Finite Group Databasebeta

← #13156329

Subgroups — #13156329

Order 1728.

556 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
1#11723668C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11722055C211yesmaximal, characteristic
4#11722040C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63572C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590805—31nomaximal
9#10590804—31nomaximal
10#10590496—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×#6293—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88813—21no—
21#88807—21no—
22#64097—31no—
23#63918—31no—
24#63918—31no—
25#63918—61no—
26#63844—31no—
27#63843—31no—
28Dic3×Dic6—31no—
29#63645—31no—
30#63643—31no—
31C3×#818—31no—
32C3×#816—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×#810—31no—
36C3×#807—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63575D311yescharacteristic
41#63572D311yescharacteristic
42#63567D311yescharacteristic
43#63566—31no—
44#63566—31no—
45#63566—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—21no—
53C3×#634Q812yescomplemented
54C2×C32×Dic3—21no—
55#6293—42no—
56C32×C3⋊C8—21no—
57C32×C3⋊C8—21no—
58#4875—91no—
59#4850—91no—
60#4793—91no—
61C2×#634—61no—
62C2×#634C2×D311yescharacteristic
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C2×#627—61no—
67C2×#627—61no—
68C2×#627—62no—
69C2×C32×C8—31no—
70C2×C32×C8—31no—
71C32×C4⋊C4—31no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—121no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80#3750—61no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4×Dic3—31no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—61no—
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—61no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—62no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic32—61no—
104Dic32—61no—
105Dic32—61no—
106C22×C33C2×D411yescharacteristic
107C33×C4C2×Q811yescharacteristic
108C33×C4#4111yescharacteristic
109C3⋊(C3⋊Dic3)—41no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3Q1612yes—
114C32×Dic3—41no—
115Q8×Dic3—91no—
116#888—91no—
117#888—91no—
118#888—181no—
119#887—91no—
120#869—91no—
121C4×Dic6—91no—
122C3×#107—91no—
123C3×C4⋊C8—91no—
124C3×#103—91no—
125#835—91no—
126#818—31no—
127#818—91no—
128#818—91no—
129#818—181no—
130#816—31no—
131#816—91no—
132#816—91no—
133#816—181no—
134Dic3⋊C8—31no—
135Dic3⋊C8—31no—
136#810—31no—
137#807—31no—
138#807—91no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—121no—
154C2×C3⋊Dic3—121no—
155C2×C3⋊Dic3—122no—
156C2×C3⋊Dic3D1211yescharacteristic
157C2×C3⋊Dic3D1211yescharacteristic
158#634—21no—
159#634—122no—
160#634Dic612yescomplemented
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—121no—
176C3×Dic6—32no—
177C3×Dic6—32no—
178C3×Dic6—62no—
179C3×Dic6—61no—
180C3×Dic6—61no—
181C3×Dic6—121no—
182#627—124no—
183#627—122no—
184#627—122no—
185C32×C8—61no—
186C32×C8—61no—
187C3×C3⋊C8—22no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—41no—
193#477—271noSylow
194C2×C33C22⋊Q811yescharacteristic
195C2×C33C2×Q1611yescharacteristic
196C2×C33AGL(1,ℤ/8)11yescharacteristic
197C2×C3×Q8—91no—
198C2×Dic6—31no—
199C2×Dic6—31no—
200C2×Dic6—61no—
201C2×Dic6—181no—
202C2×Dic6—181no—
203C2×Dic6—181no—
204C2×Dic6—182no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—61no—
210C2×C3×C8—61no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—91no—
215C3×C42—91no—
216C4⋊Dic3—31no—
217C4⋊Dic3—31no—
218C4⋊Dic3—61no—
219C4⋊Dic3—91no—
220C4⋊Dic3—91no—
221C4⋊Dic3—91no—
222C4⋊Dic3—181no—
223C4⋊Dic3—181no—
224C4⋊Dic3—181no—
225C4⋊Dic3—181no—
226C4⋊Dic3—181no—
227C4⋊Dic3—361no—
228C4⋊Dic3D3211yescharacteristic
229Dic3⋊C4—181no—
230Dic3⋊C4—181no—
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233Dic3⋊C4—182no—
234C4×Dic3—31no—
235C4×Dic3—31no—
236C4×Dic3—61no—
237C4×Dic3—181no—
238C4×Dic3—181no—
239C4×Dic3—181no—
240C2×C3⋊C8D3211yescharacteristic
241C2×C3⋊C8D3211yescharacteristic
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—41no—
246C22×C32C2×D1211yescharacteristic
247C22×C32D3×D411yescharacteristic
248C22×C32D3×D411yescharacteristic
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—121no—
258C32×C4C2×Dic611yescharacteristic
259C32×C4#23511yescharacteristic
260C32×C4#23711yescharacteristic
261C32×C4#23711yescharacteristic
262C32×C4D3×Q811yescharacteristic
263C32×C4D3×Q811yescharacteristic
264C3⋊Dic3—21no—
265C3⋊Dic3—22no—
