Finite Group Databasebeta

← #13156671

Subgroups — #13156671

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722056C211yesmaximal, characteristic
5#11722053C211yesmaximal, characteristic
6#11722019C211yesmaximal, characteristic
7C3×#63574C211yesmaximal, characteristic
8#10590421—31nomaximal
9#10590419—31nomaximal
10#10590315—31nomaximal
11C2×C3×#627—21no—
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#890—31no—
23C3×#888—31no—
24C3×#870—31no—
25C3×#869—31no—
26#63918D311yescharacteristic
27#63869D311yescharacteristic
28#63635—31no—
29#63634—31no—
30#63630—31no—
31C3×#808—31no—
32C3×#808—31no—
33#63576—31no—
34#63576—31no—
35#63576—31no—
36#63576—61no—
37#63575—31no—
38#63575—31no—
39#63574—31no—
40#63574—31no—
41#63574—61no—
42#63574D311yescharacteristic
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C3×#627—42no—
54C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
55C3×C3⋊(C3⋊C8)Q812yescomplemented
56#5081—91no—
57#5064—91no—
58#4856—91no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—61no—
61C2×#627—21no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4—31no—
64C32×C42—31no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—61no—
71C2×C3⋊(C3⋊C8)—61no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—121no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C3×C4×Dic3—21no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99#3724—21no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—21no—
102C22×C33C2×D411yescharacteristic
103C33×C4C2×Q811yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C3×C3⋊Dic3—41no—
106C3×C3⋊Dic3—41no—
107C32×Dic3—21no—
108C32×Dic3—21no—
109C32×Dic3—41no—
110C32×Dic3—41no—
111C3×C4⋊Q8—91no—
112C3×#125—91no—
113#890—31no—
114#888—31no—
115#870—31no—
116#869—31no—
117C3×#106—91no—
118#818—91no—
119#815—91no—
120#808—31no—
121#808—31no—
122#808—91no—
123#808—91no—
124#808—91no—
125#808—181no—
126#807—91no—
127#807—91no—
128#807—91no—
129#807—91no—
130#807—91no—
131#807—181no—
132#803—91no—
133#803—91no—
134#803—181no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—31no—
139C2×C32×C4—31no—
140C2×C32×C4—41no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—41no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—121no—
161C2×C3×Dic3—121no—
162C2×C3×Dic3#6711yescharacteristic
163C2×C3×Dic3#6711yescharacteristic
164C2×C3×Dic3#6711yescharacteristic
165C2×C3×Dic3#6711yescharacteristic
166C3×Dic6—122no—
167C3×Dic6—122no—
168#627—42no—
169C3⋊(C3⋊C8)—32no—
170C3⋊(C3⋊C8)—32no—
171C3⋊(C3⋊C8)—32no—
172C3⋊(C3⋊C8)—62no—
173C3⋊(C3⋊C8)—62no—
174C3⋊(C3⋊C8)—62no—
175C3⋊(C3⋊C8)—122no—
176C3⋊(C3⋊C8)Dic612yescomplemented
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—62no—
180#501—271noSylow
181C2×C33C4⋊Q811yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C2×C33#13711yescharacteristic
184C2×C3×Q8—181no—
185C2×Dic6—61no—
186C2×Dic6—61no—
187C2×C3×C8—91no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—31no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—91no—
196C3×C4⋊C4—181no—
197C3×C4⋊C4—181no—
198C3×C4⋊C4—181no—
199C3×C42—31no—
200C3×C42—31no—
201C3×C42—61no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—61no—
207C4⋊Dic3—61no—
208C4⋊Dic3—61no—
209C4⋊Dic3—61no—
210C4⋊Dic3D3211yescharacteristic
211C4⋊Dic3D3211yescharacteristic
212Dic3⋊C4—61no—
213Dic3⋊C4—61no—
214Dic3⋊C4—61no—
215Dic3⋊C4—61no—
216C4×Dic3D3211yescharacteristic
217C2×C3⋊C8—31no—
218C2×C3⋊C8—31no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8—91no—
221C2×C3⋊C8—91no—
222C2×C3⋊C8—91no—
223C2×C3⋊C8—181no—
224C2×C3⋊C8—181no—
225C2×C3⋊C8—181no—
226C2×C3⋊C8—361no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32D3×D411yescharacteristic
232C22×C32C2×#6711yescharacteristic
233C22×C32C2×#6711yescharacteristic
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—61no—
243C32×C4—61no—
244C32×C4—121no—
245C32×C4—121no—
246C32×C4C2×Dic611yescharacteristic
247C32×C4D3×Q811yescharacteristic
248C32×C4D3×Q811yescharacteristic
249C32×C4D3×Q811yescharacteristic
