Finite Group Databasebeta

← #13155251

Subgroups — #13155251

Order 1728.

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

normal subgroups only

This group has 456 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#11723336C211yesmaximal, characteristic
2C3×D3×Dic12C211yesmaximal, characteristic, complemented
3#11722068C211yesmaximal, characteristic, complemented
4#11722015C211yesmaximal, characteristic, complemented
5#11722014C211yesmaximal, characteristic, complemented
6C3×#63549C211yesmaximal, characteristic, complemented
7C3×#63543C211yesmaximal, characteristic, complemented
8#10591045—31nomaximal
9#10590642—31nomaximal
10#10590239—31nomaximal
11#89021—21no—
12C3×D3×Dic6—21no—
13#88853C2211yescharacteristic
14C3×#3746C2211yescharacteristic
15C3×#3745C2211yescharacteristic
16C32×Dic12C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88796—21no—
19C3×#3719—21no—
20C3×C3⋊(C3⋊C16)C2211yescharacteristic
21C32×C3⋊C16C2211yescharacteristic
22#64113—31no—
23C3×D3×Q16—31no—
24C2×C3×Dic12—31no—
25#63807—31no—
26#63807—31no—
27#63807—61no—
28D3×Dic12D311yescharacteristic, complemented
29#63656—31no—
30#63628—31no—
31C3×#828—31no—
32C3×#812—31no—
33C3×#800—31no—
34C3×#798—31no—
35#63551—31no—
36#63551—31no—
37#63551—31no—
38#63551—31no—
39#63551—61no—
40#63551—61no—
41#63549—31no—
42#63549D311yescharacteristic, complemented
43#63548—31no—
44#63547—31no—
45#63543D311yescharacteristic, complemented
46#6308—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×#634—21no—
49C32×C4×D3D411yescharacteristic
50C32×Dic6—21no—
51#6290—41no—
52Dic3×C3⋊D3—41no—
53C3×#627—41no—
54C3×D3×Dic3—41no—
55C33×C8C2311yescharacteristic
56C32×C3⋊C8D411yescharacteristic
57#5179—91no—
58#4915—91no—
59#4909—91no—
60#3828—61no—
61C3×D3×Q8—61no—
62C2×C3×Dic6—61no—
63#3795—62no—
64#3795—61no—
65#3795—61no—
66D3×Dic6—21no—
67C32×Q16—31no—
68C2×C32×C8—31no—
69#3746—31no—
70#3746—31no—
71#3746—31no—
72#3746—61no—
73#3746—61no—
74#3746—61no—
75#3746—121no—
76#3746C2×D311yescharacteristic
77#3745C2×D311yescharacteristic
78#3744—61no—
79C3×#216—61no—
80C3×Dic12—21no—
81C3×Dic12—31no—
82C3×Dic12—31no—
83C3×Dic12—61no—
84C3×Dic12—61no—
85C3×Dic12C2×D311yescharacteristic
86C3×Dic12C2×D311yescharacteristic
87C3×D24—31no—
88C3×D24—31no—
89C3×D24—61no—
90C3×D3×C8—21no—
91C3×D3×C8C2×D311yescharacteristic
92C3×D3×C8C2×D311yescharacteristic
93#3719—21no—
94#3716—62no—
95#3716—61no—
96#3716—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—
113D3×Q16—31no—
114#912—91no—
115C2×Dic12—31no—
116#904—91no—
117#904—91no—
118#904—181no—
119C3×SD32—91no—
120C3×M32—91no—
121#829—91no—
122#829—91no—
123#829—91no—
124#829—181no—
125#828—31no—
126#828—91no—
127#827—91no—
128#812—31no—
129Dic24—91no—
130Dic24—91no—
131Dic24—91no—
132Dic24—181no—
133#800—31no—
134#800—91no—
135#798—31no—
136C32×Q8—61no—
137C2×C32×C4—31no—
138#638—121no—
139C4⋊(C3⋊D3)—21no—
140C4×C3⋊D3—121no—
141#634—21no—
142#634—61no—
143#634—62no—
144#634—61no—
145#634—61no—
146#634—121no—
147#634—121no—
148#634—122no—
149C2×C3×Dic3—121no—
150C3×D12—62no—
151C3×D12—61no—
152C3×D12—61no—
153C3×C4×D3—21no—
154C3×C4×D3—121no—
155C3×C4×D3D1211yescharacteristic
156C3×C4×D3#6711yescharacteristic
157C3×Dic6—21no—
158C3×Dic6—21no—
159C3×Dic6—41no—
160C3×Dic6—61no—
161C3×Dic6—61no—
162C3×Dic6—121no—
163C3×Dic6—121no—
164C3×Dic6—121no—
165#627—41no—
166#625—122no—
167#625—121no—
168#625—121no—
169D3×Dic3—41no—
170D3×Dic3—122no—
171D3×Dic3—121no—
172D3×Dic3—121no—
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—
189D3×Q8—61no—
190#237—181no—
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—181no—
200C3×D8—91no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—61no—
204#216—61no—
205#214—181no—
206Dic12—31no—
207Dic12—31no—
208Dic12—61no—
209Dic12—61no—
210Dic12—91no—
211Dic12—91no—
212Dic12—91no—
213Dic12—181no—
214Dic12—181no—
215Dic12—181no—
216Dic12—361no—
217Dic12D3211yescharacteristic
218D24—31no—
219D24—31no—
220D24—61no—
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—41no—
