Finite Group Databasebeta

← #13160101

Subgroups — #13160101

Order 1728.

1152 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 158 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×#89014C211yesmaximal, characteristic
2C2×#88853C211yesmaximal, characteristic, complemented
3#11723431C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722041C211yesmaximal, characteristic, complemented
8#10592482—31nomaximal
9#10590969—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×D3×C3⋊Dic3—21no—
16#89014—22no—
17#89014—22no—
18C2×C33×C8C2211yescharacteristic
19#88853—22no—
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22Dic3⋊(C3⋊Dic3)—21no—
23#88803—21no—
24#88794—22no—
25C2×#3830—31no—
26C2×#3795—34no—
27C2×#3754—31no—
28C2×#3746—31no—
29C2×#3746—34no—
30C2×#3746—64no—
31#64106—31no—
32#63857—34no—
33C2×#3716—34no—
34C32×#102—31no—
35#63647D311yescharacteristic, complemented
36#63637—31no—
37C3×#821D314yescomplemented
38C3×(C2×D3)⋊C8—34no—
39#63567—34no—
40C22×C3×C3⋊D3D411yescharacteristic
41C22×C32×D3—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3⋊(C3⋊Dic3)—21no—
45#6308—21no—
46#6308—21no—
47#6308—21no—
48#6308D412yes—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C3×C4×C3⋊D3—21no—
51C3×C4×C3⋊D3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12D412yes—
55#6290—24no—
56D3×C3⋊Dic3—24no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#4783—94no—
60#4733—91no—
61C2×C32×D4—31no—
62C2×#638—61no—
63C22×C3⋊Dic3—61no—
64#3830—62no—
65#3830—62no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×#634—31no—
68C2×#634—34no—
69C2×#634—31no—
70C2×#634—34no—
71C2×#634—64no—
72C2×#634—64no—
73C2×C3×D12C2×D314yes—
74C2×C3×C4×D3—34no—
75C2×#625—64no—
76C2×D3×Dic3—64no—
77#3795—68no—
78#3795—68no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82C32×C4⋊C4—31no—
83#3754—62no—
84#3752—61no—
85C4⋊(C3⋊Dic3)—61no—
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87#3746—62no—
88#3746—68no—
89#3746—616no—
90C3×C4⋊Dic3C2×D314yes—
91C2×C3×C3⋊C8—34no—
92Dic3⋊Dic3—64no—
93(C2×D3)⋊Dic3—64no—
94#3716—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C32×D3—22no—
100C2×C32×D3—41no—
101C2×C32×D3—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—21no—
107C32×Dic3—41no—
108C2×#237—94no—
109C2×#235—91no—
110C2×#214—94no—
111#924—91no—
112C2×Dic12—94no—
113C2×Dic12—91no—
114C2×Dic12—184no—
115C2×#204—91no—
116#897—94no—
117C3×#102—34no—
118C3×C22⋊C8—91no—
119#821C3⋊D311yescharacteristic, complemented
120(C2×D3)⋊C8—34no—
121#816—91no—
122#810—94no—
123C23×C32—61no—
124C22×C3⋊D3D1211yescharacteristic
125C22×C3×D3—24no—
126C22×C3×D3—34no—
127C32×D4—32no—
128C32×D4—61no—
129C2×C32×C4—24no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D314yes—
132C2×C32×C4C22×D311yescharacteristic
133#638—64no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—124no—
140C2×C3⋊Dic3—124no—
141C2×C3⋊Dic3D1211yescharacteristic
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144#634—32no—
145#634—38no—
146#634—68no—
147#634—61no—
148#634—68no—
149#634—64no—
150#634—61no—
151#634—61no—
152#634—64no—
153#634—64no—
154#634—124no—
155#634—124no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—34no—
158C3×D12—24no—
159C3×D12D1218yes—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162#625—616no—
163D3×Dic3—616no—
164C32×C8—28no—
165C32×C8—24no—
166C32×C8—21no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—64no—
169#452—271noSylow
170C2×C33#12011yescharacteristic
171C2×C33#13511yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—41no—
174C32×D3—42no—
175C2×C3×D4—34no—
176C22×C3×C4—91no—
177C2×#67—181no—
178C2×#67—184no—
179C22×Dic3—184no—
180#237—188no—
181#237—188no—
182#235—182no—
183#235—182no—
184C2×D12C2×C3⋊D311yescharacteristic
185C2×C4×D3—34no—
186C2×C4×D3—181no—
187C2×Dic6—94no—
188C2×Dic6—91no—
189C2×Dic6—94no—
190C2×Dic6—91no—
191C2×Dic6—184no—
192C2×Dic6—184no—
193C2×C3×C8—24no—
194C2×C3×C8—91no—
195C2×C3×C8D3214yes—
196C2×C3×C8C2×C3⋊D311yescharacteristic
197C3×C4⋊C4—34no—
198#217—181no—
199#214—188no—
200(C2×D3)⋊C4—184no—
201C4⋊Dic3—184no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203Dic3⋊C4—181no—
204C2×C3⋊C8—34no—
205Dic12—188no—
206Dic12—182no—
207Dic12—1816no—
208#204—182no—
209C22×C32—24no—
