Finite Group Databasebeta

← #13156661

Subgroups — #13156661

Order 1728.

498 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723680C211yesmaximal, characteristic
2#11723447C211yesmaximal, characteristic
3#11722055C211yesmaximal, characteristic
4#11722052C211yesmaximal, characteristic
5C3×#63575C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×#63553C211yesmaximal, characteristic
8#10590873—31nomaximal
9#10590871—31nomaximal
10#10590417—31nomaximal
11C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88815—21no—
20#88806—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64099—31no—
23#63872—31no—
24#63869—31no—
25#63869—31no—
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63645—31no—
30#63644—31no—
31C3×#818—31no—
32C3×#817—31no—
33C3×C8×Dic3—31no—
34C3×C8×Dic3—31no—
35C3×#808—31no—
36C3×#807—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63575D311yescharacteristic
41#63573—31no—
42#63573—31no—
43#63573—61no—
44#63573D311yescharacteristic
45#63553D311yescharacteristic
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
52C2×C32×Dic3—21no—
53C32×C3⋊C8Q812yes—
54C32×C3⋊C8Q812yes—
55#4878—91no—
56#4858—91no—
57#4683—91no—
58C2×C32×C8—31no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—61no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)—61no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—121no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C3×C4⋊Dic3—61no—
80C3×C4⋊Dic3C2×D311yescharacteristic
81C3×C4⋊Dic3C2×D311yescharacteristic
82C3×C4×Dic3—31no—
83C3×C4×Dic3—31no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—21no—
87C2×C3×C3⋊C8C2×D311yescharacteristic
88C2×C3×C3⋊C8C2×D311yescharacteristic
89C2×C3×C3⋊C8C2×D311yescharacteristic
90C2×C3×C3⋊C8C2×D311yescharacteristic
91#3724—61no—
92#3724—61no—
93#3724—62no—
94#3724—61no—
95#3724—61no—
96Dic3⋊Dic3—61no—
97Dic3⋊Dic3—61no—
98Dic3⋊Dic3—62no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103C22×C33C2×D411yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C3⋊(C3⋊Dic3)—41no—
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3—21no—
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112#890—91no—
113#890—91no—
114#890—91no—
115#890—91no—
116#890—181no—
117#870—91no—
118#870—91no—
119C3×#107—91no—
120C3×#106—91no—
121C3×C4×C8—91no—
122#818—31no—
123#818—91no—
124#818—91no—
125#818—181no—
126#817—31no—
127#817—91no—
128#817—91no—
129#817—181no—
130C8×Dic3—31no—
131C8×Dic3—31no—
132#808—31no—
133#808—91no—
134#807—31no—
135#807—91no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—61no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—121no—
151C2×C3⋊Dic3—121no—
152C2×C3⋊Dic3—122no—
153C2×C3⋊Dic3D1211yescharacteristic
154C2×C3⋊Dic3D1211yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—31no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—31no—
161C2×C3×Dic3—31no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—121no—
170C32×C8—32no—
171C32×C8—32no—
172C3×C3⋊C8—22no—
173C3×C3⋊C8—22no—
174C3×C3⋊C8Dic612yes—
175C3×C3⋊C8Dic612yes—
176C3×C3⋊C8Dic612yes—
177C3×C3⋊C8Dic612yes—
178#499—271noSylow
179C2×C33C4⋊Q811yescharacteristic
180C2×C33#13511yescharacteristic
181C2×C33#13511yescharacteristic
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—31no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—91no—
192C3×C4⋊C4—91no—
193C3×C42—91no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—61no—
199C4⋊Dic3—61no—
200C4⋊Dic3—91no—
201C4⋊Dic3—91no—
202C4⋊Dic3—91no—
203C4⋊Dic3—181no—
204C4⋊Dic3—181no—
205C4⋊Dic3—181no—
206C4⋊Dic3—181no—
207C4⋊Dic3—181no—
208C4⋊Dic3—181no—
209C4⋊Dic3—181no—
210C4⋊Dic3—182no—
211C4⋊Dic3—361no—
212C4⋊Dic3D3211yescharacteristic
213Dic3⋊C4—181no—
214Dic3⋊C4—181no—
215Dic3⋊C4—181no—
216Dic3⋊C4—181no—
217Dic3⋊C4—181no—
218Dic3⋊C4—181no—
219Dic3⋊C4—181no—
220Dic3⋊C4—181no—
221Dic3⋊C4—182no—
222C4×Dic3—31no—
223C4×Dic3—31no—
224C4×Dic3—61no—
225C2×C3⋊C8D3211yescharacteristic
226C2×C3⋊C8D3211yescharacteristic
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32C2×D1211yescharacteristic
232C22×C32D3×D411yescharacteristic
233C22×C32D3×D411yescharacteristic
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—121no—
243C32×C4C2×Dic611yescharacteristic
244C32×C4C2×Dic611yescharacteristic
245C32×C4C2×Dic611yescharacteristic
246C32×C4C2×Dic611yescharacteristic
247C32×C4D3×Q811yescharacteristic
