Finite Group Databasebeta

← #13156678

Subgroups — #13156678

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723535C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6C3×#63576C211yesmaximal, characteristic
7C3×#63574C211yesmaximal, characteristic
8#10590546—31nomaximal
9#10590419—31nomaximal
10#10590418—31nomaximal
11C2×#6293—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22#64097—31no—
23C3×#890—31no—
24C3×#890—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63872D311yescharacteristic
29#63635—31no—
30#63634—31no—
31C3×#808—31no—
32C3×#808—31no—
33C3×#807—31no—
34C3×#807—31no—
35#63576D311yescharacteristic
36#63575—31no—
37#63575—31no—
38#63575—61no—
39#63574D311yescharacteristic
40#63573—31no—
41#63573—31no—
42#63573—61no—
43#63555—31no—
44#63554—31no—
45#63554—31no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53#6293—42no—
54C3×C3⋊(C3⋊C8)Q812yescomplemented
55C3×C3⋊(C3⋊C8)Q812yescomplemented
56#5064—91no—
57#4876—91no—
58#4856—91no—
59C2×#634—61no—
60C2×#627—61no—
61C2×#627—61no—
62C2×#627—62no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—31no—
65C32×C4⋊C4—31no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)—121no—
73#3750—61no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—61no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8—61no—
99Dic3⋊Dic3—21no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—62no—
104C22×C33C2×D411yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C3⋊(C3⋊Dic3)—41no—
108C3×C3⋊Dic3—21no—
109C3×C3⋊Dic3—21no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C3×#125—91no—
114#890—31no—
115#890—31no—
116#888—91no—
117#888—91no—
118#888—181no—
119#869—91no—
120C3×#107—91no—
121C3×#106—91no—
122#818—91no—
123#817—91no—
124#815—91no—
125#815—91no—
126#808—31no—
127#808—31no—
128#808—91no—
129#808—91no—
130#808—181no—
131#807—31no—
132#807—31no—
133#807—91no—
134#807—91no—
135#807—181no—
136#803—91no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—121no—
152C2×C3⋊Dic3—121no—
153C2×C3⋊Dic3—122no—
154C2×C3⋊Dic3#6711yescharacteristic
155C2×C3⋊Dic3#6711yescharacteristic
156#634—122no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
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—
172#627—124no—
173#627—122no—
174#627—122no—
175C3⋊(C3⋊C8)Dic612yescomplemented
176C3⋊(C3⋊C8)Dic612yescomplemented
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—32no—
181C3×C3⋊C8—62no—
182C3×C3⋊C8—62no—
183#501—271noSylow
184C2×C33C4⋊Q811yescharacteristic
185C2×C33AGL(1,ℤ/8)11yescharacteristic
186C2×C33#13711yescharacteristic
187C2×Dic6—181no—
188C2×Dic6—181no—
189C2×Dic6—181no—
190C2×Dic6—182no—
191C2×C3×C8—91no—
192C2×C3×C8—91no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—181no—
203C3×C4⋊C4—181no—
204C3×C42—91no—
205C4⋊Dic3—61no—
206C4⋊Dic3—61no—
207C4⋊Dic3—91no—
208C4⋊Dic3—91no—
209C4⋊Dic3—91no—
210C4⋊Dic3—181no—
211C4⋊Dic3—181no—
212C4⋊Dic3—181no—
213C4⋊Dic3—181no—
214C4⋊Dic3—361no—
215C4⋊Dic3D3211yescharacteristic
216C4⋊Dic3D3211yescharacteristic
217C4⋊Dic3D3211yescharacteristic
218Dic3⋊C4—61no—
219Dic3⋊C4—61no—
220Dic3⋊C4—181no—
221Dic3⋊C4—181no—
222Dic3⋊C4—182no—
223C4×Dic3—31no—
224C4×Dic3—31no—
225C4×Dic3—61no—
226C2×C3⋊C8—31no—
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—31no—
230C2×C3⋊C8—61no—
231C2×C3⋊C8—61no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—41no—
236C22×C32D3×D411yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32C2×#6711yescharacteristic
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—121no—
248C32×C4—121no—
249C32×C4—121no—
250C32×C4C2×Dic611yescharacteristic
251C32×C4C2×Dic611yescharacteristic
252C32×C4D3×Q811yescharacteristic
253C32×C4D3×Q811yescharacteristic
254C32×C4D3×Q811yescharacteristic
255C32×C4D3×Q811yescharacteristic
256C3⋊Dic3—21no—
257C3⋊Dic3—21no—
258C3⋊Dic3—121no—
259C3⋊Dic3—122no—
260C3⋊Dic3—121no—
261C3⋊Dic3—121no—
262C3⋊Dic3—241no—
263C3⋊Dic3—241no—
264C3⋊Dic3—242no—
265C3×Dic3—42no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—41no—
