Finite Group Databasebeta

← #13156379

Subgroups — #13156379

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723450C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722050C211yesmaximal, characteristic
5C3×#63573C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591615—31nomaximal
9#10590820—31nomaximal
10#10590495—31nomaximal
11C2×C32×Dic6C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22Q8×C3⋊Dic3—31no—
23#64099—31no—
24#63872—31no—
25#63872—31no—
26#63872—61no—
27#63843—31no—
28#63843—31no—
29#63843—61no—
30#63641—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#817—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#808—31no—
37#63576—31no—
38#63575—31no—
39#63575—31no—
40#63573D311yescharacteristic
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63566D311yescharacteristic
45#63565D311yescharacteristic
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—
53C32×Dic6—21no—
54C32×Dic6Q812yes—
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57#4857—91no—
58#4799—91no—
59#4796—91no—
60C2×C32×Q8—31no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6C2×D311yescharacteristic
63C2×C3×Dic6C2×D311yescharacteristic
64C2×C32×C8—31no—
65C32×C4⋊C4—31no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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—61no—
78C4×C3⋊Dic3—121no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
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
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—61no—
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95Dic3⋊Dic3—61no—
96Dic3⋊Dic3—61no—
97Dic3⋊Dic3—61no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—62no—
104Dic32—61no—
105Dic32—61no—
106Dic32—62no—
107C22×C33C2×D411yescharacteristic
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3⋊(C3⋊Dic3)—21no—
111C3⋊(C3⋊Dic3)—21no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—22no—
115C32×Dic3—41no—
116Q8×Dic3—91no—
117Q8×Dic3—91no—
118Q8×Dic3—181no—
119#890—91no—
120#890—91no—
121#890—91no—
122#890—181no—
123#887—91no—
124C3×#106—91no—
125C3×#103—91no—
126C3×#103—91no—
127#835—31no—
128#835—31no—
129#818—91no—
130#817—31no—
131#816—31no—
132Dic3⋊C8—91no—
133Dic3⋊C8—91no—
134Dic3⋊C8—91no—
135Dic3⋊C8—181no—
136#810—31no—
137#808—31no—
138#807—91no—
139#807—91no—
140C32×Q8—32no—
141C32×Q8—61no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—41no—
146C2×C32×C4—61no—
147C2×C32×C4—61no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—31no—
154C2×C3⋊Dic3—31no—
155C2×C3⋊Dic3—31no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—121no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C3×Dic6—22no—
180C3×Dic6—21no—
181C3×Dic6—21no—
182C3×Dic6—22no—
183C3×Dic6Dic612yes—
184C3×Dic6Dic612yes—
185C32×C8—61no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—41no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—121no—
193#479—271noSylow
194C2×C33C22⋊Q811yescharacteristic
195C2×C33#13511yescharacteristic
196C2×C33#13711yescharacteristic
197C2×C3×Q8—31no—
198C2×C3×Q8—31no—
199C2×C3×Q8—61no—
200C2×Dic6D3211yescharacteristic
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—61no—
204C2×C3×C8—91no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—91no—
210C4⋊Dic3—31no—
211C4⋊Dic3—31no—
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216C4⋊Dic3—181no—
217C4⋊Dic3—181no—
218C4⋊Dic3—181no—
219C4⋊Dic3—181no—
220C4⋊Dic3—181no—
221C4⋊Dic3—182no—
222C4⋊Dic3—182no—
223C4⋊Dic3D3211yescharacteristic
224Dic3⋊C4—181no—
225Dic3⋊C4—181no—
226Dic3⋊C4—181no—
227Dic3⋊C4—181no—
228Dic3⋊C4—182no—
229C4×Dic3—91no—
230C4×Dic3—91no—
231C4×Dic3—91no—
232C4×Dic3—181no—
233C4×Dic3—181no—
234C4×Dic3—181no—
235C4×Dic3—181no—
236C4×Dic3—181no—
237C4×Dic3—181no—
238C4×Dic3—182no—
239C4×Dic3—361no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8D3211yescharacteristic
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32D3×D411yescharacteristic
249C22×C32D3×D411yescharacteristic
250C22×C32D3×D411yescharacteristic
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—62no—
260C32×C4—121no—
261C32×C4C2×Dic611yescharacteristic
262C32×C4C2×Dic611yescharacteristic
263C32×C4#23511yescharacteristic
264C32×C4#23711yescharacteristic
265C32×C4#23711yescharacteristic
266C32×C4D3×Q811yescharacteristic
