Finite Group Databasebeta

← #13156568

Subgroups — #13156568

Order 1728.

506 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722039C211yesmaximal, characteristic
5#11722037C211yesmaximal, characteristic
6#11722019C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591613—31nomaximal
9#10590562—31nomaximal
10#10590291—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C3×#3724—21no—
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#944—31no—
23C3×#890—31no—
24C3×#870—31no—
25C3×#869—31no—
26#63869D311yescharacteristic
27#63866D311yescharacteristic
28#63662—31no—
29#63637—31no—
30#63630—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63567—31no—
34#63566—31no—
35#63565—31no—
36#63565—31no—
37#63565—31no—
38#63565—31no—
39#63565—31no—
40#63565—61no—
41#63565—61no—
42#63565D311yescharacteristic
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3⋊(C3⋊(C3⋊C8))—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5082—91no—
60#5069—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
70C32×C4⋊C4—31no—
71C32×C42—31no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—121no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101#3724—21no—
102Dic3⋊Dic3—21no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105C22×C33C2×D411yescharacteristic
106C33×C4#4111yescharacteristic
107C33×C4#4111yescharacteristic
108C3×C3⋊Dic3—41no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C32×Dic3—21no—
112C32×Dic3—41no—
113C32×Dic3—41no—
114C3×C4⋊Q8—91no—
115C3×#125—91no—
116#944—31no—
117#890—31no—
118#870—31no—
119#869—31no—
120C3×#103—91no—
121#835—31no—
122#835—91no—
123#835—91no—
124#835—91no—
125#835—91no—
126#835—181no—
127#816—91no—
128#815—91no—
129#810—31no—
130#810—91no—
131#810—91no—
132#810—91no—
133#810—91no—
134#810—181no—
135#803—91no—
136#803—91no—
137#803—181no—
138C32×Q8—61no—
139C32×Q8—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—31no—
144C2×C32×C4—31no—
145C2×C32×C4—41no—
146C2×C32×C4—61no—
147C2×C32×C4—61no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153#634—21no—
154#634—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C3×Dic6—22no—
174C3×Dic6—21no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—21no—
178C3×Dic6—41no—
179C3×Dic6—62no—
180C3×Dic6—61no—
181C3×Dic6—61no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—121no—
185C3⋊(C3⋊C8)—21no—
186C3⋊(C3⋊C8)—61no—
187C3⋊(C3⋊C8)—62no—
188C3⋊(C3⋊C8)—61no—
189C3⋊(C3⋊C8)—61no—
190C3⋊(C3⋊C8)—121no—
191C3⋊(C3⋊C8)—121no—
192C3⋊(C3⋊C8)—122no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196#491—271noSylow
197C2×C33#12411yescharacteristic
198C2×C33#13711yescharacteristic
199C2×C33#13711yescharacteristic
200C2×C3×Q8—31no—
201C2×C3×Q8—31no—
202C2×C3×Q8—61no—
203C2×C3×Q8—91no—
204C2×Dic6—31no—
205C2×Dic6—31no—
206C2×Dic6—61no—
207C2×Dic6D3211yescharacteristic
208C2×C3×C8—91no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—181no—
214C3×C4⋊C4—181no—
215C3×C4⋊C4—181no—
216C3×C4⋊C4—181no—
217C3×C42—31no—
218C3×C42—31no—
219C3×C42—61no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3D3211yescharacteristic
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229Dic3⋊C4—61no—
230Dic3⋊C4—61no—
231Dic3⋊C4—61no—
232C4×Dic3D3211yescharacteristic
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8—91no—
237C2×C3⋊C8—91no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—181no—
240C2×C3⋊C8—181no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—361no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32D3×D411yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—61no—
259C32×C4—61no—
260C32×C4—121no—
261C32×C4—121no—
262C32×C4#23511yescharacteristic
263C32×C4#23711yescharacteristic
264C32×C4#23711yescharacteristic
265C32×C4#23711yescharacteristic
266C32×C4#23911yescharacteristic
267C32×C4#23911yescharacteristic
268C3⋊Dic3—41no—
