Finite Group Databasebeta

← #13138945

Subgroups — #13138945

Order 1728.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721149C211yesmaximal, characteristic, complemented
2#11721132C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic
4#11720554C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590746—31nomaximal
9#10590741—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6199—21no—
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88450—21no—
21#88436C414yescomplemented
22#63898—31no—
23#63898—31no—
24#63894—31no—
25Dic3×D12—31no—
26C2×#3714—31no—
27C2×#3713—31no—
28#63648D311yescharacteristic, complemented
29#63642D311yescharacteristic, complemented
30C3×#821—31no—
31C3×#821—31no—
32C3×#821—31no—
33C3×#821—61no—
34C3×#815—31no—
35#63574—31no—
36#63573—31no—
37C22×#272—21no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C4⋊#272—21no—
42C4⋊#272C2×C412yescomplemented
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
48#6199—42no—
49((C3⋊D3)⋊C3)⋊C4—42no—
50C8×H(3,𝔽3)—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52#4785—91no—
53#4785—91no—
54#4711—91no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×#624—61no—
61C2×D3×Dic3—61no—
62C2×D3×Dic3—61no—
63C2×D3×Dic3—61no—
64C2×C32×C8—21no—
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×C4×Dic3—31no—
75C2×C3×C3⋊C8—31no—
76#3722—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80#3714—34no—
81#3713—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×#272—22no—
84C2×#272—41no—
85C2×#272—41no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—41no—
88C2×(C3⋊D3)⋊C3—41no—
89C4×H(3,𝔽3)C22×C411yescharacteristic
90C4×H(3,𝔽3)#4111yescharacteristic
91#1084—41no—
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—41no—
95D4×Dic3—91no—
96D4×Dic3—91no—
97C2×#213—91no—
98C2×#213—91no—
99C2×D24—91no—
100#893—91no—
101#893—91no—
102C4×D12—91no—
103C4×D12—91no—
104C3×#102—91no—
105C3×#102—91no—
106C3×#97—91no—
107#821—31no—
108#821—31no—
109#821—31no—
110#821—61no—
111#821—91no—
112#817—91no—
113#815—31no—
114#815—91no—
115#808—91no—
116#808—91no—
117C22×C3⋊D3—21no—
118C22×C3×D3—61no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—121no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3D1211yescharacteristic
127C2×C3⋊Dic3D1211yescharacteristic
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C4×D312yescomplemented
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×D12—32no—
137C3×D12—32no—
138C3×D12—32no—
139C3×D12—62no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—121no—
144#624—122no—
145D3×Dic3—122no—
146D3×Dic3—122no—
147D3×Dic3—122no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)C4×D312yescomplemented
152C3×C3⋊C8—32no—
153#442—271noSylow
154C2×H(3,𝔽3)C4×D411yescharacteristic
155C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
156C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
157#272—42no—
158(C3⋊D3)⋊C3—42no—
159C2×C3×D4—91no—
160C2×C3×D4—91no—
161C22×Dic3—181no—
162C22×Dic3—181no—
163C2×D12—31no—
164C2×D12—31no—
165C2×D12—31no—
166C2×D12—61no—
167C2×D12—91no—
168C2×C4×D3—181no—
169C2×C4×D3—181no—
170C2×C4×D3—181no—
171C2×C4×D3—181no—
172C2×C3×C8—31no—
173C2×C3×C8—31no—
174C2×C3×C8—61no—
175C2×C3×C8—91no—
176C2×C3×C8D3211yescharacteristic
177C3×C4⋊C4—91no—
178C3×C4⋊C4—91no—
179C3×C42—91no—
180#217—181no—
181#217—181no—
182#213—94no—
183#213—94no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188C4⋊Dic3—31no—
189C4⋊Dic3—31no—
190C4⋊Dic3—31no—
191C4⋊Dic3—61no—
192C4⋊Dic3—91no—
193C4×Dic3—31no—
194C4×Dic3—91no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—91no—
197D24—94no—
198C22×C32—21no—
199C22×C32C2×D1211yescharacteristic
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—22no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—62no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—122no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—241no—
215C2×C3×D3—241no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4C2×C4×D311yescharacteristic
220C32×C4#23511yescharacteristic
