Finite Group Databasebeta

← #13156541

Subgroups — #13156541

Order 1728.

558 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 98 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
1#11723677C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic, complemented
3#11722040C211yesmaximal, characteristic, complemented
4#11722034C211yesmaximal, characteristic
5C3×#63567C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic, complemented
8#10590872—31nomaximal
9#10590863—31nomaximal
10#10590539—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—21no—
19#88814—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64099—31no—
23#63869—31no—
24#63869—31no—
25#63869—61no—
26#63864—31no—
27#63863—31no—
28#63860—31no—
29#63648—31no—
30#63643—31no—
31C3×#821—31no—
32C3×#816—31no—
33C3×C8×Dic3—31no—
34C3×C8×Dic3—31no—
35C3×#810—31no—
36C3×#809—31no—
37#63567D311yescharacteristic, complemented
38#63566—31no—
39#63566—31no—
40#63566—61no—
41#63564D311yescharacteristic
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63553D311yescharacteristic, complemented
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)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59#4800—91no—
60#4768—91no—
61#4697—91no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634C2×D311yescharacteristic
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C2×C32×C8—31no—
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—121no—
81#3750—61no—
82C4×C3⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4×Dic3—31no—
87C3×C4×Dic3—31no—
88C3×C4×Dic3—61no—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—21no—
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95#3724—61no—
96#3724—61no—
97#3724—62no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—62no—
101#3722—61no—
102#3722—61no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118#947—91no—
119#936—91no—
120#890—91no—
121#890—91no—
122#890—181no—
123#875—91no—
124#870—91no—
125C3×#103—91no—
126C3×#102—91no—
127C3×C4×C8—91no—
128#835—91no—
129#832—91no—
130#821—31no—
131#821—91no—
132#821—91no—
133#821—181no—
134#816—31no—
135#816—91no—
136#816—91no—
137#816—181no—
138C8×Dic3—31no—
139C8×Dic3—31no—
140#810—31no—
141#809—31no—
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)—21no—
166#634—21no—
167#634—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—121no—
180C3×D12—62no—
181C3×D12—61no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—121no—
186C3×Dic6—62no—
187C3×Dic6—61no—
188C3×Dic6—61no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—121no—
192C32×C8—61no—
193C32×C8—61no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—41no—
199C3×C3⋊C8—41no—
200#488—271noSylow
201C2×C33#12411yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33#13511yescharacteristic
204C3×C3⋊D3—81no—
205C2×C3×Q8—91no—
206C2×C3×D4—91no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C42—91no—
223#217—181no—
224#217—181no—
225(C2×D3)⋊C4—181no—
226(C2×D3)⋊C4—181no—
227(C2×D3)⋊C4—181no—
228(C2×D3)⋊C4—181no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—181no—
236C4⋊Dic3—181no—
237C4⋊Dic3—182no—
238C4⋊Dic3—361no—
239C4⋊Dic3D3211yescharacteristic
240Dic3⋊C4—181no—
241Dic3⋊C4—181no—
242Dic3⋊C4—181no—
243Dic3⋊C4—181no—
244Dic3⋊C4—182no—
245C4×Dic3—31no—
246C4×Dic3—31no—
247C4×Dic3—61no—
248C2×C3⋊C8D3211yescharacteristic
249C2×C3⋊C8D3211yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32C2×D1211yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32D3×D411yescharacteristic
257C2×C3⋊D3—41no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C2×C3×D3—122no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—241no—
268C2×C3×D3—241no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—121no—
278C32×C4#23511yescharacteristic
279C32×C4#23511yescharacteristic
280C32×C4#23511yescharacteristic
281C32×C4#23511yescharacteristic
282C32×C4#23711yescharacteristic
283C32×C4#23711yescharacteristic
284C3⋊Dic3—21no—
285C3⋊Dic3—21no—
286C3⋊Dic3—41no—
287C3⋊Dic3—121no—
288C3⋊Dic3—122no—
289C3⋊Dic3—121no—
290C3⋊Dic3—121no—
291C3⋊Dic3—241no—
292C3⋊Dic3—241no—
293C3⋊Dic3—242no—
294C3×Dic3—42no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—61no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C3×Dic3—122no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306#125—271no—