271C3×Dic3—41no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C3×Dic3—61no—
275C3×Dic3—61no—
276C3×Dic3—61no—
277C3×Dic3—61no—
278C3×Dic3—81no—
279C3×Dic3—81no—
280C4⋊Q8—271no—
281#125—91no—
282#107—91no—
283#107—271no—
284#106—91no—
285#106—271no—
286#97—271no—
287C33#50111yescharacteristic, complemented, Sylow
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—62no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—91no—
300C2×C3×C4—91no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×C3×C4C2×D3211yescharacteristic
306C2×Dic3—21no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—31no—
310C2×Dic3—31no—
311C2×Dic3—31no—
312C2×Dic3—31no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—182no—
316C2×Dic3—181no—
317C2×Dic3—181no—
318C2×Dic3—181no—
319C2×Dic3—361no—
320C2×Dic3—361no—
321C2×Dic3—362no—
322Dic6—362no—
323Dic6—364no—
324Dic6—362no—
325Dic6—362no—
326C3×C8—92no—
327C3×C8—92no—
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—32no—
331C3⋊C8—32no—
332C3⋊C8—62no—
333C3⋊C8—62no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32#88811yescharacteristic
347C2×C32#88811yescharacteristic
348C2×C32#91111yescharacteristic
349C2×C32#91411yescharacteristic
350C2×C32#91511yescharacteristic
351C2×C32#91811yescharacteristic
352C2×C32#94411yescharacteristic
353C2×C32#94911yescharacteristic
354C2×C32#95111yescharacteristic
355C2×Q8—541no—
356C2×C8—91no—
357C2×C8—91no—
358C4⋊C4—31no—
359C4⋊C4—31no—
360C4⋊C4—31no—
361C4⋊C4—181no—
362C4⋊C4—181no—
363C4⋊C4—271no—
364C4⋊C4—541no—
365C42—91no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3D3×#6711yescharacteristic
371C22×C3D3×#6711yescharacteristic
372C22×C3#381111yescharacteristic
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—122no—
382C3×C4—121no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—121no—
388C3×C4—181no—
389C3×C4—181no—
390C3×C4—241no—
391C3×C4—241no—
392C3×C4D3×Dic611yescharacteristic
393C3×C4D3×Dic611yescharacteristic
394C3×C4D3×Dic611yescharacteristic
395C3×C4D3×Dic611yescharacteristic
396C3×C4#379711yescharacteristic
397C3×C4#379711yescharacteristic
398Dic3—41no—
399Dic3—41no—
400Dic3—41no—
401Dic3—62no—
402Dic3—62no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—61no—
407Dic3—362no—
408Dic3—361no—
409Dic3—361no—
410Dic3—361no—
411Dic3—721no—
412Dic3—721no—
413Dic3—722no—
414C32—21no—
415C32—21no—
416C32—21no—
417C32—41no—
418C32#485611yescharacteristic, complemented
419C32#487611yescharacteristic, complemented
420C32#506411yescharacteristic, complemented
421Q8—1082no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—541no—
428C2×C4D3311yescharacteristic
429C8—92no—
430C8—92no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3#6383911yescharacteristic
444C2×C3#6383911yescharacteristic
445C2×C3#6384011yescharacteristic
446C2×C3#6392511yescharacteristic
447C2×C3#6392911yescharacteristic
448C2×C3#6393711yescharacteristic
449C2×C3#6393911yescharacteristic
450C2×C3#6394311yescharacteristic
451C2×C3#6395111yescharacteristic
452C22#8895211yescharacteristic
453C4—121no—
454C4—121no—
455C4—121no—
456C4—181no—
457C4—181no—
458C4—1081no—
459C4#8895711yescharacteristic
460C4#8895711yescharacteristic
461C3—21no—
462C3—21no—
463C3—21no—
464C3—41no—
465C3#1059041811yescharacteristic, complemented
466C3#1059041911yescharacteristic, complemented
467C3#1059054611yescharacteristic, complemented
468C2#1172287411yescharacteristic
469C2#1172288811yescharacteristic
470C2#1172303411yescharacteristic
subgroupG/H
centerC22#88952
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
175. C3⋊(C3⋊C8)Dic6semidirect
176. C3⋊(C3⋊C8)Dic6semidirect
287. C33#501semidirect
418. C32#4856semidirect
419. C32#4876semidirect
420. C32#5064semidirect
465. C3#10590418semidirect
466. C3#10590419semidirect
467. C3#10590546semidirect

Maximal subgroups

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

subgroupindex
#117235352
C3×#638722
#117220712
#117220572
#117220542
C3×#635762
C3×#635742
#105905463
#105904193
#105904183