Finite Group Databasebeta

← #13156684

Subgroups — #13156684

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723536C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic
4#11722021C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×#63573C211yesmaximal, characteristic
8#10590903—31nomaximal
9#10590563—31nomaximal
10#10590546—31nomaximal
11C2×#6293—21no—
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64099—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63872—31no—
28#63872—31no—
29#63872—61no—
30#63642—31no—
31C3×#818—31no—
32C3×#817—31no—
33C3×#808—31no—
34C3×#807—31no—
35C3×#807—31no—
36C3×#807—31no—
37#63576D311yescharacteristic
38#63575D311yescharacteristic
39#63574—31no—
40#63573—31no—
41#63573—31no—
42#63573D311yescharacteristic
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—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53#6293—42no—
54C3×C3⋊(C3⋊C8)Q812yescomplemented
55C32×C3⋊C8Q812yescomplemented
56#4876—91no—
57#4876—91no—
58#4856—91no—
59C2×#634—61no—
60C2×#627—61no—
61C2×#627—61no—
62C2×#627—62no—
63C2×C32×C8—31no—
64C32×C4⋊C4—31no—
65C32×C4⋊C4—31no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70#3750—61no—
71C4×C3⋊Dic3—31no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—31no—
74C4×C3⋊Dic3—61no—
75C4×C3⋊Dic3—61no—
76C4×C3⋊Dic3—61no—
77C4×C3⋊Dic3—121no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
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—62no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—62no—
105Dic3⋊Dic3—62no—
106C22×C33C2×D411yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C33×C4C2×Q811yescharacteristic
109C3⋊(C3⋊Dic3)—21no—
110C3⋊(C3⋊Dic3)—21no—
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—41no—
114C32×Dic3—41no—
115#890—91no—
116#890—91no—
117#890—91no—
118#890—181no—
119#888—91no—
120#888—91no—
121#888—91no—
122#888—181no—
123C3×#107—91no—
124C3×#107—91no—
125C3×#106—91no—
126#818—31no—
127#817—31no—
128#817—91no—
129#815—91no—
130#815—91no—
131#815—91no—
132#815—181no—
133#808—31no—
134#808—91no—
135#808—91no—
136#807—31no—
137#807—31no—
138#807—31no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4—61no—
144C2×C32×C4—61no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—31no—
151C2×C3⋊Dic3—31no—
152C2×C3⋊Dic3—31no—
153C2×C3⋊Dic3—31no—
154C2×C3⋊Dic3—31no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—61no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—121no—
164#634—122no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177#627—124no—
178#627—122no—
179#627—122no—
180C32×C8—32no—
181C3⋊(C3⋊C8)Dic612yescomplemented
182C3×C3⋊C8—22no—
183C3×C3⋊C8—32no—
184C3×C3⋊C8—32no—
185C3×C3⋊C8—62no—
186C3×C3⋊C8Dic612yescomplemented
187C3×C3⋊C8Dic612yescomplemented
188#501—271noSylow
189C2×C33C4⋊Q811yescharacteristic
190C2×C33AGL(1,ℤ/8)11yescharacteristic
191C2×C33#13711yescharacteristic
192C2×Dic6—181no—
193C2×Dic6—181no—
194C2×Dic6—181no—
195C2×Dic6—182no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—91no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—31no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—91no—
207C3×C4⋊C4—91no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—31no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—181no—
215C4⋊Dic3—181no—
216C4⋊Dic3—181no—
217C4⋊Dic3—182no—
218C4⋊Dic3D3211yescharacteristic
219C4⋊Dic3D3211yescharacteristic
220Dic3⋊C4—181no—
221Dic3⋊C4—181no—
222Dic3⋊C4—181no—
223Dic3⋊C4—181no—
224Dic3⋊C4—181no—
225Dic3⋊C4—181no—
226Dic3⋊C4—182no—
227Dic3⋊C4—182no—
228C4×Dic3—91no—
229C4×Dic3—91no—
230C4×Dic3—91no—
231C4×Dic3—181no—
232C4×Dic3—181no—
233C4×Dic3—181no—
234C4×Dic3—361no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—31no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8D3211yescharacteristic
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—41no—
243C22×C32D3×D411yescharacteristic
244C22×C32D3×D411yescharacteristic
245C22×C32D3×D411yescharacteristic
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—41no—
253C32×C4—41no—
254C32×C4—121no—
255C32×C4—121no—
256C32×C4C2×Dic611yescharacteristic
257C32×C4C2×Dic611yescharacteristic
258C32×C4C2×Dic611yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4D3×Q811yescharacteristic
