Finite Group Databasebeta

← #13155708

Subgroups — #13155708

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2C2×#88798C211yesmaximal, characteristic
3C3×#63843C211yesmaximal, characteristic
4#11722053C211yesmaximal, characteristic
5#11722020C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63566C211yesmaximal, characteristic
8#10591541—31nomaximal
9#10590791—31nomaximal
10#10590477—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20#88798C414yescomplemented
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C2×#3756—31no—
24C3×Q8×Dic3—31no—
25C3×#887—31no—
26#63844—31no—
27#63843—31no—
28#63843D311yescharacteristic
29Dic3×Dic6—31no—
30C2×#3718—31no—
31C2×#3718—31no—
32C2×#3718—61no—
33#63642—31no—
34#63635—31no—
35C3×#835—31no—
36C3×#816—31no—
37C3×#816—31no—
38C3×#810—31no—
39#63574—31no—
40#63574—31no—
41#63574—61no—
42#63567D311yescharacteristic
43#63566D311yescharacteristic
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—21no—
53C3×#634C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yes—
58C3⋊(C3⋊(C3⋊C8))C2×C412yes—
59C32×C3⋊C8—21no—
60#5030—91no—
61#4789—91no—
62#4789—91no—
63C2×C32×Q8—31no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73#3756—34no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C4×C3⋊Dic3C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—121no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—61no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic32—21no—
104Dic32—61no—
105Dic32—61no—
106Dic32—61no—
107#3718—34no—
108#3718—34no—
109#3718—64no—
110C22×C33C2×D411yescharacteristic
111C33×C4C22×C411yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—22no—
118C32×Dic3—41no—
119C3×C4×Q8—91no—
120Q8×Dic3—31no—
121Q8×Dic3—91no—
122C2×#216—91no—
123C2×#216—91no—
124C2×#216—91no—
125C2×#216—181no—
126#887—31no—
127#887—91no—
128C4×Dic6—91no—
129C3×#103—91no—
130C3×#103—91no—
131#835—31no—
132#816—31no—
133#816—31no—
134#815—91no—
135#815—91no—
136#815—181no—
137#810—31no—
138#808—91no—
139#808—91no—
140#808—91no—
141#808—181no—
142#803—91no—
143C32×Q8—32no—
144C32×Q8—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3#6711yescharacteristic
157C2×C3⋊Dic3#6711yescharacteristic
158#634—21no—
159#634C4×D312yes—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
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—41no—
169C2×C3×Dic3—41no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—121no—
177C2×C3×Dic3—121no—
178C3×Dic6—21no—
179C3×Dic6—21no—
180C3×Dic6—22no—
181C3×Dic6—32no—
182C3×Dic6—32no—
183C3×Dic6—41no—
184C3×Dic6—62no—
185C3×Dic6—61no—
186C3×Dic6—61no—
187C3×Dic6—121no—
188C3×Dic6C4×D312yes—
189C3×Dic6C4×D312yes—
190C32×C8—61no—
191C3⋊(C3⋊C8)—32no—
192C3⋊(C3⋊C8)—32no—
193C3⋊(C3⋊C8)—32no—
194C3⋊(C3⋊C8)—62no—
195C3⋊(C3⋊C8)—62no—
196C3⋊(C3⋊C8)—62no—
197C3⋊(C3⋊C8)—122no—
198C3×C3⋊C8—22no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201#441—271noSylow
202C2×C33C4×D411yescharacteristic
203C2×C33#13711yescharacteristic
204C2×C33#13711yescharacteristic
205C2×C3×Q8—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×Q8—91no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—61no—
212C2×Dic6D3211yescharacteristic
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C42—91no—
222C3×C42—181no—
223#216—94no—
224#216—94no—
225#216—94no—
226#216—184no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3—181no—
232C4⋊Dic3D3211yescharacteristic
233Dic3⋊C4—61no—
234Dic3⋊C4—181no—
235Dic3⋊C4—181no—
236C4×Dic3—31no—
237C4×Dic3—31no—
238C4×Dic3—61no—
239C4×Dic3—61no—
240C4×Dic3—61no—
241C4×Dic3—181no—
242C4×Dic3—181no—
243C4×Dic3—181no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—181no—
249C2×C3⋊C8—181no—
250C2×C3⋊C8—361no—
251C2×C3⋊C8D3211yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32D3×D411yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—62no—
268C32×C4—121no—
269C32×C4C2×C4×D311yescharacteristic
270C32×C4C2×C4×D311yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23711yescharacteristic
274C32×C4#23711yescharacteristic
275C3⋊Dic3—22no—
276C3⋊Dic3—21no—
277C3⋊Dic3—21no—
278C3⋊Dic3—41no—
