Finite Group Databasebeta

← #13155252

Subgroups — #13155252

Order 1728.

464 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723337C211yesmaximal, characteristic
2C3×D3×Dic12C211yesmaximal, characteristic, complemented
3#11722069C211yesmaximal, characteristic, complemented
4#11722012C211yesmaximal, characteristic, complemented
5#11722010C211yesmaximal, characteristic, complemented
6C3×#63551C211yesmaximal, characteristic, complemented
7C3×#63543C211yesmaximal, characteristic, complemented
8#10591047—31nomaximal
9#10590642—31nomaximal
10#10590238—31nomaximal
11#89022—21no—
12C3×D3×Dic6—21no—
13#88852C2211yescharacteristic
14C3×#3746C2211yescharacteristic
15C3×#3746C2211yescharacteristic
16C32×Dic12C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88797—21no—
19C3×#3719—21no—
20C3×C3⋊(C3⋊C16)C2211yescharacteristic
21C32×C3⋊C16C2211yescharacteristic
22#64113—31no—
23C3×D3×Q16—31no—
24C2×C3×Dic12—31no—
25#63808—31no—
26#63808—31no—
27#63808—61no—
28D3×Dic12D311yescharacteristic, complemented
29#63655—31no—
30#63627—31no—
31C3×#829—31no—
32C3×#812—31no—
33C3×Dic24—31no—
34C3×#798—31no—
35#63551—31no—
36#63551—31no—
37#63551D311yescharacteristic, complemented
38#63550—31no—
39#63549—31no—
40#63549—31no—
41#63549—31no—
42#63549—31no—
43#63549—61no—
44#63549—61no—
45#63543D311yescharacteristic, complemented
46C4⋊(C3⋊(C3⋊D3))—21no—
47C3×#634—21no—
48C3×#634—21no—
49C32×C4×D3D411yescharacteristic
50C32×Dic6—21no—
51#6291—41no—
52#6289—41no—
53C3×#627—41no—
54C3×D3×Dic3—41no—
55C33×C8C2311yescharacteristic
56C32×C3⋊C8D411yescharacteristic
57#5179—91no—
58#4918—91no—
59#4909—91no—
60#3828—61no—
61C3×D3×Q8—61no—
62C2×C3×Dic6—61no—
63#3799—62no—
64#3799—61no—
65#3799—61no—
66D3×Dic6—21no—
67C32×Q16—31no—
68C2×C32×C8—31no—
69#3746C2×D311yescharacteristic
70#3746C2×D311yescharacteristic
71#3745—31no—
72#3745—31no—
73#3745—31no—
74#3745—61no—
75#3745—61no—
76#3745—61no—
77#3745—121no—
78#3744—61no—
79C3×#216—61no—
80C3×Dic12—21no—
81C3×Dic12—31no—
82C3×Dic12—31no—
83C3×Dic12—31no—
84C3×Dic12—31no—
85C3×Dic12—61no—
86C3×Dic12—61no—
87C3×Dic12—61no—
88C3×Dic12C2×D311yescharacteristic
89C3×Dic12C2×D311yescharacteristic
90C3×D3×C8—21no—
91C3×D3×C8C2×D311yescharacteristic
92C3×D3×C8C2×D311yescharacteristic
93#3719—21no—
94#3717—62no—
95#3717—61no—
96#3717—61no—
97C32×C16—31no—
98C3⋊(C3⋊C16)C2×D311yescharacteristic
99C3×C3⋊C16—21no—
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C3×C3⋊C16C2×D311yescharacteristic
104C3×C3⋊C16C2×D311yescharacteristic
105C2×C3⋊(C3⋊D3)—41no—
106C2×C32×D3D811yescharacteristic
107C33×C4C2×D411yescharacteristic
108C3×C3⋊Dic3—41no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—41no—
111C32×Dic3D811yescharacteristic
112C2×C3×Q16—91no—
113#919—91no—
114D3×Q16—31no—
115C2×Dic12—31no—
116#904—91no—
117#904—91no—
118#904—181no—
119C3×Q32—91no—
120C3×M32—91no—
121#829—31no—
122#829—91no—
123#829—91no—
124#828—91no—
125#828—91no—
126#828—91no—
127#828—181no—
128#812—31no—
129Dic24—31no—
130Dic24—91no—
131#800—91no—
132#800—91no—
133#800—91no—
134#800—181no—
135#798—31no—
136C32×Q8—61no—
137C2×C32×C4—31no—
138#638—121no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—62no—
