Finite Group Databasebeta

← #13155236

Subgroups — #13155236

Order 1728.

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

normal subgroups only

This group has 492 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, complemented
2C3×D3×D24C211yesmaximal, characteristic, complemented
3#11722067C211yesmaximal, characteristic, complemented
4#11722011C211yesmaximal, characteristic, complemented
5#11722009C211yesmaximal, characteristic, complemented
6C3×#63546C211yesmaximal, characteristic, complemented
7C3×#63543C211yesmaximal, characteristic, complemented
8#10591031—31nomaximal
9#10590633—31nomaximal
10#10590232—31nomaximal
11#89021—21no—
12C3×D3×D12—21no—
13#88853C2211yescharacteristic, complemented
14C3×#3745C2211yescharacteristic
15C3×#3745C2211yescharacteristic, complemented
16C32×D24C2211yescharacteristic
17C32×D3×C8C2211yescharacteristic
18#88796—21no—
19C3×#3714—21no—
20C3×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
21C32×C3⋊C16C2211yescharacteristic, complemented
22#64113—31no—
23C3×D3×D8—31no—
24C2×C3×D24—31no—
25#63807—31no—
26#63807—31no—
27#63807—61no—
28D3×D24D311yescharacteristic, complemented
29#63654—31no—
30#63627—31no—
31C3×#826—31no—
32C3×#812—31no—
33C3×D48—31no—
34C3×#798—31no—
35#63548—31no—
36#63548—31no—
37#63548—31no—
38#63548—31no—
39#63548—61no—
40#63548—61no—
41#63546—31no—
42#63546—31no—
43#63546D311yescharacteristic, complemented
44#63545—31no—
45#63543D311yescharacteristic, complemented
46C2×C3×D32—41no—
47#6308—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C32×D12—21no—
51C32×C4×D3D411yescharacteristic
52#6290—41no—
53Dic3×C3⋊D3—41no—
54C3×#626—41no—
55C33×C8C2311yescharacteristic
56C32×C3⋊C8D411yescharacteristic
57#5147—91no—
58#4911—91no—
59#4908—91no—
60#3828—61no—
61C3×D3×D4—61no—
62C2×C3×D12—61no—
63D3×D12—21no—
64#3795—62no—
65#3795—61no—
66#3795—61no—
67C32×D8—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#3745C2×D311yescharacteristic
77#3745C2×D311yescharacteristic, complemented
78#3744—61no—
79C3×#213—61no—
80C3×D24—21no—
81C3×D24—31no—
82C3×D24—31no—
83C3×D24—31no—
84C3×D24—31no—
85C3×D24—61no—
86C3×D24—61no—
87C3×D24—61no—
88C3×D24C2×D311yescharacteristic
89C3×D24C2×D311yescharacteristic
90C3×D3×C8—21no—
91C3×D3×C8C2×D311yescharacteristic
92C3×D3×C8C2×D311yescharacteristic
93#3716—62no—
94#3716—61no—
95#3716—61no—
96#3714—21no—
97C32×C16—31no—
98C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
99C3×C3⋊C16—21no—
100C3×C3⋊C16—31no—
101C3×C3⋊C16—31no—
102C3×C3⋊C16—61no—
103C3×C3⋊C16C2×D311yescharacteristic, complemented
104C3×C3⋊C16C2×D311yescharacteristic, complemented
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C32×D3—41no—
108C2×C32×D3D811yescharacteristic
109C3×D32—82no—
110C33×C4C2×D411yescharacteristic
111C3⋊(C3⋊Dic3)—41no—
112C32×Dic3D811yescharacteristic
113C2×C3×D8—91no—
114#912—91no—
115D3×D8—31no—
116#904—91no—
117#904—91no—
118#904—181no—
119C2×D24—31no—
120C3×D16—91no—
121C3×M32—91no—
122#827—91no—
123#827—91no—
124#827—91no—
125#827—181no—
126#826—31no—
127#826—91no—
128#826—91no—
129#812—31no—
130#800—91no—
131#800—91no—
132#800—91no—
133#800—181no—
134D48—31no—
135D48—91no—
136#798—31no—
137C22×C3×D3—121no—
138C22×C3×D3—121no—
139C2×D32—41no—
140C32×D4—61no—
141C2×C32×C4—31no—
142#638—121no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—121no—
146#634—61no—
147#634—62no—
148#634—61no—
149#634—61no—
