Finite Group Databasebeta

← #13138922

Subgroups — #13138922

Order 1728.

519 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C4×#6200C211yesmaximal, characteristic
2C2×#88434C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720553C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590721—31nomaximal
9#10590718—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6200C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88451C411yescharacteristic
19#88450C411yescharacteristic
20#88448C411yescharacteristic
21#88434—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23C4×#626—31no—
24C4×#625—31no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—31no—
27C2×D3×C3⋊C8—31no—
28C2×D3×C3⋊C8—31no—
29#63647D311yescharacteristic, complemented
30C8×C3⋊Dic3D311yescharacteristic, complemented
31C3×(C2×D3)⋊C8—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×(C2×D3)⋊C8—61no—
35C3×C8×Dic3—31no—
36#63572—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×#272C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
46C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#6200C814yescomplemented
50C8×H(3,𝔽3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#67⋊C8—91no—
55#67⋊C8—91no—
56C8×D12—91no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C2×#626—31no—
63C2×#625—31no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76#3724—31no—
77Dic3⋊Dic3—31no—
78#3722—31no—
79(C2×D3)⋊Dic3—31no—
80(C2×D3)⋊Dic3—31no—
81(C2×D3)⋊Dic3—31no—
82#3710—62no—
83D3×C3⋊C8—62no—
84D3×C3⋊C8—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C22×C411yescharacteristic
87C2×#272—21no—
88C2×#272—21no—
89C2×#272C2×C812yescomplemented
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3C2×C812yescomplemented
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—21no—
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3C2×C812yescomplemented
99C4×#67—91no—
100C4×#67—91no—
101C22×C3⋊C8—91no—
102C22×C3⋊C8—91no—
103C2×D3×C8—91no—
104C2×D3×C8—91no—
105C2×D3×C8—91no—
106C2×D3×C8—91no—
107C4×D12—91no—
108C3×C22⋊C8—91no—
109C3×C22⋊C8—91no—
110C3×C4×C8—91no—
111(C2×D3)⋊C8—31no—
112(C2×D3)⋊C8—31no—
113(C2×D3)⋊C8—31no—
114(C2×D3)⋊C8—61no—
115(C2×D3)⋊C8—91no—
116Dic3⋊C8—91no—
117Dic3⋊C8—91no—
118C8×Dic3—31no—
119C8×Dic3—91no—
120C4⋊(C3⋊C8)—91no—
121C22×C3⋊D3C4×D311yescharacteristic
122C22×C3×D3—31no—
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C22×C3×D3—61no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3C4×D311yescharacteristic
130C2×C3⋊Dic3C4×D311yescharacteristic
131C2×C3⋊Dic3C4×D311yescharacteristic
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146C3×C4×D3—121no—
147C3×C4×D3—121no—
148#626—34no—
149#625—34no—
150C32×C8—21no—
151C32×C8—21no—
152C32×C8—41no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)D1212yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—121no—
160C8×D4—271noSylow
161C2×H(3,𝔽3)C4×D411yescharacteristic
162C2×H(3,𝔽3)C22×C811yescharacteristic
163C2×H(3,𝔽3)#13111yescharacteristic
164#272—22no—
165(C3⋊D3)⋊C3—22no—
166C22×C3×C4—91no—
167C22×C3×C4—91no—
168C2×#67—91no—
169C2×#67—91no—
170C2×D12—91no—
171C2×C4×D3—31no—
172C2×C4×D3—31no—
173C2×C4×D3—31no—
174C2×C4×D3—61no—
175C2×C4×D3—91no—
176C2×C3×C8—31no—
177C2×C3×C8—31no—
178C2×C3×C8—61no—
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C2×C3×C8D3211yescharacteristic
183C3×C42—91no—
184#217—91no—
185#217—91no—
186(C2×D3)⋊C4—91no—
187(C2×D3)⋊C4—91no—
188(C2×D3)⋊C4—91no—
189(C2×D3)⋊C4—91no—
190C4⋊Dic3—91no—
191Dic3⋊C4—91no—
192Dic3⋊C4—91no—
193C4×Dic3—31no—
194C4×Dic3—91no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—182no—
203C2×C3⋊C8—182no—
204D3×C8—182no—
205D3×C8—182no—
206D3×C8—182no—
207D3×C8—182no—
208C22×C32—21no—
209C22×C32C2×C4×D311yescharacteristic
210C22×C32C2×C4×D311yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3D3×C812yescomplemented
214C2×C3×D3—32no—
215C2×C3×D3—32no—
216C2×C3×D3—32no—
217C2×C3×D3—62no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4C2×D1211yescharacteristic
229C32×C4#23511yescharacteristic
230C32×C4D3×D411yescharacteristic
231C32×C4#23711yescharacteristic
232C3⋊Dic3—21no—
233C3⋊Dic3—21no—