248C32×C4D3×Q811yescharacteristic
249C3⋊Dic3—21no—
250C3⋊Dic3—21no—
251C3⋊Dic3—41no—
252C3⋊Dic3—41no—
253C3⋊Dic3—121no—
254C3⋊Dic3—122no—
255C3⋊Dic3—121no—
256C3⋊Dic3—121no—
257C3⋊Dic3—241no—
258C3⋊Dic3—241no—
259C3⋊Dic3—242no—
260C3×Dic3—42no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—61no—
264C3×Dic3—61no—
265C3×Dic3—61no—
266C3×Dic3—61no—
267C3×Dic3—121no—
268C3×Dic3—121no—
269C3×Dic3—121no—
270C3×Dic3—121no—
271C3×Dic3—121no—
272C3×Dic3—121no—
273C3×Dic3—241no—
274C3×Dic3—241no—
275#125—271no—
276#125—271no—
277#107—91no—
278#107—271no—
279#106—91no—
280#106—271no—
281C4×C8—91no—
282C33#49911yescharacteristic, complemented, Sylow
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—41no—
287C2×C3×C4—62no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—91no—
291C2×C3×C4—91no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C2×D3211yescharacteristic
295C2×C3×C4C2×D3211yescharacteristic
296C2×C3×C4C2×D3211yescharacteristic
297C2×Dic3—21no—
298C2×Dic3—31no—
299C2×Dic3—31no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—121no—
309C2×Dic3—121no—
310C2×Dic3—182no—
311C2×Dic3—181no—
312C2×Dic3—181no—
313C2×Dic3—181no—
314C2×Dic3—361no—
315C2×Dic3—361no—
316C2×Dic3—362no—
317C3×C8—32no—
318C3×C8—32no—
319C3×C8—32no—
320C3×C8—32no—
321C3×C8—62no—
322C3×C8—62no—
323C3⋊C8#62712yes—
324C3⋊C8#62712yes—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32#86911yescharacteristic
338C2×C32#88811yescharacteristic
339C2×C32#88811yescharacteristic
340C2×C32#90411yescharacteristic
341C2×C32#90411yescharacteristic
342C2×C32#91211yescharacteristic
343C2×C32#91611yescharacteristic
344C2×C32#91611yescharacteristic
345C2×C32#91911yescharacteristic
346C2×C8—31no—
347C2×C8—31no—
348C4⋊C4—31no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—271no—
352C4⋊C4—541no—
353C4⋊C4—541no—
354C4⋊C4—541no—
355C4⋊C4—541no—
356C42—91no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3D3×D1211yescharacteristic
362C22×C3D3×D1211yescharacteristic
363C22×C3#381111yescharacteristic
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—122no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—181no—
376C3×C4—181no—
377C3×C4—361no—
378C3×C4—361no—
379C3×C4D3×Dic611yescharacteristic
380C3×C4D3×Dic611yescharacteristic
381C3×C4D3×Dic611yescharacteristic
382C3×C4D3×Dic611yescharacteristic
383C3×C4C2×#62711yescharacteristic
384C3×C4C2×#62711yescharacteristic
385Dic3—41no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—121no—
391Dic3—121no—
392Dic3—121no—
393Dic3—121no—
394Dic3—121no—
395Dic3—121no—
396Dic3—241no—
397Dic3—241no—
398Dic3—362no—
399Dic3—361no—
400Dic3—361no—
401Dic3—361no—
402Dic3—721no—
403Dic3—721no—
404Dic3—722no—
405C32—21no—
406C32—21no—
407C32—21no—
408C32—41no—
409C32#468311yescharacteristic, complemented
410C32#485811yescharacteristic, complemented
411C32#487811yescharacteristic, complemented
412C2×C4—61no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—181no—
416C2×C4—181no—
417C2×C4—541no—
418C2×C4D3311yescharacteristic
419C8—32no—
420C8—32no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3#6380611yescharacteristic
434C2×C3#6380611yescharacteristic
435C2×C3#6380711yescharacteristic
436C2×C3#6380811yescharacteristic
437C2×C3#6384011yescharacteristic
438C2×C3#6391811yescharacteristic
439C2×C3#6391811yescharacteristic
440C2×C3#6393611yescharacteristic
441C2×C3#6395011yescharacteristic
442C22#8895511yescharacteristic
443C4—121no—
444C4—181no—
445C4—181no—
446C4—361no—
447C4—361no—
448C4—1081no—
449C4D3×#62711yescharacteristic
450C4D3×#62711yescharacteristic
451C3—21no—
452C3—21no—
453C3—21no—
454C3—41no—
455C3#1059041711yescharacteristic, complemented
456C3#1059087111yescharacteristic, complemented
457C3#1059087311yescharacteristic, complemented
458C2#1172286411yescharacteristic
459C2#1172286511yescharacteristic
460C2#1172303611yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#499
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
282. C33#499semidirect
409. C32#4683semidirect
410. C32#4858semidirect
411. C32#4878semidirect
455. C3#10590417semidirect
456. C3#10590871semidirect
457. C3#10590873semidirect

Maximal subgroups

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

subgroupindex
#117236802
#117234472
#117220552
#117220522
C3×#635752
C3×#635732
C3×#635532
#105908733
#105908713
#105904173