Finite Group Databasebeta

← #13138795

Subgroups — #13138795

Order 1728.

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

normal subgroups only

This group has 507 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
1C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C4×#6198C211yesmaximal, characteristic
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720547C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720302C211yesmaximal, characteristic, complemented
8#10590906—31nomaximal
9#10590720—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
12C2×#6198C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4×#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C8×(C3⋊D3)⋊C3—22no—
19#88450C411yescharacteristic
20#88450C411yescharacteristic
21#88449C411yescharacteristic
22((C3⋊D3)⋊C3)⋊C8—22no—
23C2×C8×C3⋊D3D311yescharacteristic, complemented
24C2×C3×D3×C8—31no—
25C4×#626—31no—
26C4×#626—31no—
27C2×#3710—31no—
28C2×D3×C3⋊C8—31no—
29#63647D311yescharacteristic, complemented
30C3×(C2×D3)⋊C8—31no—
31C3×Dic3⋊C8—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—61no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36Dic3×C3⋊C8—31no—
37Dic3×C3⋊C8—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2×C411yescharacteristic
42C2×#1084C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—21no—
49#6198C814yescomplemented
50C8×H(3,𝔽3)—21no—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54C8×#67—91no—
55#4834—91no—
56#67⋊C8—91no—
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—31no—
61C2×#626—31no—
62C2×#626—31no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C8×C3⋊D3—22no—
69C3×C4×Dic3—31no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77C3×D3×C8—62no—
78Dic3⋊Dic3—31no—
79Dic3⋊Dic3—31no—
80#3722—31no—
81#3722—31no—
82(C2×D3)⋊Dic3—31no—
83(C2×D3)⋊Dic3—31no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)C22×C411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—21no—
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3C2×C812yescomplemented
92C2×(C3⋊D3)⋊C3C2×C812yescomplemented
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—21no—
96#1084C2×C812yescomplemented
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99C22×C3×C8—91no—
100C4×#67—91no—
101C4×#67—91no—
102C22×C3⋊C8—91no—
103C2×D3×C8—31no—
104C2×D3×C8—91no—
105C2×D3×C8—91no—
106C2×D3×C8—91no—
107C4×D12—91no—
108C3×C4⋊C8—91no—
109C3×C22⋊C8—91no—
110#830—91no—
111(C2×D3)⋊C8—31no—
112(C2×D3)⋊C8—91no—
113(C2×D3)⋊C8—91no—
114(C2×D3)⋊C8—91no—
115Dic3⋊C8—31no—
116Dic3⋊C8—31no—
117Dic3⋊C8—61no—
118C8×Dic3—91no—
119C8×Dic3—91no—
120C4×C3⋊C8—91no—
121C22×C3⋊D3C4×D311yescharacteristic
122C22×C3⋊D3C4×D311yescharacteristic
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3C4×D311yescharacteristic
129C2×C3⋊Dic3C4×D311yescharacteristic
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
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—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—61no—
146#626—34no—
147#626—34no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)#6712yescomplemented
153C3×C3⋊C8—32no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—121no—
158C8×D4—271noSylow
159C2×H(3,𝔽3)C4×D411yescharacteristic
160C2×H(3,𝔽3)C22×C811yescharacteristic
161C2×H(3,𝔽3)#13111yescharacteristic
162(C3⋊D3)⋊C3—22no—
163(C3⋊D3)⋊C3—22no—
164C22×C3×C4—91no—
165C22×C3×C4—91no—
166C2×#67—91no—
167C2×#67—91no—
168C2×D12—91no—
169C2×C4×D3—31no—
170C2×C4×D3—31no—
171C2×C4×D3—91no—
172C2×C4×D3—91no—
173C2×C3×C8—31no—
174C2×C3×C8—31no—
175C2×C3×C8—61no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C2×C3×C8—182no—
180C2×C3×C8D3211yescharacteristic
181C3×C42—91no—
182#217—91no—
183#217—91no—
184(C2×D3)⋊C4—91no—
185(C2×D3)⋊C4—91no—
186(C2×D3)⋊C4—91no—
187(C2×D3)⋊C4—91no—
188C4⋊Dic3—91no—
189Dic3⋊C4—91no—
190Dic3⋊C4—91no—
191C4×Dic3—31no—
192C4×Dic3—31no—
193C4×Dic3—61no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—91no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—182no—
202D3×C8—62no—
203D3×C8—182no—
204D3×C8—182no—
205D3×C8—182no—
206C22×C32—21no—
207C22×C32C2×C4×D311yescharacteristic
208C22×C32C2×C4×D311yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3D3×C812yescomplemented
214C2×C3⋊D3D3×C812yescomplemented
215C2×C3×D3—32no—