279C3×Dic3—22no—
280C3×Dic3—22no—
281C3×Dic3—42no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—62no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—62no—
290C3×Dic3—81no—
291C3×Dic3—122no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—241no—
297C2×Q16—271no—
298C4×Q8—91no—
299C4×Q8—271no—
300#106—271no—
301#103—91no—
302#103—91no—
303#97—271no—
304C33#44111yescharacteristic, complemented, Sylow
305C3×Q8—32no—
306C3×Q8—32no—
307C3×Q8—62no—
308C3×Q8—61no—
309C3×Q8—61no—
310C3×Q8—92no—
311C3×Q8—121no—
312C3×Q8—181no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—91no—
322C2×C3×C4—91no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—31no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—31no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—121no—
343C2×Dic3—121no—
344Dic6—21no—
345Dic6—32no—
346Dic6—32no—
347Dic6—62no—
348Dic6—61no—
349Dic6—61no—
350Dic6—121no—
351Dic6#62412yes—
352C3×C8—62no—
353C3×C8—61no—
354C3×C8—61no—
355C3⋊C8—21no—
356C3⋊C8—92no—
357C3⋊C8—92no—
358C3⋊C8—92no—
359C3⋊C8—182no—
360C3⋊C8—182no—
361C3⋊C8—182no—
362C3⋊C8—362no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32#67⋊C411yescharacteristic
376C2×C32#67⋊C411yescharacteristic
377C2×C32#91511yescharacteristic
378C2×C32#91511yescharacteristic
379C2×C32#91811yescharacteristic
380C2×C32#91811yescharacteristic
381C2×C32C4×#6711yescharacteristic
382C2×C32#94211yescharacteristic
383C2×C32#95111yescharacteristic
384C2×Q8—31no—
385C2×Q8—91no—
386Q16—274no—
387C2×C8—31no—
388C2×C8—271no—
389C4⋊C4—31no—
390C4⋊C4—91no—
391C4⋊C4—181no—
392C4⋊C4—541no—
393C42—91no—
394C42—181no—
395C42—541no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—41no—
400C22×C3#380211yescharacteristic
401C22×C3D3×#6711yescharacteristic
402C22×C3D3×#6711yescharacteristic
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—62no—
412C3×C4—62no—
413C3×C4—122no—
414C3×C4—121no—
415C3×C4—121no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4—182no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4#379511yescharacteristic
422C3×C4C4×D3211yescharacteristic
423C3×C4C4×D3211yescharacteristic
424C3×C4#380411yescharacteristic
425C3×C4#380511yescharacteristic
426C3×C4C2×#62411yescharacteristic
427Dic3—22no—
428Dic3—41no—
429Dic3—62no—
430Dic3—62no—
431Dic3—61no—
432Dic3—61no—
433Dic3—61no—
434Dic3—61no—
435Dic3—122no—
436Dic3—121no—
437Dic3—121no—
438Dic3—121no—
439Dic3—121no—
440Dic3—241no—
441C32—21no—
442C32—21no—
443C32—21no—
444C32—41no—
445C32#478911yescharacteristic, complemented
446C32#478911yescharacteristic, complemented
447C32#503011yescharacteristic, complemented
448Q8—32no—
449Q8—61no—
450Q8—92no—
451Q8—181no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—91no—
455C2×C4—91no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4D3311yescharacteristic
459C8—61no—
460C8—272no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3#6381111yescharacteristic
474C2×C3#6381111yescharacteristic
475C2×C3#6387911yescharacteristic
476C2×C3#6390211yescharacteristic
477C2×C3#6392911yescharacteristic
478C2×C3#6394311yescharacteristic
479C2×C3#6394611yescharacteristic
480C2×C3#6394811yescharacteristic
481C2×C3#6395211yescharacteristic
482C22#8895311yescharacteristic
483C4—62no—
484C4—121no—
485C4—181no—
486C4—181no—
487C4—182no—
488C4—361no—
489C4D3×#62411yescharacteristic
490C4#8896111yescharacteristic
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#1059047711yescharacteristic, complemented
496C3#1059079111yescharacteristic, complemented
497C3#1059154111yescharacteristic, complemented
498C2#1172289111yescharacteristic
499C2#1172289211yescharacteristic
500C2#1172293611yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
20. #88798C4semidirect
304. C33#441semidirect
445. C32#4789semidirect
446. C32#4789semidirect
447. C32#5030semidirect
495. C3#10590477semidirect
496. C3#10590791semidirect
497. C3#10591541semidirect

Maximal subgroups

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

subgroupindex
#117236682
C2×#887982
C3×#638432
#117220532
#117220202
C3×#635672
C3×#635662
#105915413
#105907913
#105904773