141C4⋊(C3⋊D3)—61no—
142C4⋊(C3⋊D3)—61no—
143C4⋊(C3⋊D3)—121no—
144C4⋊(C3⋊D3)—121no—
145C4⋊(C3⋊D3)—122no—
146C4×C3⋊D3—121no—
147#634—21no—
148#634—21no—
149C2×C3×Dic3—121no—
150C3×C4×D3—21no—
151C3×C4×D3—121no—
152C3×C4×D3D1211yescharacteristic
153C3×C4×D3#6711yescharacteristic
154C3×Dic6—21no—
155C3×Dic6—21no—
156C3×Dic6—41no—
157C3×Dic6—62no—
158C3×Dic6—61no—
159C3×Dic6—61no—
160C3×Dic6—61no—
161C3×Dic6—61no—
162C3×Dic6—121no—
163C3×Dic6—121no—
164C3×Dic6—121no—
165#627—41no—
166#626—122no—
167#626—121no—
168#626—121no—
169#624—122no—
170#624—121no—
171#624—121no—
172D3×Dic3—41no—
173C32×C8—21no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—31no—
177C32×C8—41no—
178C32×C8C22×D311yescharacteristic
179C32×C8C22×D311yescharacteristic
180C32×C8C22×D311yescharacteristic
181C3×C3⋊C8—21no—
182C3×C3⋊C8D1211yescharacteristic
183C3×C3⋊C8#6711yescharacteristic
184#510—271noSylow
185C2×C33C2×D811yescharacteristic
186C3⋊(C3⋊D3)—81no—
187C32×D3—21no—
188C2×C3×Q8—181no—
189#239—181no—
190D3×Q8—61no—
191#235—181no—
192#235—181no—
193#235—182no—
194C2×Dic6—61no—
195C3×Q16—31no—
196C3×Q16—31no—
197C3×Q16—61no—
198C3×Q16—91no—
199C3×Q16—91no—
200C3×Q16—181no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—61no—
204#216—61no—
205#215—181no—
206Dic12—31no—
207Dic12—31no—
208Dic12—31no—
209Dic12—31no—
210Dic12—61no—
211Dic12—61no—
212Dic12—61no—
213Dic12D3211yescharacteristic
214D24—91no—
215D24—91no—
216D24—91no—
217D24—181no—
218D24—181no—
219D24—181no—
220D24—361no—
221#204—181no—
222#204—181no—
223#204—182no—
224D3×C8D3211yescharacteristic
225C3×C16—31no—
226C3×C16—31no—
227C3×C16—61no—
228C3×C16—91no—
229C3⋊C16—31no—
230C3⋊C16—31no—
231C3⋊C16—61no—
232C3⋊C16D3211yescharacteristic
233C22×C32—31no—
234C2×C3⋊D3—121no—
235C2×C3⋊D3—122no—
236C2×C3⋊D3—121no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—241no—
239C2×C3⋊D3—241no—
240C2×C3⋊D3—242no—
241C2×C3×D3—21no—
242C2×C3×D3D2411yescharacteristic
243C2×C3×D3#21311yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—31no—
248C32×C4—41no—
249C32×C4—121no—
250C32×C4C2×D1211yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4C2×#6711yescharacteristic
253C3⋊Dic3—41no—
254C3⋊Dic3—41no—
255C3×Dic3—21no—
256C3×Dic3—41no—
257C3×Dic3—41no—
258C3×Dic3—81no—
259C3×Dic3—122no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C3×Dic3—121no—
263C3×Dic3—121no—
264C3×Dic3—241no—
265C3×Dic3D2411yescharacteristic
266C3×Dic3#21311yescharacteristic
267#135—271no—
268C2×Q16—91no—
269Q32—91no—
270Q32—271no—
271SD32—271no—
272SD32—271no—
273M32—91no—
274C33#51011yescharacteristic, complemented, Sylow
275C3×Q8—61no—
276C3×Q8—61no—
277C3×Q8—121no—
278C3×Q8—181no—
279C3×Q8—181no—
280C3×Q8—361no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—61no—
284C2×C3×C4—361no—
285#67—361no—
286#67—361no—
287#67—362no—
288C2×Dic3—121no—
289D12—182no—
290D12—181no—
291D12—181no—
292D12—181no—
293D12—361no—
294D12—361no—
295D12—362no—
296D12—361no—
297C4×D3—121no—
298C4×D3—361no—
299C4×D3—361no—
300C4×D3—362no—
301C4×D3—361no—
302C4×D3#62611yescharacteristic
303Dic6—21no—
304Dic6—62no—
305Dic6—61no—
306Dic6—61no—
307Dic6—61no—
308Dic6—61no—