210C22×C32—62no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32C2×D1211yescharacteristic
214C22×C32D3×D414yes—
215C2×C3⋊D3—21no—
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3×D3—28no—
219C2×C3×D3—44no—
220C2×C3×D3—44no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C2×C3×D3—64no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—121no—
227C32×C4C2×D1214yes—
228C32×C4D3×D414yes—
229C32×C4D3×D411yescharacteristic
230C32×C4D3×D411yescharacteristic
231C3⋊Dic3—21no—
232C3⋊Dic3—62no—
233C3⋊Dic3—68no—
234C3⋊Dic3—128no—
235C3⋊Dic3—128no—
236C3⋊Dic3—124no—
237C3⋊Dic3—121no—
238C3×Dic3—44no—
239C3×Dic3—64no—
240C2×#41—271no—
241C2×Q16—271no—
242C2×SD16—271no—
243C22⋊Q8—271no—
244#103—271no—
245#102—31no—
246C22⋊C8—91no—
247C33#45211yescharacteristic, complemented, Sylow
248C23×C3—64no—
249C23×C3—91no—
250C22×D3—21no—
251C22×D3—34no—
252C3×D4—38no—
253C3×D4—64no—
254C2×C3×C4—24no—
255C2×C3×C4—64no—
256C2×C3×C4—91no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C2×D3214yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—184no—
262#67—1816no—
263C2×Dic3—21no—
264C2×Dic3—34no—
265C2×Dic3—1816no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—184no—
269C2×Dic3—181no—
270C2×Dic3—364no—
271C2×Dic3—364no—
272D12—21no—
273D12C4⋊(C3⋊D3)12yes—
274C4×D3—64no—
275C4×D3—64no—
276C4×D3—184no—
277Dic6—98no—
278Dic6—92no—
279Dic6—188no—
280Dic6—184no—
281Dic6—181no—
282Dic6—188no—
283Dic6—184no—
284Dic6—184no—
285Dic6—181no—
286Dic6—181no—
287Dic6—364no—
288Dic6—364no—
289C3×C8—28no—
290C3×C8—24no—
291C3×C8—21no—
292C3×C8—181no—
293C3⋊C8—64no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32—122no—
298C2×C32#88214yes—
299C2×C32#88211yescharacteristic
300C2×C32#90411yescharacteristic
301C2×C32#90911yescharacteristic
302C2×C32#91214yes—
303C2×C32#91514yes—
304C3⋊D3—41no—
305C3×D3—48no—
306C3×D3—124no—
307#41—542no—
308#41—542no—
309C2×Q8—271no—
310C2×Q8—271no—
311C2×D4—31no—
312C2×D4—541no—
313C22×C4—91no—
314C22×C4—541no—
315Q16—542no—
316SD16—542no—
317C2×C8—91no—
318C2×C8D3×C3⋊D311yescharacteristic
319C4⋊C4—31no—
320C4⋊C4—541no—
321C22⋊C4—541no—
322C22×C3—24no—
323C22×C3—68no—
324C22×C3—124no—
325C22×C3—124no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3—181no—
329C22×C3D3×D1214yes—
330C22×C3D4×C3⋊D311yescharacteristic
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—64no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—124no—
340C3×C4—181no—
341C3×C4D3×D1214yes—
342C3×C4#381114yes—
343C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
344C3×C4D4×C3⋊D311yescharacteristic
345Dic3—41no—
346Dic3—64no—
347Dic3—182no—
348Dic3—188no—
349Dic3—368no—
350Dic3—368no—
351Dic3—361no—
352Dic3—364no—
353C32—24no—
354C32#473311yescharacteristic, complemented
355C32#478314yescomplemented
356C23—61no—
357C23—91no—
358Q8—272no—
359Q8—541no—
360Q8—541no—
361Q8—541no—
362D4—32no—
363D4—61no—
364D4—544no—
365C2×C4—61no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×C3⋊D311yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—128no—
379C2×C3—361no—
380C2×C3#6380014yes—
381C2×C3#6380714yes—
382C2×C3#6392314yes—
383C2×C3#6409111yescharacteristic
384C2×C3#6412111yescharacteristic
385C2×C3#6412411yescharacteristic
386D3—42no—
387D3—124no—
388C22—62no—
389C22—121no—
390C22—121no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22C3⋊D3×D1211yescharacteristic
395C4—121no—
396C4—181no—
397C4—542no—
398C4—1081no—
399C4D3×C4⋊(C3⋊D3)11yescharacteristic
400C4#8903911yescharacteristic
401C3—24no—
402C3#1059096914yescomplemented
403C3#1059248211yescharacteristic, complemented
404C2—122no—
405C2—361no—
406C2#1172333911yescharacteristic
407C2#1172334011yescharacteristic
408C2#1172354211yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89014C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#452
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
3. #11723431C2semidirect
4. C2×#88794C2semidirect
5. C3×#63647C2semidirect
6. C32×#821C2semidirect
7. #11722041C2semidirect
35. #63647D3semidirect
37. C3×#821D3semidirect
119. #821C3⋊D3semidirect
247. C33#452semidirect
354. C32#4733semidirect
355. C32#4783semidirect
402. C3#10590969semidirect
403. C3#10592482semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
C2×#888532
#117234312
C2×#887942
C3×#636472
C32×#8212
#117220412
#105924823
#105909693