Finite Group Databasebeta

← #13156313

Subgroups — #13156313

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic, complemented
3#11722052C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590759—31nomaximal
9#10590757—31nomaximal
10#10590472—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×#6294—21no—
13C2×#6293—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88814—21no—
21#88807—21no—
22#64097—31no—
23#63918—31no—
24#63918—31no—
25#63918—61no—
26#63898—31no—
27#63894—31no—
28Dic3×D12—31no—
29#63648—31no—
30#63644—31no—
31C3×#821—31no—
32C3×#817—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×#809—31no—
36C3×#808—31no—
37#63573—31no—
38#63573—31no—
39#63573—61no—
40#63573D311yescharacteristic, complemented
41#63572D311yescharacteristic, complemented
42#63564D311yescharacteristic
43#63563—31no—
44#63563—31no—
45#63563—61no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)Q812yescomplemented
54C2×C32×Dic3—21no—
55#6294—42no—
56#6293—42no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59#4870—91no—
60#4842—91no—
61#4761—91no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634—61no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×#627—61no—
68C2×#627—61no—
69C2×#627—62no—
70C2×#624—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73C2×C32×C8—31no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—121no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84#3750—61no—
85C4×C3⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3—61no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—62no—
104#3722—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3SD1612yes—
117C32×Dic3—41no—
118D4×Dic3—91no—
119#893—91no—
120#888—91no—
121#888—91no—
122#888—181no—
123C4×D12—91no—
124#869—91no—
125C3×#106—91no—
126C3×C4⋊C8—91no—
127C3×#102—91no—
128#832—91no—
129#821—31no—
130#821—91no—
131#821—91no—
132#821—181no—
133#817—31no—
134#817—91no—
135#817—91no—
136#817—181no—
137Dic3⋊C8—31no—
138Dic3⋊C8—31no—
139#809—31no—
140#808—31no—
141#808—91no—
142C22×C3⋊D3—21no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—121no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—61no—
156C2×C3⋊Dic3—62no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—121no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—122no—
162C2×C3⋊Dic3D1211yescharacteristic
163C2×C3⋊Dic3D1211yescharacteristic
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)Dic612yescomplemented
166#634—122no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—121no—
179C3×D12—32no—
180C3×D12—32no—
181C3×D12—62no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—121no—
185#627—124no—
186#627—122no—
187#627—122no—
188#624—122no—
189D3×Dic3—122no—
190D3×Dic3—122no—
191C32×C8—61no—
192C32×C8—61no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—41no—
199#476—271noSylow
200C2×C33C22⋊Q811yescharacteristic
201C2×C33C2×SD1611yescharacteristic
202C2×C33AGL(1,ℤ/8)11yescharacteristic
203C3×C3⋊D3—42no—
204C2×C3×D4—91no—
205C22×Dic3—181no—
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—61no—
209C2×C4×D3—181no—
210C2×C4×D3—181no—
211C2×Dic6—181no—
212C2×Dic6—181no—
213C2×Dic6—181no—
214C2×Dic6—182no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—91no—
225C3×C42—91no—
226#217—181no—
227(C2×D3)⋊C4—181no—
228(C2×D3)⋊C4—181no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—181no—
236C4⋊Dic3—181no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—361no—
240C4⋊Dic3D3211yescharacteristic
241Dic3⋊C4—181no—
242Dic3⋊C4—181no—
243Dic3⋊C4—182no—
244C4×Dic3—31no—
245C4×Dic3—31no—
246C4×Dic3—61no—
247C2×C3⋊C8D3211yescharacteristic
248C2×C3⋊C8D3211yescharacteristic
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32C2×D1211yescharacteristic
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C2×C3⋊D3—22no—
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—122no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—241no—
267C2×C3×D3—241no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—121no—
277C32×C4C2×Dic611yescharacteristic
278C32×C4#23511yescharacteristic
279C32×C4#23711yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4D3×Q811yescharacteristic
282C32×C4D3×Q811yescharacteristic
283C3⋊Dic3—21no—
284C3⋊Dic3—41no—
285C3⋊Dic3—121no—
286C3⋊Dic3—122no—
287C3⋊Dic3—121no—
288C3⋊Dic3—121no—
289C3⋊Dic3—241no—
290C3⋊Dic3—241no—
291C3⋊Dic3—242no—
292C3⋊Dic3#20412yes—
293C3×Dic3—32no—
294C3×Dic3—32no—
295C3×Dic3—42no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—62no—
