Finite Group Databasebeta

← #13156680

Subgroups — #13156680

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#6296C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722056C211yesmaximal, characteristic
4#11722053C211yesmaximal, characteristic
5#11722020C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×#63573C211yesmaximal, characteristic
8#10590903—31nomaximal
9#10590419—31nomaximal
10#10590418—31nomaximal
11C2×#6296—21no—
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#890—31no—
23C3×#890—31no—
24#63919—31no—
25#63918—31no—
26#63918—31no—
27#63872D311yescharacteristic
28#63642—31no—
29#63635—31no—
30#63634—31no—
31C3×#818—31no—
32C3×#817—31no—
33C3×#808—31no—
34C3×#807—31no—
35#63576—31no—
36#63576—31no—
37#63576—61no—
38#63575D311yescharacteristic
39#63574—31no—
40#63574—31no—
41#63574—61no—
42#63573D311yescharacteristic
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53#6296—42no—
54C3⋊(C3⋊(C3⋊C8))Q812yescomplemented
55C32×C3⋊C8Q812yescomplemented
56#5064—91no—
57#4876—91no—
58#4856—91no—
59C2×#627—61no—
60C2×#627—61no—
61C2×#627—61no—
62C2×C32×C8—31no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4—31no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—61no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—61no—
74C2×C3⋊(C3⋊C8)—121no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—61no—
98Dic3⋊Dic3—21no—
99Dic3⋊Dic3—21no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102C22×C33C2×D411yescharacteristic
103C33×C4C2×Q811yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—41no—
108C3×C3⋊Dic3—41no—
109C32×Dic3—41no—
110C32×Dic3—41no—
111C3×#125—91no—
112#890—31no—
113#890—31no—
114#888—91no—
115#888—91no—
116#869—91no—
117C3×#107—91no—
118C3×#106—91no—
119#818—31no—
120#817—31no—
121#815—91no—
122#815—91no—
123#815—181no—
124#808—31no—
125#808—91no—
126#808—91no—
127#808—91no—
128#808—181no—
129#807—31no—
130#807—91no—
131#807—91no—
132#807—91no—
133#807—181no—
134#803—91no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4—61no—
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#6711yescharacteristic
147C2×C3⋊Dic3#6711yescharacteristic
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—31no—
153C2×C3×Dic3—31no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—41no—
157C2×C3×Dic3—41no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—121no—
165C2×C3×Dic3—121no—
166#627—122no—
167#627—122no—
168#627—122no—
169C32×C8—32no—
170C3⋊(C3⋊C8)—32no—
171C3⋊(C3⋊C8)—32no—
172C3⋊(C3⋊C8)—32no—
173C3⋊(C3⋊C8)—62no—
174C3⋊(C3⋊C8)—62no—
175C3⋊(C3⋊C8)—62no—
176C3⋊(C3⋊C8)—122no—
177C3×C3⋊C8—22no—
178C3×C3⋊C8Dic612yescomplemented
179C3×C3⋊C8Dic612yescomplemented
180#501—271noSylow
181C2×C33C4⋊Q811yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C2×C33#13711yescharacteristic
184C2×Dic6—181no—
185C2×Dic6—181no—
186C2×Dic6—181no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—61no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—31no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4—91no—
198C3×C4⋊C4—181no—
199C3×C4⋊C4—181no—
200C3×C42—91no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—61no—
206C4⋊Dic3—61no—
207C4⋊Dic3—61no—
208C4⋊Dic3—61no—
209C4⋊Dic3—181no—
210C4⋊Dic3D3211yescharacteristic
211C4⋊Dic3D3211yescharacteristic
212Dic3⋊C4—61no—
213Dic3⋊C4—61no—
214Dic3⋊C4—181no—
215Dic3⋊C4—181no—
216C4×Dic3—31no—
217C4×Dic3—31no—
218C4×Dic3—61no—
219C2×C3⋊C8—91no—
220C2×C3⋊C8—91no—
221C2×C3⋊C8—91no—
222C2×C3⋊C8—181no—
223C2×C3⋊C8—181no—
224C2×C3⋊C8—181no—
225C2×C3⋊C8—361no—
226C2×C3⋊C8D3211yescharacteristic
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32D3×D411yescharacteristic
232C22×C32D3×D411yescharacteristic
233C22×C32C2×#6711yescharacteristic
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×C4—121no—
244C32×C4C2×Dic611yescharacteristic
245C32×C4C2×Dic611yescharacteristic
246C32×C4D3×Q811yescharacteristic
247C32×C4D3×Q811yescharacteristic
248C32×C4D3×Q811yescharacteristic
249C32×C4D3×Q811yescharacteristic
250C3⋊Dic3—21no—
251C3⋊Dic3—21no—
252C3⋊Dic3—41no—