216C2×C3×D3—32no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4#23511yescharacteristic
224C32×C4D3×D411yescharacteristic
225C32×C4#23711yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3—21no—
228C3⋊Dic3—21no—
229C3×Dic3—32no—
230C3×Dic3—32no—
231C3×Dic3—62no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—121no—
237C22×C8—91no—
238C22×C8—271no—
239C4×D4—271no—
240C4⋊C8—91no—
241C22⋊C8—91no—
242C22⋊C8—271no—
243C4×C8—271no—
244H(3,𝔽3)C8×D411yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C23×C3—91no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—91no—
250C22×D3—91no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263#67—94no—
264#67—94no—
265C2×Dic3—31no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—31no—
270C2×Dic3—31no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—91no—
274C2×Dic3—91no—
275D12—94no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—181no—
281C4×D3—181no—
282C4×D3—181no—
283C4×D3—181no—
284C3×C8—21no—
285C3×C8—61no—
286C3×C8—61no—
287C3×C8—92no—
288C3×C8—121no—
289C3×C8—181no—
290C3×C8—181no—
291C3⋊C8—32no—
292C3⋊C8—61no—
293C3⋊C8—61no—
294C3⋊C8—61no—
295C3⋊C8—92no—
296C3⋊C8—121no—
297C3⋊C8—181no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32#67⋊C411yescharacteristic
302C2×C32C2×D3×C811yescharacteristic
303C2×C32C2×D3×C811yescharacteristic
304C2×C32#90111yescharacteristic
305C2×C32#90711yescharacteristic
306C2×C32C4×#6711yescharacteristic
307C3⋊D3—22no—
308C3⋊D3—22no—
309C3×D3—62no—
310C3×D3—62no—
311C2×D4—271no—
312C22×C4—91no—
313C22×C4—91no—
314C2×C8—91no—
315C2×C8—91no—
316C2×C8—91no—
317C2×C8—182no—
318C2×C8—542no—
319C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C4⋊C4—271no—
321C22⋊C4—271no—
322C22⋊C4—271no—
323C42—91no—
324C22×C3—31no—
325C22×C3—31no—
326C22×C3—61no—
327C22×C3—92no—
328C22×C3—92no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3C4×D3211yescharacteristic
334C2×D3—32no—
335C2×D3—32no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—92no—
341C2×D3—92no—
342C2×D3—181no—
343C2×D3—181no—
344C2×D3—181no—
345C2×D3—181no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—31no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—92no—
353C3×C4—181no—
354C3×C4—181no—
355C3×C4—181no—
356C3×C4#379511yescharacteristic
357C3×C4D3×#6711yescharacteristic
358Dic3—32no—
359Dic3—32no—
360Dic3—62no—
361Dic3—61no—
362Dic3—61no—
363Dic3—61no—
364Dic3—61no—
365Dic3—121no—
366Dic3—181no—
367Dic3—181no—
368C32—21no—
369C32#67⋊C811yescharacteristic, complemented
370C32C8×#6711yescharacteristic, complemented
371C23—91no—
372C23—91no—
373D4—274no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—91no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—21no—
384C8—92no—
385C8—181no—
386C8—181no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—182no—
397C2×C3—182no—
398C2×C3D32×C811yescharacteristic
399C2×C3#6380411yescharacteristic
400C2×C3#6387911yescharacteristic
401D3—62no—
402D3—62no—
403D3—182no—
404D3—182no—
405C22—92no—
406C22—92no—
407C22—181no—
408C22—181no—
409C22—181no—
410C22—181no—
411C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
412C4—92no—
413C4—181no—
414C4—181no—
415C4—181no—
416C4#8864811yescharacteristic
417C4#8867611yescharacteristic
418C3—31no—
419C3—31no—
420C3—61no—
421C3#1059071711yescharacteristic
422C2—182no—
423C2—182no—
424C2C8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C2#1172084111yescharacteristic
426C2#1172112211yescharacteristic
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×#6198C4
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
1. C2×C8×(C3⋊D3)⋊C3C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720547C2semidirect
5. #63647⋊C3C2semidirect
6. #11720317C2semidirect
7. #11720302C2semidirect
23. C2×C8×C3⋊D3D3semidirect
29. #63647D3semidirect
49. #6198C8semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
91. C2×(C3⋊D3)⋊C3C2×C8semidirect
92. C2×(C3⋊D3)⋊C3C2×C8semidirect
96. #1084C2×C8semidirect
152. C3⋊(C3⋊C8)#67semidirect
213. C2×C3⋊D3D3×C8semidirect
214. C2×C3⋊D3D3×C8semidirect
244. H(3,𝔽3)C8×D4semidirect
369. C32#67⋊C8semidirect
370. C32C8×#67semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
C4×#61982
C2×((C3⋊D3)⋊C3)⋊C82
#117205472
#63647⋊C32
#117203172
#117203022
#105909063
#105907203