261C32×C4D3×Q811yescharacteristic
262C3⋊Dic3—41no—
263C3⋊Dic3—41no—
264C3⋊Dic3—61no—
265C3⋊Dic3—61no—
266C3⋊Dic3—62no—
267C3⋊Dic3—62no—
268C3⋊Dic3—61no—
269C3⋊Dic3—61no—
270C3⋊Dic3—61no—
271C3⋊Dic3—61no—
272C3⋊Dic3—121no—
273C3⋊Dic3—121no—
274C3⋊Dic3—121no—
275C3⋊Dic3—121no—
276C3⋊Dic3—122no—
277C3⋊Dic3—122no—
278C3×Dic3—42no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—122no—
285C3×Dic3—122no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290C4⋊Q8—271no—
291#125—271no—
292#107—91no—
293#107—91no—
294#106—91no—
295#106—271no—
296#97—271no—
297C33#50111yescharacteristic, complemented, Sylow
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—91no—
322C2×Dic3—91no—
323C2×Dic3—91no—
324C2×Dic3—91no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—181no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—361no—
334C2×Dic3—361no—
335Dic6—362no—
336Dic6—364no—
337Dic6—362no—
338Dic6—362no—
339C3×C8—32no—
340C3×C8—32no—
341C3×C8—62no—
342C3×C8—92no—
343C3⋊C8—32no—
344C3⋊C8—32no—
345C3⋊C8—62no—
346C3⋊C8#62712yescomplemented
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—41no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32#88811yescharacteristic
360C2×C32#88811yescharacteristic
361C2×C32#88811yescharacteristic
362C2×C32#91111yescharacteristic
363C2×C32#91111yescharacteristic
364C2×C32#91411yescharacteristic
365C2×C32#91511yescharacteristic
366C2×C32#91811yescharacteristic
367C2×C32#91811yescharacteristic
368C2×Q8—541no—
369C2×C8—31no—
370C2×C8—91no—
371C4⋊C4—31no—
372C4⋊C4—31no—
373C4⋊C4—91no—
374C4⋊C4—91no—
375C4⋊C4—541no—
376C4⋊C4—541no—
377C4⋊C4—541no—
378C42—271no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—41no—
383C22×C3#380211yescharacteristic
384C22×C3#381111yescharacteristic
385C22×C3#381111yescharacteristic
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—122no—
395C3×C4—122no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4—121no—
399C3×C4—121no—
400C3×C4—361no—
401C3×C4—361no—
402C3×C4D3×Dic611yescharacteristic
403C3×C4D3×Dic611yescharacteristic
404C3×C4D3×Dic611yescharacteristic
405C3×C4D3×Dic611yescharacteristic
406C3×C4#379711yescharacteristic
407C3×C4C2×#62711yescharacteristic
408Dic3—41no—
409Dic3—41no—
410Dic3—122no—
411Dic3—122no—
412Dic3—121no—
413Dic3—121no—
414Dic3—121no—
415Dic3—121no—
416Dic3—182no—
417Dic3—182no—
418Dic3—181no—
419Dic3—181no—
420Dic3—181no—
421Dic3—181no—
422Dic3—181no—
423Dic3—181no—
424Dic3—361no—
425Dic3—361no—
426Dic3—361no—
427Dic3—361no—
428Dic3—362no—
429Dic3—362no—
430C32—21no—
431C32—21no—
432C32—21no—
433C32—41no—
434C32#485611yescharacteristic, complemented
435C32#487611yescharacteristic, complemented
436C32#487611yescharacteristic, complemented
437Q8—1082no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—271no—
443C2×C4—271no—
444C2×C4D3311yescharacteristic
445C8—32no—
446C8—92no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3#6379811yescharacteristic
460C2×C3#6381111yescharacteristic
461C2×C3#6384011yescharacteristic
462C2×C3#6384011yescharacteristic
463C2×C3#6391911yescharacteristic
464C2×C3#6392711yescharacteristic
465C2×C3#6393711yescharacteristic
466C2×C3#6394511yescharacteristic
467C2×C3#6395111yescharacteristic
468C22#8895411yescharacteristic
469C4—121no—
470C4—121no—
471C4—361no—
472C4—361no—
473C4—541no—
474C4—541no—
475C4D3×#62711yescharacteristic
476C4#8895711yescharacteristic
477C3—21no—
478C3—21no—
479C3—21no—
480C3—41no—
481C3#1059054611yescharacteristic, complemented
482C3#1059056311yescharacteristic, complemented
483C3#1059090311yescharacteristic, complemented
484C2#1172287711yescharacteristic
485C2#1172289411yescharacteristic
486C2#1172303711yescharacteristic
subgroupG/H
centerC22#88954
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. C32×C3⋊C8Q8semidirect
181. C3⋊(C3⋊C8)Dic6semidirect
186. C3×C3⋊C8Dic6semidirect
187. C3×C3⋊C8Dic6semidirect
297. C33#501semidirect
346. C3⋊C8#627semidirect
434. C32#4856semidirect
435. C32#4876semidirect
436. C32#4876semidirect
481. C3#10590546semidirect
482. C3#10590563semidirect
483. C3#10590903semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117234502
#117220792
#117220212
C3×#635762
C3×#635752
C3×#635732
#105909033
#105905633
#105905463