Finite Group Databasebeta

← #13156535

Subgroups — #13156535

Order 1728.

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

normal subgroups only

This group has 542 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#11723680C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic
3#11722037C211yesmaximal, characteristic, complemented
4#11722031C211yesmaximal, characteristic
5C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8#10591614—31nomaximal
9#10591478—31nomaximal
10#10590868—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88815—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#947—31no—
23C3×#936—31no—
24#63872—31no—
25#63869—31no—
26#63869—31no—
27#63863D311yescharacteristic
28#63662—31no—
29#63659—31no—
30C8×C3⋊Dic3—31no—
31C3×#835—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#816—31no—
35#63566D311yescharacteristic, complemented
36#63565—31no—
37#63565—31no—
38#63565—61no—
39#63563D311yescharacteristic
40#63562—31no—
41#63562—31no—
42#63562—61no—
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C32×C3⋊C8—21no—
59#5056—91no—
60#4800—91no—
61#4768—91no—
62C2×C32×Q8—31no—
63C2×C32×D4—31no—
64C2×C3×D12—21no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×D12C2×D311yescharacteristic
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4×C3⋊Dic3C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—121no—
81C3×#217—61no—
82C3×#217—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—61no—
98#3724—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103(C2×D3)⋊Dic3—21no—
104(C2×D3)⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3—41no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—21no—
113C3×C3⋊Dic3—41no—
114C3×C3⋊Dic3—41no—
115C32×Dic3—41no—
116C3×#124—91no—
117#947—31no—
118#936—31no—
119#890—91no—
120#890—91no—
121#870—91no—
122C3×#103—91no—
123C3×#102—91no—
124#835—31no—
125#835—91no—
126#835—91no—
127#835—181no—
128#832—31no—
129#832—91no—
130#832—91no—
131#832—181no—
132#821—31no—
133#816—31no—
134C8×Dic3—91no—
135C8×Dic3—91no—
136C8×Dic3—181no—
137#810—91no—
138#809—91no—
139C4×C3⋊C8—91no—
140C23×C32—61no—
141C22×C3×D3—21no—
142C22×C3×D3—21no—
143C22×C3×D3—41no—
144C32×Q8—61no—
145C32×Q8—61no—
146C32×D4—61no—
147C32×D4—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3#6711yescharacteristic
159C2×C3⋊Dic3#6711yescharacteristic
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—31no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3—121no—
175C3×D12—22no—
176C3×D12—21no—
177C3×D12—21no—
178C3×D12—21no—
179C3×D12—21no—
180C3×D12—41no—
181C3×Dic6—22no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—41no—
187C32×C8—61no—
188C3⋊(C3⋊C8)—61no—
189C3⋊(C3⋊C8)—62no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—61no—
192C3⋊(C3⋊C8)—121no—
193C3⋊(C3⋊C8)—121no—
194C3⋊(C3⋊C8)—122no—
195C3×C3⋊C8—22no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198#488—271noSylow
199C2×C33#12411yescharacteristic
200C2×C33#13511yescharacteristic
201C2×C33#13511yescharacteristic
202C32×D3—81no—
203C2×C3×Q8—31no—
204C2×C3×Q8—31no—
205C2×C3×Q8—61no—
206C2×C3×D4—31no—
207C2×C3×D4—31no—
208C2×C3×D4—61no—
209C2×D12D3211yescharacteristic
210C2×Dic6D3211yescharacteristic
211C2×C3×C8—31no—
212C2×C3×C8—31no—
213C2×C3×C8—61no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—91no—
216C3×C22⋊C4—181no—
217C3×C22⋊C4—181no—
218C3×C42—91no—
219#217—61no—
220#217—61no—
221(C2×D3)⋊C4—61no—
222(C2×D3)⋊C4—61no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—31no—
226C4⋊Dic3—31no—
227C4⋊Dic3—61no—
228C4⋊Dic3—61no—
229C4⋊Dic3—181no—
230C4⋊Dic3—181no—
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233Dic3⋊C4—181no—
234Dic3⋊C4—181no—
235C4×Dic3—31no—
236C4×Dic3—31no—
237C4×Dic3—61no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—91no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—181no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—361no—
245C2×C3⋊C8D3211yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32—121no—
251C22×C32—121no—
252C22×C32—121no—
253C22×C32D3×D411yescharacteristic
254C22×C32D3×D411yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C2×C3×D3—42no—
257C2×C3×D3—41no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—41no—
261C2×C3×D3—41no—
262C2×C3×D3—41no—
263C2×C3×D3—81no—
264C2×C3×D3—81no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4—121no—