235C2×C3×D3—21no—
236C2×C3×D3—122no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3D2411yescharacteristic
240C2×C3×D3#21311yescharacteristic
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—31no—
245C32×C4—41no—
246C32×C4—121no—
247C32×C4C2×D1211yescharacteristic
248C32×C4D3×D411yescharacteristic
249C32×C4C2×#6711yescharacteristic
250C3⋊Dic3—41no—
251C3⋊Dic3—121no—
252C3⋊Dic3—122no—
253C3⋊Dic3—121no—
254C3⋊Dic3—121no—
255C3⋊Dic3—241no—
256C3⋊Dic3—241no—
257C3⋊Dic3—242no—
258C3×Dic3—21no—
259C3×Dic3—41no—
260C3×Dic3—41no—
261C3×Dic3—81no—
262C3×Dic3—121no—
263C3×Dic3—121no—
264C3×Dic3—241no—
265C3×Dic3D2411yescharacteristic
266C3×Dic3#21311yescharacteristic
267#135—271no—
268C2×Q16—91no—
269Q32—271no—
270Q32—271no—
271SD32—91no—
272SD32—271no—
273M32—91no—
274C33#51011yescharacteristic, complemented, Sylow
275C3×Q8—61no—
276C3×Q8—61no—
277C3×Q8—121no—
278C3×Q8—181no—
279C3×Q8—361no—
280C3×D4—181no—
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—
288#67—361no—
289C2×Dic3—121no—
290C2×Dic3—361no—
291D12—62no—
292D12—61no—
293D12—61no—
294C4×D3—121no—
295C4×D3—361no—
296C4×D3—361no—
297C4×D3—362no—
298C4×D3#62611yescharacteristic
299Dic6—21no—
300Dic6—61no—
301Dic6—61no—
302Dic6—121no—
303Dic6—121no—
304Dic6—121no—
305Dic6—182no—
306Dic6—181no—
307Dic6—181no—
308Dic6—181no—
309Dic6—361no—
310Dic6—361no—
311Dic6—362no—
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—81no—
332C3×D3—21no—
333C3×D3—21no—
334C3×D3—41no—
335C3×D3—242no—
336C3×D3—241no—
337C3×D3—241no—
338#41—541no—
339C2×Q8—181no—
340Q16—31no—
341Q16—91no—
342Q16—181no—
343Q16—271no—
344SD16—541no—
345D8—91no—
346C2×C8—31no—
347C16—31no—
348C16—91no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—61no—
352C22×C3—361no—
353C2×D3—122no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3#371411yescharacteristic
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—41no—
363C3×C4—61no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4—241no—
367C3×C4—361no—
368C3×C4D3×D1211yescharacteristic
369C3×C4C2×#62611yescharacteristic
370C3×C4D3×#6711yescharacteristic
371Dic3—41no—
372Dic3—121no—
373Dic3—121no—
374Dic3—241no—
375Dic3—362no—
376Dic3—361no—
377Dic3—361no—
378Dic3—361no—
379Dic3—721no—
380Dic3—721no—
381Dic3—722no—
382Dic3#371411yescharacteristic
383C32—21no—
384C32—21no—
385C32—21no—
386C32—41no—
387C32#490911yescharacteristic, complemented
388C32#491511yescharacteristic, complemented
389C32#517911yescharacteristic, complemented
390Q8—61no—
391Q8—181no—
392Q8—361no—
393Q8—541no—
394D4—181no—
395D4—1081no—
396C2×C4—31no—
397C2×C4—361no—
398C2×C4—1081no—
399C8—31no—
400C8D3311yescharacteristic
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—41no—
405C2×C3—61no—
406C2×C3—61no—
407C2×C3—121no—
408C2×C3—721no—
409C2×C3D3×D2411yescharacteristic
410C2×C3C2×#371411yescharacteristic
411C2×C3D3×#21311yescharacteristic
412D3—21no—
413D3—242no—
414D3—241no—
415D3—241no—
416C22—31no—
417C22—361no—
418C4—31no—
419C4—121no—
420C4—361no—
421C4—1081no—
422C4D3×#62611yescharacteristic
423C3—21no—
424C3—21no—
425C3—21no—
426C3—41no—
427C3#1059023911yescharacteristic, complemented
428C3#1059064211yescharacteristic, complemented
429C3#1059104511yescharacteristic, complemented
430C2—61no—
431C2—721no—
432C2D3×#371411yescharacteristic
subgroupG/H
centerC2D3×#3714
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
prime radical#11723336C2
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. #11722068C2semidirect
4. #11722015C2semidirect
5. #11722014C2semidirect
6. C3×#63549C2semidirect
7. C3×#63543C2semidirect
28. D3×Dic12D3semidirect
42. #63549D3semidirect
45. #63543D3semidirect
274. C33#510semidirect
387. C32#4909semidirect
388. C32#4915semidirect
389. C32#5179semidirect
427. C3#10590239semidirect
428. C3#10590642semidirect
429. C3#10591045semidirect

Maximal subgroups

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

subgroupindex
#117233362
C3×D3×Dic122
#117220682
#117220152
#117220142
C3×#635492
C3×#635432
#105910453
#105906423
#105902393