266C3⋊Dic3—41no—
267C3⋊Dic3—121no—
268C3⋊Dic3—122no—
269C3⋊Dic3—121no—
270C3⋊Dic3—121no—
271C3⋊Dic3—241no—
272C3⋊Dic3—241no—
273C3⋊Dic3—242no—
274C3⋊Dic3Dic1212yes—
275C3×Dic3—32no—
276C3×Dic3—32no—
277C3×Dic3—42no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—122no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C3×Dic3—241no—
290C4⋊Q8—271no—
291C4×Q8—271no—
292#107—91no—
293#107—271no—
294C4⋊C8—91no—
295#103—91no—
296#103—271no—
297C33#47711yescharacteristic, complemented, Sylow
298C3×Q8—92no—
299C3×Q8—181no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—62no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—91no—
308C2×C3×C4—91no—
309C2×C3×C4—181no—
310C2×C3×C4—181no—
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313C2×C3×C4C2×D3211yescharacteristic
314C2×Dic3—21no—
315C2×Dic3—31no—
316C2×Dic3—31no—
317C2×Dic3—31no—
318C2×Dic3—31no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—121no—
326C2×Dic3—121no—
327C2×Dic3—182no—
328C2×Dic3—181no—
329C2×Dic3—181no—
330C2×Dic3—181no—
331C2×Dic3—361no—
332C2×Dic3—361no—
333C2×Dic3—362no—
334Dic6—32no—
335Dic6—32no—
336Dic6—62no—
337Dic6—61no—
338Dic6—61no—
339Dic6—121no—
340Dic6—362no—
341Dic6—364no—
342Dic6—362no—
343Dic6—362no—
344C3×C8—62no—
345C3×C8—61no—
346C3×C8—61no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—121no—
350C3⋊C8—21no—
351C3⋊C8—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32#87811yescharacteristic
365C2×C32#89611yescharacteristic
366C2×C32#89711yescharacteristic
367C2×C32C2×Dic1211yescharacteristic
368C2×C32#90811yescharacteristic
369C2×C32#91111yescharacteristic
370C2×C32#91411yescharacteristic
371C2×C32D3×Q1611yescharacteristic
372C2×C32D3×Q1611yescharacteristic
373C2×Q8—91no—
374C2×Q8—541no—
375C2×C8—31no—
376C2×C8—31no—
377C4⋊C4—31no—
378C4⋊C4—91no—
379C4⋊C4—271no—
380C4⋊C4—541no—
381C4⋊C4—541no—
382C42—91no—
383C42—541no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3D3×D1211yescharacteristic
389C22×C3D3×D1211yescharacteristic
390C22×C3#381111yescharacteristic
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—92no—
400C3×C4—122no—
401C3×C4—121no—
402C3×C4—121no—
403C3×C4—182no—
404C3×C4—181no—
405C3×C4—361no—
406C3×C4D3×Dic611yescharacteristic
407C3×C4D3×Dic611yescharacteristic
408C3×C4#379511yescharacteristic
409C3×C4#379711yescharacteristic
410C3×C4#380411yescharacteristic
411C3×C4#380511yescharacteristic
412Dic3—32no—
413Dic3—32no—
414Dic3—41no—
415Dic3—62no—
416Dic3—62no—
417Dic3—61no—
418Dic3—61no—
419Dic3—62no—
420Dic3—62no—
421Dic3—122no—
422Dic3—121no—
423Dic3—121no—
424Dic3—241no—
425Dic3—362no—
426Dic3—361no—
427Dic3—361no—
428Dic3—361no—
429Dic3—721no—
430Dic3—721no—
431Dic3—722no—
432C32—21no—
433C32—21no—
434C32—21no—
435C32—41no—
436C32#479311yescharacteristic, complemented
437C32#485011yescharacteristic, complemented
438C32#487511yescharacteristic, complemented
439Q8—92no—
440Q8—181no—
441Q8—1082no—
442C2×C4—61no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—541no—
448C2×C4D3311yescharacteristic
449C8—61no—
450C8—61no—
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#6379411yescharacteristic
464C2×C3#6379511yescharacteristic
465C2×C3D3×Dic1211yescharacteristic
466C2×C3D3×Dic1211yescharacteristic
467C2×C3#6389311yescharacteristic
468C2×C3#6389511yescharacteristic
469C2×C3#6389611yescharacteristic
470C2×C3#6393711yescharacteristic
471C2×C3#6394411yescharacteristic
472C22#8895511yescharacteristic
473C4—92no—
474C4—121no—
475C4—181no—
476C4—182no—
477C4—361no—
478C4—1081no—
479C4#8895711yescharacteristic
480C4#8896111yescharacteristic
481C3—21no—
482C3—21no—
483C3—21no—
484C3—41no—
485C3#1059049611yescharacteristic, complemented
486C3#1059080411yescharacteristic, complemented
487C3#1059080511yescharacteristic, complemented
488C2#1172284911yescharacteristic
489C2#1172287911yescharacteristic
490C2#1172298411yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
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
53. C3×#634Q8semidirect
160. #634Dic6semidirect
297. C33#477semidirect
436. C32#4793semidirect
437. C32#4850semidirect
438. C32#4875semidirect
485. C3#10590496semidirect
486. C3#10590804semidirect
487. C3#10590805semidirect

Maximal subgroups

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

subgroupindex
#117236682
#117235352
#117220552
#117220402
C3×#635752
C3×#635722
C3×#635672
#105908053
#105908043
#105904963