C3×C3⋊D3AGL(1,ℤ/8)11yescharacteristic
191C32×D3—41no—
192C32×D3—42no—
193C3×#41—61no—
194C3×#41—61no—
195C3×#41—121no—
196C3×#41—182no—
197C3×#41—181no—
198C2×C3×Q8—91no—
199C2×C3×D4—91no—
200C2×C3×D4—181no—
201C22×C3×C4—91no—
202C22×C3×C4—181no—
203C2×#67—181no—
204#239—21no—
205D3×Q8—61no—
206D3×Q8—61no—
207#237—62no—
208#237—61no—
209D3×D4—61no—
210D3×D4—62no—
211D3×D4—181no—
212#235—62no—
213#235—61no—
214C2×D12—91no—
215C2×C4×D3—61no—
216C2×C4×D3—62no—
217C3×SD16—181no—
218#215—61no—
219#215—61no—
220#215—121no—
221#215—181no—
222#214—181no—
223#213—182no—
224(C2×D3)⋊C4—181no—
225C4⋊Dic3—91no—
226C2×C3⋊C8—91no—
227C2×C3⋊C8—91no—
228#204—181no—
229D3×C8—181no—
230C22×C32—62no—
231C22×C32—61no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3C2×#6711yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—21no—
236C2×C3×D3—21no—
237C2×C3×D3—22no—
238C2×C3×D3—41no—
239C2×C3×D3—42no—
240C2×C3×D3—61no—
241C2×C3×D3—61no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—241no—
249D32—21no—
250D32—21no—
251D32—41no—
252D32—41no—
253D32—121no—
254D32—121no—
255D32—241no—
256D32—241no—
257#157—121no—
258C32×C4—22no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—41no—
265C32×C4—42no—
266C32×C4—61no—
267C32×C4—81no—
268C32×C4C2×#6711yescharacteristic
269C3⋊Dic3—21no—
270C3⋊Dic3C2×#6711yescharacteristic
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—41no—
274C3×Dic3—61no—
275C3×Dic3—61no—
276C3×Dic3—62no—
277C3×Dic3—121no—
278C2×#41—91no—
279C2×SD16—271no—
280C2×D8—271no—
281#121—271no—
282#103—271no—
283#102—271no—
284C22⋊C8—271no—
285C33#44911yescharacteristic, complemented, Sylow
286C23×C3—91no—
287C23×C3—181no—
288C22×D3—61no—
289C22×D3—62no—
290C22×D3—181no—
291C22×D3—361no—
292C3×Q8—21no—
293C3×Q8—21no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—91no—
297C3×Q8—181no—
298C3×Q8GO+(4,𝔽2)11yescharacteristic, complemented
299C3×D4—62no—
300C3×D4—61no—
301C3×D4—62no—
302C3×D4—61no—
303C3×D4—92no—
304C3×D4—122no—
305C3×D4—121no—
306C3×D4—181no—
307C3×D4—182no—
308C2×C3×C4—62no—
309C2×C3×C4—61no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—91no—
313C2×C3×C4—91no—
314C2×C3×C4—122no—
315C2×C3×C4—121no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4—182no—
319#67—62no—
320#67—62no—
321#67—62no—
322#67—361no—
323#67—362no—
324C2×Dic3—61no—
325C2×Dic3—62no—
326C2×Dic3—181no—
327D12—22no—
328D12—21no—
329D12—61no—
330D12—61no—
331D12—121no—
332D12—181no—
333C4×D3—22no—
334C4×D3—21no—
335C4×D3—61no—
336C4×D3—61no—
337C4×D3—61no—
338C4×D3—62no—
339C4×D3—61no—
340C4×D3—62no—
341C4×D3—121no—
342Dic6—61no—
343Dic6—61no—
344Dic6—62no—
345Dic6—121no—
346C3×C8—181no—
347C3⋊C8—61no—
348C3⋊C8—61no—
349C3⋊C8—121no—
350C3⋊C8—181no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—121no—
356C2×C32—122no—
357C2×C32#95311yescharacteristic
358C3⋊D3—81no—
359C3⋊D3#94911yescharacteristic
360C3⋊D3#95011yescharacteristic
361C3×D3—41no—
362C3×D3—42no—
363C3×D3—41no—
364C3×D3—42no—
365C3×D3—61no—
366C3×D3—61no—
367C3×D3—61no—
368C3×D3—61no—
369C3×D3—81no—
370C3×D3—82no—
371C3×D3—241no—
372C3×D3—241no—
