Finite Group Databasebeta

← #13161029

Subgroups — #13161029

Order 1728.

1140 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723981C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592459—31nomaximal
9#10591402—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×#6312—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×#6293—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64149—31no—
22#64149—34no—
23#64149—64no—
24#64097—31no—
25#63919—34no—
26C32×#102—31no—
27#63659D311yescharacteristic, complemented
28#63648—31no—
29C8×C3⋊Dic3—31no—
30#63636—31no—
31C3×#832—34no—
32C3×#832D314yescomplemented
33#63564—34no—
34#63564—34no—
35#63553—34no—
36C23×C33—21no—
37C22×C3⋊(C3⋊D3)—21no—
38C33×D4—21no—
39C33×D4—21no—
40C2×C33×C4C2311yescharacteristic
41#6312—42no—
42#6312—42no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C32×Dic3—21no—
49#6293—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C32×C3⋊C8—21no—
52#4760—94no—
53#4760—91no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D314yes—
56C2×C32×D4C2×D311yescharacteristic
57C2×#638—61no—
58C2×#638—61no—
59C2×#638—64no—
60C2×#638—64no—
61C2×#638—68no—
62C2×#638—68no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×#634—61no—
67C2×#627—64no—
68C2×C32×C8—31no—
69C32×C4⋊C4—31no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72C4×C3⋊Dic3—31no—
73C4×C3⋊Dic3—34no—
74C4×C3⋊Dic3—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×C4⋊Dic3—34no—
77C3×C4⋊Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80#3724—64no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C2×D411yescharacteristic
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C33×C4#4111yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)D812yes—
91C3⋊(C3⋊Dic3)SD1612yes—
92C3×C3⋊Dic3—41no—
93C32×Dic3—41no—
94#940—94no—
95#940—91no—
96#940—184no—
97#888—94no—
98#888—91no—
99C3×#102—34no—
100C3×#102—91no—
101#832—34no—
102#832C3⋊D311yescharacteristic, complemented
103#821—94no—
104#821—91no—
105C8×Dic3—94no—
106C8×Dic3—91no—
107#809—94no—
108#809—91no—
109C23×C32—28no—
110C23×C32—24no—
111C23×C32—21no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—64no—
114C22×C3⋊D3—61no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—24no—
118C32×D4—24no—
119C32×D4—44no—
120C32×D4—44no—
121C2×C32×C4—24no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125#638—122no—
126#638—122no—
127#638—128no—
128#638—128no—
129#638—1216no—
130#638—1216no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—34no—
135C2×C3⋊Dic3—34no—
136C2×C3⋊Dic3—64no—
137C2×C3⋊Dic3—64no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C4⋊(C3⋊D3)—124no—
144#634—122no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—64no—
147#627—128no—
148C32×C8—61no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—24no—
151C3×C3⋊C8—64no—
152#486—271noSylow
153C2×C33—81no—
154C2×C33#12411yescharacteristic
155C2×C33C2×D811yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C3⋊(C3⋊D3)—81no—
158C2×C3×D4—24no—
159C2×C3×D4D3214yes—
160C2×C3×D4C2×C3⋊D311yescharacteristic
161C2×#67—184no—
162C2×#67—184no—
163C2×#67—181no—
164C2×#67—181no—
165C2×#67—188no—
166C2×#67—188no—
167C2×D12—94no—
168C2×D12—91no—
169C2×D12—184no—
170C2×Dic6—181no—
171C2×Dic6—184no—
172C2×C3×C8—34no—
173C2×C3×C8—91no—
174C3×C4⋊C4—34no—
175C3×C4⋊C4—91no—
176C4⋊Dic3—34no—
177C4⋊Dic3C2×C3⋊D311yescharacteristic
178Dic3⋊C4—181no—
179Dic3⋊C4—184no—
180C4×Dic3—94no—
181C4×Dic3—91no—
182C4×Dic3—184no—
183C2×C3⋊C8—34no—
184C2×C3⋊C8C2×C3⋊D311yescharacteristic
185C22×C32—24no—
186C22×C32—48no—
187C22×C32—48no—
188C22×C32—48no—
189C22×C32—44no—
190C22×C32—41no—
191C22×C32—44no—
192C22×C32—41no—
193C22×C32—44no—
194C22×C32—41no—
195C22×C32D3×D414yes—
196C22×C32D3×D411yescharacteristic
197C2×C3⋊D3—128no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—128no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—121no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—121no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—121no—
