Finite Group Databasebeta

← #13160062

Subgroups — #13160062

Order 1728.

1128 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×D3×#634C211yesmaximal, characteristic
2C2×#88853C211yesmaximal, characteristic, complemented
3#11723426C211yesmaximal, characteristic
4C2×#88800C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic
7#11722042C211yesmaximal, characteristic, complemented
8#10592438—31nomaximal
9#10590952—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6293—21no—
15C2×D3×C3⋊Dic3—21no—
16D3×#634—22no—
17D3×#634—22no—
18C2×C33×C8C2211yescharacteristic
19#88853—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88807—21no—
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88800—22no—
25C22×#634—31no—
26C2×D3×Dic6—34no—
27#64133—31no—
28C2×#3746—31no—
29C2×#3746—31no—
30C2×#3746—34no—
31C2×#3746—64no—
32#63857—34no—
33C2×#3719—34no—
34C32×C22⋊C8—31no—
35#63643—31no—
36#63643D311yescharacteristic, complemented
37C3×(C2×D3)⋊C8D314yes—
38C3×#816—34no—
39#63567—34no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46#6308—21no—
47#6308D412yes—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634D412yes—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55#6293—24no—
56D3×C3⋊Dic3—24no—
57C33×C8—21no—
58C32×C3⋊C8—21no—
59#4809—91no—
60#4739—94no—
61C22×C32×C4—31no—
62C22×C3⋊Dic3—61no—
63C2×#634—31no—
64C2×#634—34no—
65C2×#634—31no—
66C2×#634—34no—
67C2×#634—62no—
68C2×#634—61no—
69C2×#634—62no—
70C2×#634—64no—
71C2×#634—64no—
72C2×#634C2×D311yescharacteristic
73C2×C3×C4×D3C2×D314yes—
74C2×C3×Dic6—34no—
75C2×#627—64no—
76C2×D3×Dic3—64no—
77D3×Dic6—68no—
78D3×Dic6—68no—
79C2×C32×C8—24no—
80C2×C32×C8—31no—
81C2×C32×C8C2×D314yes—
82C2×C32×C8C2×D311yescharacteristic
83#3757—61no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86#3746—62no—
87#3746—68no—
88#3746—616no—
89#3746—62no—
90C3×C4⋊Dic3—34no—
91C2×C3×C3⋊C8C2×D314yes—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3719—68no—
95C22×C33C2×D411yescharacteristic
96C2×C32×D3—21no—
97C2×C32×D3—21no—
98C2×C32×D3Q1612yescomplemented
99C33×C4C2×D411yescharacteristic
100C33×C4C2×D411yescharacteristic
101C3⋊(C3⋊Dic3)—22no—
102C3⋊(C3⋊Dic3)—41no—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—41no—
105C32×Dic3—21no—
106C2×D3×Q8—91no—
107C22×Dic6—94no—
108C2×#216—91no—
109#924—94no—
110C2×Dic12—94no—
111C2×Dic12—94no—
112C2×Dic12—91no—
113C2×Dic12—184no—
114#897—91no—
115C3×#103—91no—
116C3×C22⋊C8—34no—
117(C2×D3)⋊C8C3⋊D311yescharacteristic
118#816—34no—
119#816—94no—
120#810—91no—
121C23×C32—31no—
122C22×C3×D3D1214yes—
123C2×C32×C4—24no—
124C2×C32×C4—31no—
125C2×C32×C4—61no—
126C2×C32×C4—61no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—64no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—124no—
137C2×C3⋊Dic3—124no—
138#634—21no—
139#634—32no—
140#634—38no—
141#634—68no—
142#634—64no—
143#634—61no—
144#634—68no—
145#634—61no—
146#634—64no—
147#634—61no—
148#634—64no—
149#634—64no—
150#634—124no—
151#634—124no—
152#634D1212yes—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—64no—
155C2×C3×Dic3D1214yes—
156C3×C4×D3—24no—
157C3×C4×D3—24no—
158C3×Dic6—38no—
159C3×Dic6—64no—
160#627—616no—
161D3×Dic3—616no—
162C32×C8—28no—
163C32×C8—24no—
164C32×C8—21no—
165C32×C8—61no—
166C3×C3⋊C8—24no—
167#451—271noSylow
168C2×C33#12011yescharacteristic
169C2×C33C2×Q1611yescharacteristic
170C2×C33#13711yescharacteristic
171C32×D3—22no—
172C2×C3×Q8—91no—
173C22×C3×C4—34no—
174C22×Dic3—184no—
175D3×Q8—182no—
176D3×Q8—182no—
177C2×C4×D3—181no—
178C2×C4×D3C2×C3⋊D311yescharacteristic
179C2×Dic6—34no—
180C2×Dic6—94no—
181C2×Dic6—91no—
182C2×Dic6—94no—
183C2×Dic6—91no—
184C2×Dic6—184no—
185C2×Dic6—184no—
186C2×Dic6—188no—
187C2×Dic6—188no—
188C2×Dic6—184no—
189C2×Dic6—181no—
190C2×C3×C8—24no—
191C2×C3×C8—34no—
192C2×C3×C8D3214yes—
193C2×C3×C8C2×C3⋊D311yescharacteristic
194C3×C4⋊C4—91no—
195#217—184no—
196#216—182no—
197(C2×D3)⋊C4—181no—
198C4⋊Dic3—34no—
199C4⋊Dic3—181no—
200Dic3⋊C4—184no—
201C2×C3⋊C8C2×C3⋊D311yescharacteristic
202Dic12—188no—
203Dic12—182no—
204Dic12—1816no—
205Dic12—188no—
206C22×C32—24no—
207C22×C32—32no—
208C22×C32—61no—
209C22×C32—61no—