250C32×C4D3×Q811yescharacteristic
251C32×C4D3×Q811yescharacteristic
252C3⋊Dic3—41no—
253C3⋊Dic3—41no—
254C3×Dic3—21no—
255C3×Dic3—21no—
256C3×Dic3—21no—
257C3×Dic3—21no—
258C3×Dic3—41no—
259C3×Dic3—41no—
260C3×Dic3—41no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C3×Dic3—81no—
265C3×Dic3—81no—
266C3×Dic3—121no—
267C3×Dic3—121no—
268C3×Dic3—121no—
269C3×Dic3—121no—
270C3×Dic3—241no—
271C3×Dic3—241no—
272C4⋊Q8—91no—
273#125—91no—
274#107—271no—
275#107—271no—
276#106—91no—
277#106—271no—
278#97—271no—
279C33#50111yescharacteristic, complemented, Sylow
280C3×Q8—362no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—41no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4C2×D3211yescharacteristic
300C2×C3×C4C2×D3211yescharacteristic
301C2×C3×C4C2×D3211yescharacteristic
302C2×Dic3—21no—
303C2×Dic3—21no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—121no—
309C2×Dic3—121no—
310C2×Dic3#62511yescharacteristic
311C2×Dic3#62511yescharacteristic
312Dic6—122no—
313Dic6—122no—
314C3×C8—92no—
315C3⋊C8—32no—
316C3⋊C8—32no—
317C3⋊C8—62no—
318C3⋊C8—92no—
319C3⋊C8—92no—
320C3⋊C8—92no—
321C3⋊C8—182no—
322C3⋊C8—182no—
323C3⋊C8—182no—
324C3⋊C8—362no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32#88811yescharacteristic
338C2×C32#91411yescharacteristic
339C2×C32#91511yescharacteristic
340C2×C32#93211yescharacteristic
341C2×C32#94211yescharacteristic
342C2×C32#94411yescharacteristic
343C2×C32#94411yescharacteristic
344C2×C32#94911yescharacteristic
345C2×C32#95111yescharacteristic
346C2×Q8—181no—
347C2×C8—91no—
348C2×C8—271no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—181no—
354C4⋊C4—181no—
355C4⋊C4—181no—
356C42—31no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3C2×#62511yescharacteristic
362C22×C3D3×#6711yescharacteristic
363C22×C3D3×#6711yescharacteristic
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—121no—
377C3×C4—121no—
378C3×C4—121no—
379C3×C4—121no—
380C3×C4—121no—
381C3×C4—121no—
382C3×C4—241no—
383C3×C4—241no—
384C3×C4—361no—
385C3×C4—361no—
386C3×C4D3×Dic611yescharacteristic
387C3×C4D3×Dic611yescharacteristic
388C3×C4#379711yescharacteristic
389C3×C4#379711yescharacteristic
390C3×C4#379711yescharacteristic
391C3×C4#379711yescharacteristic
392Dic3—21no—
393Dic3—21no—
394Dic3—41no—
395Dic3—41no—
396Dic3—121no—
397Dic3—121no—
398Dic3—121no—
399Dic3—121no—
400Dic3—241no—
401Dic3—241no—
402C32—21no—
403C32—21no—
404C32—21no—
405C32—41no—
406C32#485611yescharacteristic, complemented
407C32#506411yescharacteristic, complemented
408C32#508111yescharacteristic, complemented
409Q8—362no—
410C2×C4—31no—
411C2×C4—31no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4D3311yescharacteristic
417C8—92no—
418C8—272no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3#6382311yescharacteristic
432C2×C3#6382711yescharacteristic
433C2×C3#6383911yescharacteristic
434C2×C3#6383911yescharacteristic
435C2×C3#6387311yescharacteristic
436C2×C3#6392911yescharacteristic
437C2×C3#6393411yescharacteristic
438C2×C3#6393911yescharacteristic
439C2×C3#6394611yescharacteristic
440C22D3×#62511yescharacteristic
441C4—61no—
442C4—61no—
443C4—121no—
444C4—121no—
445C4—361no—
446C4—361no—
447C4#8895711yescharacteristic
448C4#8895811yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3#1059031511yescharacteristic, complemented
454C3#1059041911yescharacteristic, complemented
455C3#1059042111yescharacteristic, complemented
456C2#1172286711yescharacteristic
457C2#1172288211yescharacteristic
458C2#1172303211yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#501
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
54. C3⋊(C3⋊(C3⋊C8))Q8semidirect
55. C3×C3⋊(C3⋊C8)Q8semidirect
176. C3⋊(C3⋊C8)Dic6semidirect
279. C33#501semidirect
406. C32#4856semidirect
407. C32#5064semidirect
408. C32#5081semidirect
453. C3#10590315semidirect
454. C3#10590419semidirect
455. C3#10590421semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#638692
#117220802
#117220562
#117220532
#117220192
C3×#635742
#105904213
#105904193
#105903153