299C3×Dic3—62no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—121no—
303C3×Dic3—121no—
304C3×Dic3—241no—
305C4⋊Q8—271no—
306C4×D4—271no—
307#106—91no—
308#106—271no—
309C4⋊C8—91no—
310#102—91no—
311#102—271no—
312C33#47611yescharacteristic, complemented, Sylow
313C23×C3—181no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—121no—
317C3×D4—92no—
318C3×D4—181no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—41no—
323C2×C3×C4—62no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—181no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—31no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—121no—
342C2×Dic3—182no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—361no—
347C2×Dic3—361no—
348C2×Dic3—362no—
349C2×Dic3—362no—
350D12—32no—
351D12—32no—
352D12—62no—
353D12—61no—
354D12—61no—
355D12—121no—
356C4×D3—362no—
357C4×D3—362no—
358Dic6—362no—
359Dic6—364no—
360Dic6—362no—
361Dic6—362no—
362C3×C8—62no—
363C3×C8—61no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—121no—
368C3⋊C8—21no—
369C3⋊C8—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32#87811yescharacteristic
383C2×C32#89611yescharacteristic
384C2×C32#89711yescharacteristic
385C2×C32C2×#20411yescharacteristic
386C2×C32#90811yescharacteristic
387C2×C32#91111yescharacteristic
388C2×C32D3×SD1611yescharacteristic
389C2×C32D3×SD1611yescharacteristic
390C2×C32#91411yescharacteristic
391C3⋊D3—42no—
392C3×D3—122no—
393C3×D3—122no—
394C3×D3—242no—
395C2×Q8—541no—
396C2×D4—91no—
397C22×C4—541no—
398C2×C8—31no—
399C2×C8—31no—
400C4⋊C4—31no—
401C4⋊C4—91no—
402C4⋊C4—271no—
403C4⋊C4—541no—
404C22⋊C4—541no—
405C42—91no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3—182no—
411C22×C3—361no—
412C22×C3—361no—
413C22×C3D3×D1211yescharacteristic
414C22×C3D3×D1211yescharacteristic
415C22×C3#381111yescharacteristic
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—241no—
424C2×D3—241no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—92no—
434C3×C4—122no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—181no—
438C3×C4—361no—
439C3×C4D3×Dic611yescharacteristic
440C3×C4D3×Dic611yescharacteristic
441C3×C4#379511yescharacteristic
442C3×C4#379711yescharacteristic
443C3×C4#380411yescharacteristic
444C3×C4#380511yescharacteristic
445Dic3—32no—
446Dic3—32no—
447Dic3—41no—
448Dic3—62no—
449Dic3—62no—
450Dic3—61no—
451Dic3—61no—
452Dic3—121no—
453Dic3—121no—
454Dic3—241no—
455Dic3—362no—
456Dic3—361no—
457Dic3—361no—
458Dic3—361no—
459Dic3—721no—
460Dic3—721no—
461Dic3—722no—
462C32—21no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32#476111yescharacteristic, complemented
467C32#484211yescharacteristic, complemented
468C32#487011yescharacteristic, complemented
469C23—181no—
470Q8—1082no—
471D4—92no—
472D4—181no—
473C2×C4—61no—
474C2×C4—91no—
475C2×C4—91no—
476C2×C4—181no—
477C2×C4—541no—
478C2×C4—1082no—
479C2×C4D3311yescharacteristic
480C8—61no—
481C8—61no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—362no—
495C2×C3D3×#20411yescharacteristic
496C2×C3D3×#20411yescharacteristic
497C2×C3#6379411yescharacteristic
498C2×C3#6379511yescharacteristic
499C2×C3#6389311yescharacteristic
500C2×C3#6389511yescharacteristic
501C2×C3#6389611yescharacteristic
502C2×C3#6393711yescharacteristic
503C2×C3#6394011yescharacteristic
504D3—122no—
505D3—122no—
506D3—242no—
507C22—182no—
508C22—361no—
509C22—361no—
510C22#8895511yescharacteristic
511C4—92no—
512C4—121no—
513C4—181no—
514C4—361no—
515C4—1081no—
516C4#8895711yescharacteristic
517C4#8896111yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059047211yescharacteristic, complemented
523C3#1059075711yescharacteristic, complemented
524C3#1059075911yescharacteristic, complemented
525C2—362no—
526C2#1172284311yescharacteristic
527C2#1172287911yescharacteristic
528C2#1172298411yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723535C2semidirect
3. #11722052C2semidirect
5. C3×#63573C2semidirect
6. C3×#63572C2semidirect
40. #63573D3semidirect
41. #63572D3semidirect
53. C3×C4⋊(C3⋊D3)Q8semidirect
165. C4⋊(C3⋊D3)Dic6semidirect
312. C33#476semidirect
466. C32#4761semidirect
467. C32#4842semidirect
468. C32#4870semidirect
522. C3#10590472semidirect
523. C3#10590757semidirect
524. C3#10590759semidirect

Maximal subgroups

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

subgroupindex
#117236912
#117235352
#117220522
#117220342
C3×#635732
C3×#635722
C3×#635642
#105907593
#105907573
#105904723