307#124—271no—
308#103—91no—
309#103—271no—
310#102—91no—
311#102—271no—
312C4×C8—91no—
313C33#48811yescharacteristic, complemented, Sylow
314C23×C3—181no—
315C22×D3—61no—
316C22×D3—61no—
317C22×D3—121no—
318C3×Q8—181no—
319C3×Q8—181no—
320C3×D4—181no—
321C3×D4—181no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—62no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—181no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—121no—
345C2×Dic3—182no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351C2×Dic3—362no—
352D12—62no—
353D12—61no—
354D12—61no—
355D12—61no—
356D12—61no—
357D12—121no—
358Dic6—62no—
359Dic6—61no—
360Dic6—61no—
361Dic6—61no—
362Dic6—61no—
363Dic6—121no—
364C3×C8—61no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—61no—
368C3×C8—121no—
369C3×C8—121no—
370C3⋊C8—21no—
371C3⋊C8—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32#87511yescharacteristic
385C2×C32#88511yescharacteristic
386C2×C32#88511yescharacteristic
387C2×C32#90411yescharacteristic
388C2×C32#90411yescharacteristic
389C2×C32#91211yescharacteristic
390C2×C32#91611yescharacteristic
391C2×C32#91611yescharacteristic
392C2×C32#91911yescharacteristic
393C3⋊D3—81no—
394C3×D3—242no—
395C3×D3—241no—
396C3×D3—241no—
397C2×Q8—91no—
398C2×D4—91no—
399C2×C8—31no—
400C2×C8—31no—
401C4⋊C4—31no—
402C4⋊C4—271no—
403C4⋊C4—541no—
404C4⋊C4—541no—
405C22⋊C4—541no—
406C22⋊C4—541no—
407C42—91no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—361no—
413C22×C3—361no—
414C22×C3—361no—
415C22×C3D3×D1211yescharacteristic
416C22×C3D3×D1211yescharacteristic
417C22×C3#381111yescharacteristic
418C2×D3—122no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—241no—
426C2×D3—241no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—122no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—181no—
439C3×C4—181no—
440C3×C4—361no—
441C3×C4#379511yescharacteristic
442C3×C4#379511yescharacteristic
443C3×C4#379811yescharacteristic
444C3×C4#379811yescharacteristic
445C3×C4#380411yescharacteristic
446C3×C4#380411yescharacteristic
447Dic3—41no—
448Dic3—61no—
449Dic3—61no—
450Dic3—61no—
451Dic3—61no—
452Dic3—121no—
453Dic3—121no—
454Dic3—122no—
455Dic3—121no—
456Dic3—121no—
457Dic3—362no—
458Dic3—361no—
459Dic3—361no—
460Dic3—361no—
461Dic3—721no—
462Dic3—721no—
463Dic3—722no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—41no—
468C32#469711yescharacteristic, complemented
469C32#476811yescharacteristic, complemented
470C32#480011yescharacteristic, complemented
471C23—181no—
472Q8—181no—
473Q8—181no—
474D4—181no—
475D4—181no—
476C2×C4—61no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—181no—
480C2×C4—541no—
481C2×C4D3311yescharacteristic
482C8—61no—
483C8—61no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—721no—
497C2×C3#6380611yescharacteristic
498C2×C3#6380611yescharacteristic
499C2×C3#6380711yescharacteristic
500C2×C3#6380811yescharacteristic
501C2×C3#6385211yescharacteristic
502C2×C3#6385211yescharacteristic
503C2×C3#6391311yescharacteristic
504C2×C3#6393611yescharacteristic
505C2×C3#6395011yescharacteristic
506D3—242no—
507D3—241no—
508D3—241no—
509C22—361no—
510C22—361no—
511C22—361no—
512C22#8895511yescharacteristic
513C4—121no—
514C4—181no—
515C4—181no—
516C4—361no—
517C4—1081no—
518C4#8896011yescharacteristic
519C4#8896011yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059053911yescharacteristic, complemented
525C3#1059086311yescharacteristic, complemented
526C3#1059087211yescharacteristic, complemented
527C2—721no—
528C2#1172286411yescharacteristic
529C2#1172286511yescharacteristic
530C2#1172301011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#488
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. #11723447C2semidirect
3. #11722040C2semidirect
5. C3×#63567C2semidirect
7. C3×#63553C2semidirect
37. #63567D3semidirect
45. #63553D3semidirect
313. C33#488semidirect
468. C32#4697semidirect
469. C32#4768semidirect
470. C32#4800semidirect
524. C3#10590539semidirect
525. C3#10590863semidirect
526. C3#10590872semidirect

Maximal subgroups

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

subgroupindex
#117236772
#117234472
#117220402
#117220342
C3×#635672
C3×#635642
C3×#635532
#105908723
#105908633
#105905393