234C3⋊Dic3D3×C812yescomplemented
235C3×Dic3—32no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—121no—
241C22×C8—271no—
242C22×C8—271no—
243C4×D4—271no—
244C4⋊C8—271no—
245C22⋊C8—91no—
246C22⋊C8—91no—
247C4×C8—91no—
248H(3,𝔽3)C8×D411yescharacteristic, complemented, Sylow
249C23×C3—91no—
250C23×C3—91no—
251C22×D3—31no—
252C22×D3—31no—
253C22×D3—31no—
254C22×D3—61no—
255C22×D3—91no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4C2×D3211yescharacteristic
268#67—94no—
269#67—94no—
270C2×Dic3—31no—
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—61no—
276C2×Dic3—91no—
277C2×Dic3—91no—
278C2×Dic3—91no—
279D12—94no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—121no—
287C4×D3—121no—
288C4×D3—181no—
289C4×D3—181no—
290C3×C8—21no—
291C3×C8—61no—
292C3×C8—61no—
293C3×C8—92no—
294C3×C8—121no—
295C3×C8—181no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—61no—
301C3⋊C8—92no—
302C3⋊C8—121no—
303C3⋊C8—181no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32C4×D1211yescharacteristic
308C2×C32#67⋊C411yescharacteristic
309C2×C32C2×D3×C811yescharacteristic
310C2×C32C2×D3×C811yescharacteristic
311C2×C32#90111yescharacteristic
312C2×C32#90711yescharacteristic
313C3⋊D3—22no—
314C3×D3—62no—
315C3×D3—62no—
316C3×D3—62no—
317C3×D3—122no—
318C2×D4—271no—
319C22×C4—91no—
320C22×C4—91no—
321C2×C8—91no—
322C2×C8—91no—
323C2×C8—91no—
324C2×C8—542no—
325C2×C8—542no—
326C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
327C4⋊C4—271no—
328C22⋊C4—271no—
329C22⋊C4—271no—
330C42—91no—
331C22×C3—31no—
332C22×C3—31no—
333C22×C3—61no—
334C22×C3—92no—
335C22×C3—92no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3—181no—
339C22×C3—181no—
340C22×C3C4×D3211yescharacteristic
341C2×D3—32no—
342C2×D3—32no—
343C2×D3—32no—
344C2×D3—62no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—92no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—181no—
355C2×D3—181no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—31no—
359C3×C4—31no—
360C3×C4—61no—
361C3×C4—61no—
362C3×C4—92no—
363C3×C4—181no—
364C3×C4—181no—
365C3×C4—181no—
366C3×C4D3×D1211yescharacteristic
367C3×C4#380411yescharacteristic
368Dic3—32no—
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—92no—
374Dic3—121no—
375Dic3—181no—
376Dic3—181no—
377C32—21no—
378C32C8×D1211yescharacteristic, complemented
379C32#67⋊C811yescharacteristic, complemented
380C23—91no—
381C23—91no—
382D4—274no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C8—21no—
393C8—92no—
394C8—181no—
395C8—181no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—61no—
403C2×C3—61no—
404C2×C3—61no—
405C2×C3—182no—
406C2×C3—182no—
407C2×C3D32×C811yescharacteristic
408C2×C3#6380411yescharacteristic
409C2×C3#6388011yescharacteristic
410D3—62no—
411D3—62no—
412D3—62no—
413D3—122no—
414D3—182no—
415C22—92no—
416C22—92no—
417C22—181no—
418C22—181no—
419C22—181no—
420C22—181no—
421C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
422C4—92no—
423C4—181no—
424C4—181no—
425C4—181no—
426C4#8865411yescharacteristic
427C4#8867111yescharacteristic
428C3—31no—
429C3—31no—
430C3—61no—
431C3#1059072111yescharacteristic
432C2—182no—
433C2—182no—
434C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
435C2#1172084111yescharacteristic
436C2#1172112311yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC2×#6200C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C8×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88434C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720553C2semidirect
5. #63647⋊C3C2semidirect
6. C8×(C3⋊Dic3)⋊C3C2semidirect
7. #11720311C2semidirect
29. #63647D3semidirect
30. C8×C3⋊Dic3D3semidirect
49. #6200C8semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
89. C2×#272C2×C8semidirect
92. C2×(C3⋊D3)⋊C3C2×C8semidirect
98. (C3⋊Dic3)⋊C3C2×C8semidirect
154. C3⋊(C3⋊C8)D12semidirect
213. C2×C3⋊D3D3×C8semidirect
234. C3⋊Dic3D3×C8semidirect
248. H(3,𝔽3)C8×D4semidirect
378. C32C8×D12semidirect
379. C32#67⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#62002
C2×#884342
C2×((C3⋊D3)⋊C3)⋊C82
#117205532
#63647⋊C32
C8×(C3⋊Dic3)⋊C32
#117203112
#105907213
#105907183