269C3⋊Dic3—41no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—81no—
282C3×Dic3—122no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—121no—
287C3×Dic3—241no—
288C4⋊Q8—91no—
289#125—91no—
290#103—91no—
291#103—271no—
292#103—271no—
293#103—271no—
294#97—271no—
295C33#49111yescharacteristic, complemented, Sylow
296C3×Q8—62no—
297C3×Q8—61no—
298C3×Q8—61no—
299C3×Q8—61no—
300C3×Q8—61no—
301C3×Q8—121no—
302C3×Q8—181no—
303C3×Q8—181no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—41no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4C2×D3211yescharacteristic
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3#62511yescharacteristic
334C2×Dic3#62511yescharacteristic
335Dic6—21no—
336Dic6—21no—
337Dic6—62no—
338Dic6—61no—
339Dic6—61no—
340Dic6—61no—
341Dic6—61no—
342Dic6—121no—
343C3×C8—181no—
344C3⋊C8—62no—
345C3⋊C8—61no—
346C3⋊C8—61no—
347C3⋊C8—182no—
348C3⋊C8—181no—
349C3⋊C8—181no—
350C3⋊C8—181no—
351C3⋊C8—361no—
352C3⋊C8—361no—
353C3⋊C8—362no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#88511yescharacteristic
367C2×C32#91511yescharacteristic
368C2×C32#91811yescharacteristic
369C2×C32#93611yescharacteristic
370C2×C32#94211yescharacteristic
371C2×C32#94211yescharacteristic
372C2×C32#94711yescharacteristic
373C2×C32#95111yescharacteristic
374C2×C32#95111yescharacteristic
375C2×Q8—31no—
376C2×Q8—91no—
377C2×C8—91no—
378C2×C8—271no—
379C4⋊C4—31no—
380C4⋊C4—91no—
381C4⋊C4—181no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C4⋊C4—181no—
385C42—31no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3C2×#62511yescharacteristic
391C22×C3D3×#6711yescharacteristic
392C22×C3D3×#6711yescharacteristic
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—41no—
400C3×C4—41no—
401C3×C4—61no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—122no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—121no—
411C3×C4—121no—
412C3×C4—241no—
413C3×C4—361no—
414C3×C4—361no—
415C3×C4#379611yescharacteristic
416C3×C4#379611yescharacteristic
417C3×C4#379611yescharacteristic
418C3×C4#379911yescharacteristic
419C3×C4#380411yescharacteristic
420C3×C4#380511yescharacteristic
421Dic3—21no—
422Dic3—21no—
423Dic3—41no—
424Dic3—41no—
425Dic3—122no—
426Dic3—121no—
427Dic3—121no—
428Dic3—121no—
429Dic3—121no—
430Dic3—241no—
431C32—21no—
432C32—21no—
433C32—21no—
434C32—41no—
435C32#479711yescharacteristic, complemented
436C32#506911yescharacteristic, complemented
437C32#508211yescharacteristic, complemented
438Q8—61no—
439Q8—61no—
440Q8—181no—
441Q8—181no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4D3311yescharacteristic
449C8—181no—
450C8—541no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3#6382711yescharacteristic
464C2×C3#6382711yescharacteristic
465C2×C3#6386311yescharacteristic
466C2×C3#6390711yescharacteristic
467C2×C3#6392911yescharacteristic
468C2×C3#6394311yescharacteristic
469C2×C3#6394611yescharacteristic
470C2×C3#6394811yescharacteristic
471C2×C3#6395911yescharacteristic
472C22D3×#62511yescharacteristic
473C4—61no—
474C4—61no—
475C4—121no—
476C4—121no—
477C4—361no—
478C4—361no—
479C4#8896211yescharacteristic
480C4#8896411yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059029111yescharacteristic, complemented
486C3#1059056211yescharacteristic, complemented
487C3#1059161311yescharacteristic, complemented
488C2#1172288211yescharacteristic
489C2#1172288311yescharacteristic
490C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
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
295. C33#491semidirect
435. C32#4797semidirect
436. C32#5069semidirect
437. C32#5082semidirect
485. C3#10590291semidirect
486. C3#10590562semidirect
487. C3#10591613semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×#638662
#117220762
#117220392
#117220372
#117220192
C3×#635652
#105916133
#105905623
#105902913