253C3⋊Dic3—41no—
254C3×Dic3—41no—
255C3×Dic3—41no—
256C3×Dic3—41no—
257C3×Dic3—41no—
258C3×Dic3—61no—
259C3×Dic3—61no—
260C3×Dic3—61no—
261C3×Dic3—61no—
262C3×Dic3—81no—
263C3×Dic3—81no—
264C3×Dic3—121no—
265C3×Dic3—121no—
266C3×Dic3—121no—
267C3×Dic3—121no—
268C3×Dic3—121no—
269C3×Dic3—121no—
270C3×Dic3—241no—
271C3×Dic3—241no—
272C4⋊Q8—271no—
273#125—91no—
274#107—91no—
275#107—271no—
276#106—91no—
277#106—271no—
278#97—271no—
279C33#50111yescharacteristic, complemented, Sylow
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—41no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—91no—
289C2×C3×C4—91no—
290C2×C3×C4—121no—
291C2×C3×C4—121no—
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—21no—
299C2×Dic3—31no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3—61no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—61no—
308C2×Dic3—61no—
309C2×Dic3—121no—
310C2×Dic3—121no—
311Dic6—362no—
312Dic6—362no—
313Dic6—362no—
314C3×C8—32no—
315C3×C8—32no—
316C3×C8—62no—
317C3⋊C8—92no—
318C3⋊C8—92no—
319C3⋊C8—92no—
320C3⋊C8—182no—
321C3⋊C8—182no—
322C3⋊C8—182no—
323C3⋊C8—362no—
324C3⋊C8#62712yescomplemented
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#88811yescharacteristic
338C2×C32#88811yescharacteristic
339C2×C32#91111yescharacteristic
340C2×C32#91411yescharacteristic
341C2×C32#91511yescharacteristic
342C2×C32#91811yescharacteristic
343C2×C32#94411yescharacteristic
344C2×C32#94911yescharacteristic
345C2×C32#95111yescharacteristic
346C2×Q8—541no—
347C2×C8—31no—
348C2×C8—271no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—181no—
354C4⋊C4—181no—
355C4⋊C4—541no—
356C42—91no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—41no—
361C22×C3#380211yescharacteristic
362C22×C3D3×#6711yescharacteristic
363C22×C3D3×#6711yescharacteristic
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—121no—
373C3×C4—121no—
374C3×C4—121no—
375C3×C4—121no—
376C3×C4—181no—
377C3×C4—181no—
378C3×C4—241no—
379C3×C4—241no—
380C3×C4—361no—
381C3×C4—361no—
382C3×C4D3×Dic611yescharacteristic
383C3×C4D3×Dic611yescharacteristic
384C3×C4#379711yescharacteristic
385C3×C4#379711yescharacteristic
386C3×C4#379711yescharacteristic
387C3×C4C2×#62711yescharacteristic
388Dic3—41no—
389Dic3—41no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393Dic3—61no—
394Dic3—121no—
395Dic3—121no—
396Dic3—121no—
397Dic3—121no—
398Dic3—121no—
399Dic3—121no—
400Dic3—241no—
401Dic3—241no—
402C32—21no—
403C32—21no—
404C32—21no—
405C32—41no—
406C32#485611yescharacteristic, complemented
407C32#487611yescharacteristic, complemented
408C32#506411yescharacteristic, complemented
409Q8—1082no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4D3311yescharacteristic
417C8—32no—
418C8—272no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3#6379811yescharacteristic
432C2×C3#6381111yescharacteristic
433C2×C3#6383911yescharacteristic
434C2×C3#6383911yescharacteristic
435C2×C3#6391911yescharacteristic
436C2×C3#6392511yescharacteristic
437C2×C3#6392911yescharacteristic
438C2×C3#6393911yescharacteristic
439C2×C3#6394311yescharacteristic
440C22#8895311yescharacteristic
441C4—121no—
442C4—121no—
443C4—181no—
444C4—181no—
445C4—361no—
446C4—361no—
447C4D3×#62711yescharacteristic
448C4#8895811yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3#1059041811yescharacteristic, complemented
454C3#1059041911yescharacteristic, complemented
455C3#1059090311yescharacteristic, complemented
456C2#1172287611yescharacteristic
457C2#1172289111yescharacteristic
458C2#1172303511yescharacteristic
subgroupG/H
centerC22#88953
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. C32×C3⋊C8Q8semidirect
178. C3×C3⋊C8Dic6semidirect
179. C3×C3⋊C8Dic6semidirect
279. C33#501semidirect
324. C3⋊C8#627semidirect
406. C32#4856semidirect
407. C32#4876semidirect
408. C32#5064semidirect
453. C3#10590418semidirect
454. C3#10590419semidirect
455. C3#10590903semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
C3×#638722
#117220562
#117220532
#117220202
C3×#635752
C3×#635732
#105909033
#105904193
#105904183