150#634—121no—
151#634—121no—
152#634—122no—
153C3×#67—121no—
154C3×D12—21no—
155C3×D12—21no—
156C3×D12—41no—
157C3×D12—62no—
158C3×D12—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—61no—
162C3×D12—121no—
163C3×D12—121no—
164C3×C4×D3—21no—
165C3×C4×D3D1211yescharacteristic
166C3×C4×D3#6711yescharacteristic
167#626—41no—
168#625—122no—
169#625—121no—
170#625—121no—
171D3×Dic3—122no—
172D3×Dic3—121no—
173D3×Dic3—121no—
174C32×C8—21no—
175C32×C8—21no—
176C32×C8—21no—
177C32×C8—31no—
178C32×C8—41no—
179C32×C8C22×D311yescharacteristic
180C32×C8C22×D311yescharacteristic
181C32×C8C22×D311yescharacteristic
182C3×C3⋊C8—21no—
183C3×C3⋊C8D1211yescharacteristic
184C3×C3⋊C8#6711yescharacteristic
185#509—271noSylow
186C2×C33C2×D811yescharacteristic
187C3×C3⋊D3—81no—
188C3×C3⋊D3—81no—
189C32×D3—21no—
190C32×D3—81no—
191C2×C3×D4—181no—
192#237—181no—
193D3×D4—61no—
194#235—181no—
195#235—181no—
196#235—182no—
197C2×D12—61no—
198C3×D8—31no—
199C3×D8—31no—
200C3×D8—61no—
201C3×D8—91no—
202C3×D8—91no—
203C3×D8—181no—
204C2×C3×C8—31no—
205C2×C3×C8—31no—
206C2×C3×C8—61no—
207#214—181no—
208#213—61no—
209Dic12—91no—
210Dic12—91no—
211Dic12—91no—
212Dic12—181no—
213Dic12—181no—
214Dic12—181no—
215Dic12—361no—
216D24—31no—
217D24—31no—
218D24—31no—
219D24—31no—
220D24—61no—
221D24—61no—
222D24—61no—
223D24D3211yescharacteristic
224#204—181no—
225#204—181no—
226#204—182no—
227D3×C8D3211yescharacteristic
228C3×C16—31no—
229C3×C16—31no—
230C3×C16—61no—
231C3×C16—91no—
232C3⋊C16—31no—
233C3⋊C16—31no—
234C3⋊C16—61no—
235C3⋊C16D3211yescharacteristic, complemented
236C22×C32—31no—
237C22×C32—121no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—41no—
240C2×C3×D3—21no—
241C2×C3×D3—41no—
242C2×C3×D3—41no—
243C2×C3×D3—81no—
244C2×C3×D3—122no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—241no—
250C2×C3×D3—242no—
251C2×C3×D3—242no—
252C2×C3×D3D2411yescharacteristic
253C2×C3×D3#21311yescharacteristic
254D32—82no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—31no—
259C32×C4—41no—
260C32×C4C2×D1211yescharacteristic
261C32×C4D3×D411yescharacteristic
262C32×C4C2×#6711yescharacteristic
263C3⋊Dic3—121no—
264C3⋊Dic3—122no—
265C3⋊Dic3—121no—
266C3⋊Dic3—121no—
267C3⋊Dic3—241no—
268C3⋊Dic3—241no—
269C3⋊Dic3—242no—
270C3×Dic3—21no—
271C3×Dic3D2411yescharacteristic
272C3×Dic3#21311yescharacteristic
273#135—271no—
274C2×D8—91no—
275SD32—271no—
276SD32—271no—
277D16—91no—
278D16—271no—
279M32—91no—
280C33#50911yescharacteristic, complemented, Sylow
281C23×C3—361no—
282C22×D3—121no—
283C22×D3—121no—
284C3×D4—61no—
285C3×D4—61no—
286C3×D4—121no—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—361no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—61no—
293#67—121no—
294#67—361no—
295#67—361no—
296#67—362no—
297#67—361no—
298C2×Dic3—361no—
299D12—21no—
300D12—62no—
301D12—61no—
302D12—61no—
303D12—61no—
304D12—61no—
305D12—121no—
306D12—121no—
307C4×D3—361no—
308C4×D3—361no—
309C4×D3—362no—
310C4×D3#62611yescharacteristic
311Dic6—182no—
312Dic6—181no—
313Dic6—181no—
314Dic6—181no—
315Dic6—361no—
316Dic6—361no—
317Dic6—362no—
318C3×C8—21no—
319C3×C8—21no—
320C3×C8—21no—
321C3×C8—31no—
322C3×C8—31no—
323C3×C8—41no—
324C3×C8—61no—
325C3×C8C2×D3211yescharacteristic
326C3×C8C2×D3211yescharacteristic
327C3×C8C2×D3211yescharacteristic
328C3⋊C8#62611yescharacteristic
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—41no—