309Dic6—121no—
310Dic6—121no—
311Dic6—121no—
312C3×C8—21no—
313C3×C8—21no—
314C3×C8—21no—
315C3×C8—31no—
316C3×C8—31no—
317C3×C8—41no—
318C3×C8—61no—
319C3×C8C2×D3211yescharacteristic
320C3×C8C2×D3211yescharacteristic
321C3×C8C2×D3211yescharacteristic
322C3⋊C8#62611yescharacteristic
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—41no—
327C2×C32—61no—
328C2×C32C2×D2411yescharacteristic
329C2×C32D3×D811yescharacteristic
330C2×C32C2×#21311yescharacteristic
331C3⋊D3—241no—
332C3⋊D3—242no—
333C3⋊D3—241no—
334C3⋊D3—241no—
335C3⋊D3—481no—
336C3⋊D3—481no—
337C3⋊D3—482no—
338C3×D3—21no—
339C3×D3—21no—
340C3×D3—41no—
341#41—541no—
342C2×Q8—181no—
343Q16—31no—
344Q16—91no—
345Q16—91no—
346Q16—181no—
347SD16—541no—
348D8—271no—
349C2×C8—31no—
350C16—31no—
351C16—91no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—61no—
355C2×D3—362no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—721no—
360C2×D3—721no—
361C2×D3—722no—
362C2×D3#371411yescharacteristic
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—41no—
369C3×C4—61no—
370C3×C4—121no—
371C3×C4—121no—
372C3×C4—241no—
373C3×C4—361no—
374C3×C4—361no—
375C3×C4D3×D1211yescharacteristic
376C3×C4C2×#62611yescharacteristic
377C3×C4D3×#6711yescharacteristic
378Dic3—41no—
379Dic3—122no—
380Dic3—121no—
381Dic3—121no—
382Dic3—121no—
383Dic3—121no—
384Dic3—241no—
385Dic3#371411yescharacteristic
386C32—21no—
387C32—21no—
388C32—21no—
389C32—41no—
390C32#490911yescharacteristic, complemented
391C32#491811yescharacteristic, complemented
392C32#517911yescharacteristic, complemented
393Q8—61no—
394Q8—181no—
395Q8—181no—
396Q8—361no—
397D4—541no—
398D4—1081no—
399C2×C4—31no—
400C2×C4—361no—
401C2×C4—1081no—
402C8—31no—
403C8D3311yescharacteristic
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—121no—
411C2×C3D3×D2411yescharacteristic
412C2×C3C2×#371411yescharacteristic
413C2×C3D3×#21311yescharacteristic
414D3—21no—
415D3—722no—
416D3—721no—
417D3—721no—
418D3—721no—
419D3—1441no—
420D3—1441no—
421D3—1442no—
422C22—31no—
423C22—1081no—
424C4—31no—
425C4—121no—
426C4—361no—
427C4—361no—
428C4D3×#62611yescharacteristic
429C3—21no—
430C3—21no—
431C3—21no—
432C3—41no—
433C3#1059023811yescharacteristic, complemented
434C3#1059064211yescharacteristic, complemented
435C3#1059104711yescharacteristic, complemented
436C2—61no—
437C2—2161no—
438C2D3×#371411yescharacteristic
subgroupG/H
centerC2D3×#3714
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
prime radical#11723337C2
hypocenterC33#510
LCSG ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C3×D3×Dic12C2semidirect
3. #11722069C2semidirect
4. #11722012C2semidirect
5. #11722010C2semidirect
6. C3×#63551C2semidirect
7. C3×#63543C2semidirect
28. D3×Dic12D3semidirect
37. #63551D3semidirect
45. #63543D3semidirect
274. C33#510semidirect
390. C32#4909semidirect
391. C32#4918semidirect
392. C32#5179semidirect
433. C3#10590238semidirect
434. C3#10590642semidirect
435. C3#10591047semidirect

Maximal subgroups

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

subgroupindex
#117233372
C3×D3×Dic122
#117220692
#117220122
#117220102
C3×#635512
C3×#635432
#105910473
#105906423
#105902383