274C32×C4#23511yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4#23711yescharacteristic
279C32×C4#23711yescharacteristic
280C3⋊Dic3—21no—
281C3⋊Dic3—21no—
282C3⋊Dic3—41no—
283C3⋊Dic3—41no—
284C3×Dic3—42no—
285C3×Dic3—41no—
286C3×Dic3—41no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—241no—
298C3×Dic3—241no—
299#125—271no—
300#124—91no—
301#103—91no—
302#103—271no—
303#102—91no—
304#102—271no—
305C4×C8—271no—
306C33#48811yescharacteristic, complemented, Sylow
307C23×C3—61no—
308C23×C3—61no—
309C23×C3—121no—
310C22×D3—21no—
311C3×Q8—62no—
312C3×Q8—61no—
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—121no—
317C3×D4—62no—
318C3×D4—61no—
319C3×D4—61no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—121no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—121no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337C2×Dic3—21no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—121no—
349C2×Dic3—121no—
350D12—21no—
351D12—21no—
352Dic6—21no—
353Dic6—21no—
354C3×C8—62no—
355C3×C8—61no—
356C3×C8—61no—
357C3⋊C8—21no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—241no—
378C2×C32#88511yescharacteristic
379C2×C32#88511yescharacteristic
380C2×C32#91211yescharacteristic
381C2×C32#91611yescharacteristic
382C2×C32#91611yescharacteristic
383C2×C32#91911yescharacteristic
384C2×C32#93611yescharacteristic
385C2×C32#95011yescharacteristic
386C2×C32#95011yescharacteristic
387C3×D3—82no—
388C3×D3—81no—
389C3×D3—81no—
390C2×Q8—31no—
391C2×D4—31no—
392C2×C8—31no—
393C2×C8—271no—
394C4⋊C4—91no—
395C4⋊C4—91no—
396C4⋊C4—541no—
397C4⋊C4—541no—
398C22⋊C4—181no—
399C22⋊C4—181no—
400C42—91no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—122no—
406C22×C3—121no—
407C22×C3—121no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—241no—
413C22×C3—241no—
414C22×C3#380211yescharacteristic
415C22×C3D3×#6711yescharacteristic
416C22×C3D3×#6711yescharacteristic
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—41no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—122no—
429C3×C4—121no—
430C3×C4—121no—
431C3×C4—181no—
432C3×C4—181no—
433C3×C4—361no—
434C3×C4—361no—
435C3×C4#379811yescharacteristic
436C3×C4#380411yescharacteristic
437C3×C4#380411yescharacteristic
438C3×C4#380511yescharacteristic
439C3×C4#380511yescharacteristic
440C3×C4#380511yescharacteristic
441Dic3—41no—
442Dic3—62no—
443Dic3—62no—
444Dic3—61no—
445Dic3—61no—
446Dic3—61no—
447Dic3—61no—
448Dic3—121no—
449Dic3—121no—
450Dic3—121no—
451Dic3—121no—
452Dic3—241no—
453Dic3—241no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#476811yescharacteristic, complemented
459C32#480011yescharacteristic, complemented
460C32#505611yescharacteristic, complemented
461C23—61no—
462Q8—61no—
463Q8—61no—
464D4—61no—
465D4—61no—
466C2×C4—61no—
467C2×C4—91no—
468C2×C4—91no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4D3311yescharacteristic
472C8—61no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—242no—
487C2×C3—241no—
488C2×C3—241no—
489C2×C3#6380911yescharacteristic
490C2×C3#6381011yescharacteristic
491C2×C3#6391511yescharacteristic
492C2×C3#6393311yescharacteristic
493C2×C3#6393511yescharacteristic
494C2×C3#6394711yescharacteristic
495C2×C3#6394911yescharacteristic
496C2×C3#6395911yescharacteristic
497C2×C3#6395911yescharacteristic
498D3—81no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22#8895311yescharacteristic
503C4—121no—
504C4—181no—
505C4—181no—
506C4—361no—
507C4—361no—
508C4#8895911yescharacteristic
509C4#8896311yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059086811yescharacteristic, complemented
515C3#1059147811yescharacteristic, complemented
516C3#1059161411yescharacteristic, complemented
517C2—241no—
518C2#1172285911yescharacteristic
519C2#1172286011yescharacteristic
520C2#1172300611yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
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
1. #11723680C2semidirect
3. #11722037C2semidirect
5. C3⋊C8×C3⋊Dic3C2semidirect
6. C3×#63566C2semidirect
35. #63566D3semidirect
306. C33#488semidirect
458. C32#4768semidirect
459. C32#4800semidirect
460. C32#5056semidirect
514. C3#10590868semidirect
515. C3#10591478semidirect
516. C3#10591614semidirect

Maximal subgroups

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

subgroupindex
#117236802
C3×#638632
#117220372
#117220312
C3⋊C8×C3⋊Dic32
C3×#635662
C3×#635632
#105916143
#105914783
#105908683