267C3⋊Dic3—41no—
268C3⋊Dic3—41no—
269C3⋊Dic3—61no—
270C3⋊Dic3—61no—
271C3⋊Dic3—62no—
272C3⋊Dic3—62no—
273C3⋊Dic3—61no—
274C3⋊Dic3—61no—
275C3⋊Dic3—61no—
276C3⋊Dic3—61no—
277C3⋊Dic3—121no—
278C3⋊Dic3—121no—
279C3⋊Dic3—121no—
280C3⋊Dic3—121no—
281C3⋊Dic3—122no—
282C3⋊Dic3—122no—
283C3×Dic3—24no—
284C3×Dic3—22no—
285C3×Dic3—22no—
286C3×Dic3—42no—
287C3×Dic3—41no—
288C3×Dic3—41no—
289C3×Dic3—122no—
290C3×Dic3—122no—
291C3×Dic3—121no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295#125—271no—
296C4×Q8—271no—
297#107—271no—
298#106—91no—
299C4⋊C8—271no—
300#103—91no—
301#103—91no—
302C33#47911yescharacteristic, complemented, Sylow
303C3×Q8—32no—
304C3×Q8—32no—
305C3×Q8—62no—
306C3×Q8—62no—
307C3×Q8—61no—
308C3×Q8—61no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—62no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—62no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4C2×D3211yescharacteristic
322C2×C3×C4C2×D3211yescharacteristic
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—91no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—361no—
345C2×Dic3—361no—
346Dic6—21no—
347Dic6#62712yes—
348C3×C8—61no—
349C3×C8—61no—
350C3×C8—121no—
351C3×C8—181no—
352C3⋊C8—21no—
353C3⋊C8—61no—
354C3⋊C8—61no—
355C3⋊C8—121no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32—41no—
368C2×C32#87811yescharacteristic
369C2×C32#87811yescharacteristic
370C2×C32#89611yescharacteristic
371C2×C32#91511yescharacteristic
372C2×C32#91511yescharacteristic
373C2×C32#91611yescharacteristic
374C2×C32#91611yescharacteristic
375C2×C32#91811yescharacteristic
376C2×C32#91911yescharacteristic
377C2×Q8—31no—
378C2×C8—31no—
379C2×C8—91no—
380C4⋊C4—31no—
381C4⋊C4—91no—
382C4⋊C4—91no—
383C4⋊C4—541no—
384C4⋊C4—541no—
385C4⋊C4—541no—
386C42—271no—
387C42—541no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3#380211yescharacteristic
393C22×C3#381111yescharacteristic
394C22×C3#381111yescharacteristic
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4—64no—
404C3×C4—62no—
405C3×C4—62no—
406C3×C4—122no—
407C3×C4—121no—
408C3×C4—121no—
409C3×C4—361no—
410C3×C4—361no—
411C3×C4D3×Dic611yescharacteristic
412C3×C4D3×Dic611yescharacteristic
413C3×C4#379511yescharacteristic
414C3×C4#380411yescharacteristic
415C3×C4#380511yescharacteristic
416C3×C4C2×#62711yescharacteristic
417Dic3—22no—
418Dic3—41no—
419Dic3—122no—
420Dic3—122no—
421Dic3—121no—
422Dic3—121no—
423Dic3—121no—
424Dic3—121no—
425Dic3—182no—
426Dic3—182no—
427Dic3—181no—
428Dic3—181no—
429Dic3—181no—
430Dic3—181no—
431Dic3—181no—
432Dic3—181no—
433Dic3—361no—
434Dic3—361no—
435Dic3—361no—
436Dic3—361no—
437Dic3—362no—
438Dic3—362no—
439C32—21no—
440C32—21no—
441C32—21no—
442C32—41no—
443C32#479611yescharacteristic, complemented
444C32#479911yescharacteristic, complemented
445C32#485711yescharacteristic, complemented
446Q8—32no—
447Q8—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—271no—
453C2×C4—271no—
454C2×C4D3311yescharacteristic
455C8—61no—
456C8—181no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3#6380911yescharacteristic
470C2×C3#6381111yescharacteristic
471C2×C3#6389611yescharacteristic
472C2×C3#6389911yescharacteristic
473C2×C3#6393111yescharacteristic
474C2×C3#6395011yescharacteristic
475C2×C3#6395011yescharacteristic
476C2×C3#6395111yescharacteristic
477C2×C3#6398211yescharacteristic
478C22#8895411yescharacteristic
479C4—62no—
480C4—121no—
481C4—361no—
482C4—361no—
483C4—541no—
484C4—541no—
485C4D3×#62711yescharacteristic
486C4#8896111yescharacteristic
487C3—21no—
488C3—21no—
489C3—21no—
490C3—41no—
491C3#1059049511yescharacteristic, complemented
492C3#1059082011yescharacteristic, complemented
493C3#1059161511yescharacteristic, complemented
494C2#1172286311yescharacteristic
495C2#1172289311yescharacteristic
496C2#1172298011yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#479
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
302. C33#479semidirect
443. C32#4796semidirect
444. C32#4799semidirect
445. C32#4857semidirect
491. C3#10590495semidirect
492. C3#10590820semidirect
493. C3#10591615semidirect

Maximal subgroups

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

subgroupindex
#117234502
#117233972
#117220802
#117220502
C3×#635732
C3×#635662
C3×#635652
#105916153
#105908203
#105904953