Finite Group Databasebeta

← #13156576

Subgroups — #13156576

Order 1728.

554 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 96 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#11723451C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722021C211yesmaximal, characteristic
5C3×#63567C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591616—31nomaximal
9#10590904—31nomaximal
10#10590568—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64153—31no—
23#64099—31no—
24#63873—31no—
25#63873—31no—
26#63873—61no—
27#63872—31no—
28#63872—31no—
29#63872—61no—
30#63642—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#816—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#810—31no—
37#63567—31no—
38#63567D311yescharacteristic
39#63566—31no—
40#63566D311yescharacteristic
41#63565—31no—
42#63565D311yescharacteristic
43#63555—31no—
44#63555—31no—
45#63555—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)D411yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#4797—91no—
60#4797—91no—
61#4797—91no—
62C2×C32×Q8—31no—
63C2×#634C2×D311yescharacteristic
64C2×C3×Dic6—21no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—61no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74#3750—61no—
75#3750—61no—
76#3750—61no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—31no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—61no—
83C4×C3⋊Dic3—121no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—61no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—61no—
98#3724—61no—
99#3724—61no—
100#3724—61no—
101#3724—62no—
102#3724—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—62no—
109C22×C33C2×D411yescharacteristic
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3⋊(C3⋊Dic3)—21no—
113C3⋊(C3⋊Dic3)—21no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118#944—91no—
119#944—91no—
120#944—91no—
121#944—181no—
122#890—91no—
123#890—91no—
124#890—91no—
125#890—181no—
126C3×#103—91no—
127C3×#103—91no—
128C3×#103—91no—
129#835—31no—
130#835—31no—
131#835—91no—
132#816—31no—
133#816—31no—
134#816—91no—
135#815—91no—
136#815—91no—
137#815—91no—
138#815—181no—
139#810—31no—
140#810—31no—
141#810—91no—
142C32×Q8—61no—
143C32×Q8—61no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—61no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—121no—
168C2×C3⋊Dic3—121no—
169#634—21no—
170#634—21no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—41no—
188C3×Dic6—41no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—121no—
194C3×Dic6—121no—
195C32×C8—61no—
196C3⋊(C3⋊C8)—21no—
197C3×C3⋊C8—22no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203#491—271noSylow
204C2×C33#12411yescharacteristic
205C2×C33#13711yescharacteristic
206C2×C33#13711yescharacteristic
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C2×C3×Q8—91no—
211C2×Dic6—31no—
212C2×Dic6—31no—
213C2×Dic6—61no—
214C2×Dic6D3211yescharacteristic
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—91no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—91no—
223C4⋊Dic3—31no—
224C4⋊Dic3—31no—
225C4⋊Dic3—61no—
226C4⋊Dic3—181no—
227C4⋊Dic3—181no—
228C4⋊Dic3—181no—
229C4⋊Dic3—182no—
230C4⋊Dic3D3211yescharacteristic
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233Dic3⋊C4—181no—
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236Dic3⋊C4—181no—
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—182no—
241Dic3⋊C4—182no—
242Dic3⋊C4—182no—
243C4×Dic3—91no—
244C4×Dic3—91no—
245C4×Dic3—91no—
246C4×Dic3—181no—
247C4×Dic3—181no—
248C4×Dic3—181no—
249C4×Dic3—361no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—61no—
253C2×C3⋊C8D3211yescharacteristic
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32D3×D411yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—121no—
270C32×C4—121no—
271C32×C4#23511yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23511yescharacteristic
274C32×C4#23711yescharacteristic
275C32×C4#23711yescharacteristic
276C32×C4#23711yescharacteristic
277C3⋊Dic3—41no—
278C3⋊Dic3—41no—
279C3⋊Dic3—61no—
280C3⋊Dic3—61no—
281C3⋊Dic3—62no—
282C3⋊Dic3—62no—