209C32×C4#23514yes—
210C32×C4#23511yescharacteristic
211C32×C4#23714yes—
212C32×C4#23711yescharacteristic
213C3⋊Dic3—32no—
214C3⋊Dic3—32no—
215C3⋊Dic3—38no—
216C3⋊Dic3—38no—
217C3⋊Dic3—41no—
218C3⋊Dic3—68no—
219C3⋊Dic3—68no—
220C3×Dic3—44no—
221C3×Dic3—124no—
222C4⋊Q8—271no—
223#127—271no—
224#102—31no—
225#102—91no—
226#102—271no—
227#102—271no—
228C4×C8—271no—
229C33#48611yescharacteristic, complemented, Sylow
230C23×C3—28no—
231C23×C3—24no—
232C23×C3—21no—
233C22×D3—188no—
234C22×D3—184no—
235C22×D3—181no—
236C3×D4—24no—
237C3×D4—24no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—44no—
241C3×D4—44no—
242C2×C3×C4—24no—
243C2×C3×C4—64no—
244C2×C3×C4—181no—
245C2×C3×C4C2×D3214yes—
246C2×C3×C4C22×C3⋊D311yescharacteristic
247#67—362no—
248#67—362no—
249#67—368no—
250#67—368no—
251#67—3616no—
252#67—3616no—
253C2×Dic3—21no—
254C2×Dic3—64no—
255C2×Dic3—94no—
256C2×Dic3—94no—
257C2×Dic3—91no—
258C2×Dic3—91no—
259C2×Dic3—184no—
260C2×Dic3—184no—
261D12—184no—
262D12—184no—
263D12—181no—
264D12—181no—
265D12—364no—
266D12—364no—
267Dic6—362no—
268Dic6—368no—
269C3×C8—64no—
270C3×C8—181no—
271C3⋊C8—21no—
272C3⋊C8—64no—
273C2×C32—24no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—88no—
277C2×C32—84no—
278C2×C32—81no—
279C2×C32#88514yes—
280C2×C32#88511yescharacteristic
281C2×C32D3×D814yes—
282C2×C32D3×D811yescharacteristic
283C2×C32D3×SD1614yes—
284C2×C32D3×SD1611yescharacteristic
285C3⋊D3—248no—
286C3⋊D3—244no—
287C3⋊D3—241no—
288C2×Q8—541no—
289C2×D4—271no—
290C2×D4—541no—
291C2×D4—541no—
292C2×D4D3×C3⋊D311yescharacteristic
293C2×C8—31no—
294C2×C8—91no—
295C4⋊C4—31no—
296C4⋊C4—91no—
297C4⋊C4—541no—
298C42—271no—
299C22×C3—24no—
300C22×C3—48no—
301C22×C3—48no—
302C22×C3—48no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3#381114yes—
310C22×C3D4×C3⋊D311yescharacteristic
311C2×D3—368no—
312C2×D3—368no—
313C2×D3—368no—
314C2×D3—361no—
315C2×D3—361no—
316C2×D3—361no—
317C2×D3—364no—
318C2×D3—364no—
319C2×D3—364no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—124no—
323C3×C4—361no—
324C3×C4#380414yes—
325C3×C4#380414yes—
326C3×C4#382811yescharacteristic
327C3×C4#383011yescharacteristic
328Dic3—41no—
329Dic3—92no—
330Dic3—92no—
331Dic3—98no—
332Dic3—98no—
333Dic3—124no—
334Dic3—188no—
335Dic3—188no—
336C32—24no—
337C32#476014yescomplemented
338C32#476011yescharacteristic, complemented
339C23—21no—
340C23—541no—
341Q8—1082no—
342D4—21no—
343D4—21no—
344D4—541no—
345D4—541no—
346D4—1082no—
347D4—1082no—
348C2×C4—61no—
349C2×C4—181no—
350C2×C4—271no—
351C2×C4—271no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—88no—
359C2×C3—84no—
360C2×C3—81no—
361C2×C3#6392614yes—
362C2×C3#6393014yes—
363C2×C3#6395314yes—
364C2×C3#6409411yescharacteristic
365C2×C3D8×C3⋊D311yescharacteristic
366C2×C3SD16×C3⋊D311yescharacteristic
367D3—728no—
368D3—724no—
369D3—721no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—1081no—
374C22—1081no—
375C22—1081no—
376C22#8903911yescharacteristic
377C4—121no—
378C4—272no—
379C4—272no—
380C4—361no—
381C4#8902811yescharacteristic
382C4#8903111yescharacteristic
383C3—24no—
384C3#1059140214yescomplemented
385C3#1059245911yescharacteristic, complemented
386C2—81no—
387C2—2161no—
388C2#1172354711yescharacteristic
389C2#1172355311yescharacteristic
390C2#1172359211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#486
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723536C2semidirect
3. C3×#63659C2semidirect
4. C32×#832C2semidirect
5. #11722036C2semidirect
6. #11722035C2semidirect
7. #11722018C2semidirect
27. #63659D3semidirect
32. C3×#832D3semidirect
102. #832C3⋊D3semidirect
229. C33#486semidirect
337. C32#4760semidirect
338. C32#4760semidirect
384. C3#10591402semidirect
385. C3#10592459semidirect

Maximal subgroups

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

subgroupindex
#117239812
#117235362
C3×#636592
C32×#8322
#117220362
#117220352
#117220182
#105924593
#105914023