333C2×C32—61no—
334C2×C32—241no—
335C2×C32C2×D2411yescharacteristic
336C2×C32D3×D811yescharacteristic
337C2×C32C2×#21311yescharacteristic
338C3⋊D3—81no—
339C3⋊D3—81no—
340C3×D3—21no—
341C3×D3—21no—
342C3×D3—41no—
343C3×D3—81no—
344C3×D3—81no—
345C3×D3—161no—
346C3×D3—242no—
347C3×D3—241no—
348C3×D3—241no—
349C3×D3—241no—
350C3×D3—241no—
351C3×D3—481no—
352#41—541no—
353C2×D4—181no—
354Q16—271no—
355SD16—541no—
356D8—31no—
357D8—91no—
358D8—91no—
359D8—181no—
360C2×C8—31no—
361C16—31no—
362C16—91no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—61no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—241no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3—722no—
372C2×D3—41no—
373C2×D3—122no—
374C2×D3—121no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—241no—
379C2×D3—242no—
380C2×D3—242no—
381C2×D3#371411yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—41no—
388C3×C4—61no—
389C3×C4D3×D1211yescharacteristic
390C3×C4C2×#62611yescharacteristic
391C3×C4D3×#6711yescharacteristic
392Dic3—362no—
393Dic3—361no—
394Dic3—361no—
395Dic3—361no—
396Dic3—721no—
397Dic3—721no—
398Dic3—722no—
399Dic3#371411yescharacteristic
400C32—21no—
401C32—21no—
402C32—21no—
403C32—41no—
404C32#490811yescharacteristic, complemented
405C32#491111yescharacteristic, complemented
406C32#514711yescharacteristic, complemented
407C23—361no—
408Q8—541no—
409D4—61no—
410D4—181no—
411D4—181no—
412D4—361no—
413D4—1081no—
414C2×C4—31no—
415C2×C4—1081no—
416C8—31no—
417C8D3311yescharacteristic
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—41no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—121no—
425C2×C3—241no—
426C2×C3—241no—
427C2×C3—481no—
428C2×C3—721no—
429C2×C3—721no—
430C2×C3D3×D2411yescharacteristic
431C2×C3C2×#371411yescharacteristic
432C2×C3D3×#21311yescharacteristic
433D3—21no—
434D3—81no—
435D3—242no—
436D3—241no—
437D3—241no—
438D3—241no—
439D3—241no—
440D3—481no—
441C22—31no—
442C22—121no—
443C22—361no—
444C22—361no—
445C22—722no—
446C4—31no—
447C4—1081no—
448C4D3×#62611yescharacteristic
449C3—21no—
450C3—21no—
451C3—21no—
452C3—41no—
453C3#1059023211yescharacteristic, complemented
454C3#1059063311yescharacteristic, complemented
455C3#1059103111yescharacteristic, complemented
456C2—61no—
457C2—241no—
458C2—721no—
459C2—721no—
460C2D3×#371411yescharacteristic
subgroupG/H
centerC2D3×#3714
commutator, Fitting, cosocleC33×C8C23
Frattini, hypercenterC8D33
socleC2×C33C2×D8
hypocenterC33#509
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
1. #11723336C2semidirect
2. C3×D3×D24C2semidirect
3. #11722067C2semidirect
4. #11722011C2semidirect
5. #11722009C2semidirect
6. C3×#63546C2semidirect
7. C3×#63543C2semidirect
13. #88853C22semidirect
15. C3×#3745C22semidirect
20. C3×C3⋊(C3⋊C16)C22semidirect
21. C32×C3⋊C16C22semidirect
28. D3×D24D3semidirect
43. #63546D3semidirect
45. #63543D3semidirect
77. #3745C2×D3semidirect
98. C3⋊(C3⋊C16)C2×D3semidirect
103. C3×C3⋊C16C2×D3semidirect
104. C3×C3⋊C16C2×D3semidirect
235. C3⋊C16D32semidirect
280. C33#509semidirect
404. C32#4908semidirect
405. C32#4911semidirect
406. C32#5147semidirect
453. C3#10590232semidirect
454. C3#10590633semidirect
455. C3#10591031semidirect

Maximal subgroups

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

subgroupindex
#117233362
C3×D3×D242
#117220672
#117220112
#117220092
C3×#635462
C3×#635432
#105910313
#105906333
#105902323