283C3⋊Dic3—61no—
284C3⋊Dic3—61no—
285C3⋊Dic3—61no—
286C3⋊Dic3—61no—
287C3⋊Dic3—121no—
288C3⋊Dic3—121no—
289C3⋊Dic3—121no—
290C3⋊Dic3—121no—
291C3⋊Dic3—122no—
292C3⋊Dic3—122no—
293C3×Dic3—42no—
294C3×Dic3—42no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—122no—
300C3×Dic3—122no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C3×Dic3—121no—
304C3×Dic3—121no—
305C4⋊Q8—271no—
306#125—271no—
307#103—91no—
308#103—91no—
309#103—91no—
310#103—271no—
311#97—271no—
312C33#49111yescharacteristic, complemented, Sylow
313C3×Q8—61no—
314C3×Q8—61no—
315C3×Q8—61no—
316C3×Q8—61no—
317C3×Q8—121no—
318C3×Q8—121no—
319C3×Q8—181no—
320C3×Q8—181no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—91no—
349C2×Dic3—91no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358Dic6—21no—
359Dic6—21no—
360Dic6—61no—
361Dic6—61no—
362Dic6—61no—
363Dic6—61no—
364Dic6—121no—
365Dic6—121no—
366C3×C8—62no—
367C3×C8—61no—
368C3×C8—61no—
369C3×C8—181no—
370C3⋊C8—21no—
371C3⋊C8—62no—
372C3⋊C8—61no—
373C3⋊C8—61no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32#88511yescharacteristic
387C2×C32#88511yescharacteristic
388C2×C32#88511yescharacteristic
389C2×C32#91511yescharacteristic
390C2×C32#91511yescharacteristic
391C2×C32#91511yescharacteristic
392C2×C32#91811yescharacteristic
393C2×C32#91811yescharacteristic
394C2×C32#91811yescharacteristic
395C2×Q8—31no—
396C2×Q8—91no—
397C2×C8—31no—
398C2×C8—91no—
399C4⋊C4—31no—
400C4⋊C4—91no—
401C4⋊C4—541no—
402C4⋊C4—541no—
403C4⋊C4—541no—
404C4⋊C4—541no—
405C42—271no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3#380211yescharacteristic
411C22×C3#381111yescharacteristic
412C22×C3#381111yescharacteristic
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—122no—
422C3×C4—122no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—361no—
428C3×C4—361no—
429C3×C4#379511yescharacteristic
430C3×C4#379511yescharacteristic
431C3×C4#379811yescharacteristic
432C3×C4#380411yescharacteristic
433C3×C4#380411yescharacteristic
434C3×C4#380511yescharacteristic
435Dic3—41no—
436Dic3—41no—
437Dic3—122no—
438Dic3—122no—
439Dic3—121no—
440Dic3—121no—
441Dic3—121no—
442Dic3—121no—
443Dic3—182no—
444Dic3—182no—
445Dic3—181no—
446Dic3—181no—
447Dic3—181no—
448Dic3—181no—
449Dic3—181no—
450Dic3—181no—
451Dic3—361no—
452Dic3—361no—
453Dic3—361no—
454Dic3—361no—
455Dic3—362no—
456Dic3—362no—
457C32—21no—
458C32—21no—
459C32—21no—
460C32—41no—
461C32#479711yescharacteristic, complemented
462C32#479711yescharacteristic, complemented
463C32#479711yescharacteristic, complemented
464Q8—61no—
465Q8—61no—
466Q8—181no—
467Q8—181no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—271no—
473C2×C4—271no—
474C2×C4D3311yescharacteristic
475C8—61no—
476C8—181no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—41no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3#6381111yescharacteristic
490C2×C3#6381111yescharacteristic
491C2×C3#6391311yescharacteristic
492C2×C3#6391511yescharacteristic
493C2×C3#6393111yescharacteristic
494C2×C3#6394511yescharacteristic
495C2×C3#6395111yescharacteristic
496C2×C3#6395111yescharacteristic
497C2×C3#6395311yescharacteristic
498C22#8895411yescharacteristic
499C4—121no—
500C4—121no—
501C4—361no—
502C4—361no—
503C4—541no—
504C4—541no—
505C4#8896011yescharacteristic
506C4#8896111yescharacteristic
507C3—21no—
508C3—21no—
509C3—21no—
510C3—41no—
511C3#1059056811yescharacteristic, complemented
512C3#1059090411yescharacteristic, complemented
513C3#1059161611yescharacteristic, complemented
514C2#1172289311yescharacteristic
515C2#1172289411yescharacteristic
516C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
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
312. C33#491semidirect
461. C32#4797semidirect
462. C32#4797semidirect
463. C32#4797semidirect
511. C3#10590568semidirect
512. C3#10590904semidirect
513. C3#10591616semidirect

Maximal subgroups

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

subgroupindex
#117234512
#117234502
#117220762
#117220212
C3×#635672
C3×#635662
C3×#635652
#105916163
#105909043
#105905683