373C3×D3—481no—
374#41—61no—
375#41—182no—
376#41—181no—
377C2×Q8—91no—
378C2×D4—91no—
379C2×D4—181no—
380C2×D4—271no—
381C2×D4—541no—
382C22×C4—91no—
383C22×C4—181no—
384SD16—181no—
385SD16—541no—
386D8—542no—
387C2×C8—271no—
388C2×C8—271no—
389C4⋊C4—271no—
390C22⋊C4—541no—
391C22×C3—62no—
392C22×C3—61no—
393C22×C3—62no—
394C22×C3—61no—
395C22×C3—91no—
396C22×C3—122no—
397C22×C3—121no—
398C22×C3—181no—
399C22×C3—181no—
400C22×C3—361no—
401C22×C3—361no—
402C22×C3—361no—
403C2×D3—22no—
404C2×D3—21no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—121no—
408C2×D3—121no—
409C2×D3—122no—
410C2×D3—122no—
411C2×D3—121no—
412C2×D3—121no—
413C2×D3—241no—
414C2×D3—361no—
415C2×D3—361no—
416C2×D3—721no—
417C2×D3—721no—
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—42no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—81no—
429C3×C4—91no—
430C3×C4—121no—
431C3×C4—181no—
432C3×C4C2×GO+(4,𝔽2)11yescharacteristic
433Dic3—21no—
434Dic3—61no—
435Dic3—61no—
436Dic3—62no—
437Dic3—121no—
438Dic3—361no—
439C32—21no—
440C32—21no—
441C32—41no—
442C32—41no—
443C32#523811yescharacteristic, complemented
444C23—91no—
445C23—181no—
446C23—541no—
447Q8—91no—
448Q8—181no—
449Q8D32⋊D311yescharacteristic, complemented
450D4—62no—
451D4—61no—
452D4—92no—
453D4—182no—
454D4—181no—
455D4—541no—
456D4—1082no—
457C2×C4—62no—
458C2×C4—61no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—182no—
464C2×C4—541no—
465C8—181no—
466C8—541no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—91no—
472C2×C3—91no—
473C2×C3—121no—
474C2×C3—122no—
475C2×C3—121no—
476C2×C3—122no—
477C2×C3—241no—
478C2×C3—242no—
479C2×C3—721no—
480C2×C3#6424111yescharacteristic
481D3—41no—
482D3—42no—
483D3—61no—
484D3—61no—
485D3—61no—
486D3—61no—
487D3—241no—
488D3—241no—
489D3—481no—
490C22—62no—
491C22—61no—
492C22—91no—
493C22—181no—
494C22—181no—
495C22—361no—
496C22—361no—
497C22—361no—
498C22—1081no—
499C22—1081no—
500C4—21no—
501C4—61no—
502C4—91no—
503C4—181no—
504C4—1081no—
505C4C2×D32⋊D311yescharacteristic
506C3—21no—
507C3—21no—
508C3—41no—
509C3—41no—
510C3#1059347811yescharacteristic, complemented
511C2—91no—
512C2—91no—
513C2—121no—
514C2—122no—
515C2—721no—
516C2#1172433111yescharacteristic
subgroupG/H
centerC2#11724331
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×D32⋊D3
FittingC33×Q8D4
socleC2×C33#120
hypercenterQ8D32⋊D3
hypocenterC33#449
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64319C2semidirect
2. #11724314C2semidirect
3. #11724306C2semidirect
4. #11723708C2semidirect
5. #11723109C2semidirect
6. #11722758C2semidirect
7. D32⋊(C3⋊C8)C2semidirect
27. #64319D3semidirect
40. C33×Q8D4semidirect
45. C3×#634D4semidirect
52. C3×#625D4semidirect
133. C32×Q8#67semidirect
142. #634#67semidirect
177. #625#67semidirect
285. C33#449semidirect
298. C3×Q8GO+(4,𝔽2)semidirect
443. C32#5238semidirect
449. Q8D32⋊D3semidirect
510. C3#10593478semidirect

Maximal subgroups

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

subgroupindex
C3×#643192
#117243142
#117243062
#117237082
#117231092
#117227582
D32⋊(C3⋊C8)2
#105934783
#52389