210C22×C32C2×D1214yes—
211C22×C32D3×D411yescharacteristic
212C2×C3×D3—24no—
213C2×C3×D3—24no—
214C2×C3×D3Dic1218yescomplemented
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—61no—
218C32×C4C2×D1211yescharacteristic
219C32×C4D3×D414yes—
220C32×C4D3×D414yes—
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—41no—
224C3⋊Dic3—62no—
225C3⋊Dic3—68no—
226C3⋊Dic3—128no—
227C3⋊Dic3—128no—
228C3⋊Dic3—124no—
229C3⋊Dic3—121no—
230C3×Dic3—24no—
231C3×Dic3—68no—
232C3×Dic3—124no—
233C22×Q8—271no—
234C2×Q16—271no—
235C2×Q16—271no—
236C22⋊Q8—271no—
237#103—91no—
238#103—271no—
239C22⋊C8—31no—
240C33#45111yescharacteristic, complemented, Sylow
241C23×C3—34no—
242C22×D3C4⋊(C3⋊D3)11yescharacteristic
243C3×Q8—92no—
244C3×Q8—181no—
245C2×C3×C4—24no—
246C2×C3×C4—34no—
247C2×C3×C4—64no—
248C2×C3×C4—64no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253C2×Dic3—64no—
254C2×Dic3—64no—
255C2×Dic3—1816no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C2×Dic3—184no—
259C2×Dic3—181no—
260C2×Dic3—364no—
261C2×Dic3—364no—
262C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
263C4×D3—21no—
264C4×D3—21no—
265C4×D3—184no—
266Dic6—38no—
267Dic6—64no—
268Dic6—98no—
269Dic6—92no—
270Dic6—188no—
271Dic6—184no—
272Dic6—1816no—
273Dic6—184no—
274Dic6—181no—
275Dic6—188no—
276Dic6—184no—
277Dic6—184no—
278Dic6—181no—
279Dic6—181no—
280Dic6—364no—
281Dic6—364no—
282C3×C8—28no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—64no—
286C3⋊C8—21no—
287C2×C32—24no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—62no—
291C2×C32#88214yes—
292C2×C32#88211yescharacteristic
293C2×C32C2×Dic1214yes—
294C2×C32#90914yes—
295C2×C32#91511yescharacteristic
296C2×C32D3×Q1611yescharacteristic
297C3×D3—28no—
298C2×Q8—91no—
299C2×Q8—271no—
300C2×Q8—271no—
301C2×Q8—541no—
302C2×Q8—542no—
303C2×Q8—542no—
304C22×C4—31no—
305C22×C4—541no—
306Q16—542no—
307Q16—542no—
308C2×C8—31no—
309C2×C8D3×C3⋊D311yescharacteristic
310C4⋊C4—91no—
311C4⋊C4—541no—
312C22⋊C4—541no—
313C22×C3—24no—
314C22×C3—38no—
315C22×C3—64no—
316C22×C3—64no—
317C22×C3D3×D1214yes—
318C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
319C2×D3—21no—
320C2×D3—21no—
321C2×D3#374612yescomplemented
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—64no—
325C3×C4—182no—
326C3×C4—361no—
327C3×C4D3×D1214yes—
328C3×C4#381114yes—
329C3×C4D4×C3⋊D311yescharacteristic
330C3×C4D4×C3⋊D311yescharacteristic
331Dic3—21no—
332Dic3—68no—
333Dic3—124no—
334Dic3—182no—
335Dic3—188no—
336Dic3—368no—
337Dic3—368no—
338Dic3—361no—
339Dic3—364no—
340C32—24no—
341C32#473914yescomplemented
342C32#480911yescharacteristic, complemented
343C23—31no—
344Q8—92no—
345Q8—181no—
346Q8—272no—
347Q8—544no—
348Q8—541no—
349Q8—541no—
350Q8—541no—
351C2×C4—31no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—544no—
357C2×C4—541no—
358C2×C4—541no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—21no—
361C8—61no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—68no—
366C2×C3D3×Dic1214yes—
367C2×C3#6380014yes—
368C2×C3#6392314yes—
369C2×C3#6409111yescharacteristic
370C2×C3C2×#374611yescharacteristic
371C2×C3#6411811yescharacteristic
372D3—22no—
373C22—32no—
374C22—61no—
375C22—61no—
376C22D3×C4⋊(C3⋊D3)11yescharacteristic
377C4—61no—
378C4—182no—
379C4—361no—
380C4—542no—
381C4—1081no—
382C4C3⋊D3×D1211yescharacteristic
383C4#8903911yescharacteristic
384C3—24no—
385C3#1059095214yescomplemented
386C3#1059243811yescharacteristic, complemented
387C2—62no—
388C2D3×#374611yescharacteristic
389C2#1172332611yescharacteristic
390C2#1172354211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#451
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88853C2semidirect
4. C2×#88800C2semidirect
5. C3×#63643C2semidirect
7. #11722042C2semidirect
36. #63643D3semidirect
98. C2×C32×D3Q16semidirect
214. C2×C3×D3Dic12semidirect
240. C33#451semidirect
321. C2×D3#3746semidirect
341. C32#4739semidirect
342. C32#4809semidirect
385. C3#10590952semidirect
386. C3#10592438semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#6342
C2×#888532
#117234262
C2×#888002
C3×#636432
C32×(C2×D3)⋊C82
#117220422
#105924383
#105909523