221C32×C4#23711yescharacteristic
222C3⋊Dic3—21no—
223C3⋊Dic3—21no—
224C3⋊Dic3—41no—
225C3×Dic3—61no—
226C3×Dic3—61no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—241no—
231C2×D8—271no—
232C4×D4—271no—
233C4×D4—271no—
234#106—271no—
235#102—91no—
236#102—91no—
237#97—91no—
238H(3,𝔽3)#44211yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C23×C3—181no—
241C22×D3—61no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—121no—
245C22×D3—181no—
246C3×D4—92no—
247C3×D4—92no—
248C3×D4—181no—
249C3×D4—181no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4C2×D3211yescharacteristic
258C2×Dic3—31no—
259C2×Dic3—31no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—91no—
264C2×Dic3—91no—
265C2×Dic3—121no—
266C2×Dic3—181no—
267C2×Dic3—362no—
268C2×Dic3—362no—
269D12—32no—
270D12—32no—
271D12—32no—
272D12—62no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—92no—
277D12—121no—
278D12—181no—
279C4×D3—362no—
280C4×D3—362no—
281C4×D3—362no—
282C4×D3—362no—
283C3×C8—21no—
284C3×C8—61no—
285C3×C8—61no—
286C3×C8—92no—
287C3×C8—121no—
288C3⋊C8—32no—
289C3⋊C8—92no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32C4×D1211yescharacteristic
294C2×C32#67⋊C411yescharacteristic
295C2×C32#90811yescharacteristic
296C2×C32#90811yescharacteristic
297C2×C32#91111yescharacteristic
298C2×C32#91411yescharacteristic
299C3⋊D3—42no—
300C3×D3—122no—
301C3×D3—122no—
302C3×D3—122no—
303C3×D3—242no—
304C2×D4—91no—
305C2×D4—91no—
306C22×C4—541no—
307C22×C4—541no—
308D8—274no—
309C2×C8—91no—
310C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4⋊C4—91no—
312C4⋊C4—91no—
313C22⋊C4—541no—
314C22⋊C4—541no—
315C42—91no—
316C22×C3—31no—
317C22×C3—31no—
318C22×C3—61no—
319C22×C3—182no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—361no—
325C22×C3D3×D1211yescharacteristic
326C2×D3—62no—
327C2×D3—62no—
328C2×D3—62no—
329C2×D3—122no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—182no—
337C2×D3—241no—
338C2×D3—241no—
339C2×D3—361no—
340C2×D3—361no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—181no—
348C3×C4—181no—
349C3×C4—361no—
350C3×C4—361no—
351C3×C4C4×D3211yescharacteristic
352C3×C4#380411yescharacteristic
353Dic3—61no—
354Dic3—61no—
355Dic3—121no—
356Dic3—121no—
357Dic3—121no—
358Dic3—181no—
359Dic3—181no—
360Dic3—241no—
361Dic3—361no—
362C32—21no—
363C32#471111yescharacteristic, complemented
364C32#478511yescharacteristic, complemented
365C23—181no—
366C23—181no—
367D4—92no—
368D4—92no—
369D4—181no—
370D4—181no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—1082no—
376C2×C4—1082no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C8—21no—
379C8—92no—
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—362no—
390C2×C3—362no—
391C2×C3#6379411yescharacteristic
392C2×C3#6379511yescharacteristic
393C2×C3#6388011yescharacteristic
394D3—122no—
395D3—122no—
396D3—122no—
397D3—242no—
398D3—362no—
399C22—182no—
400C22—182no—
401C22—361no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#8865411yescharacteristic
406C4—181no—
407C4—181no—
408C4—361no—
409C4—361no—
410C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
411C4#8867111yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059074611yescharacteristic
416C2—362no—
417C2—362no—
418C2#1172084811yescharacteristic
419C2#1172084911yescharacteristic
420C2#1172112311yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×#88436C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#442
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721149C2semidirect
2. #11721132C2semidirect
4. #11720554C2semidirect
5. #63648⋊C3C2semidirect
6. #63642⋊C3C2semidirect
7. #11720307C2semidirect
21. #88436C4semidirect
28. #63648D3semidirect
29. #63642D3semidirect
42. C4⋊#272C2×C4semidirect
47. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
129. C4⋊(C3⋊D3)C4×D3semidirect
151. C3⋊(C3⋊C8)C4×D3semidirect
238. H(3,𝔽3)#442semidirect
363. C32#4711semidirect
364. C32#4785semidirect

Maximal subgroups

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

subgroupindex
#117211492
#117211322
C2×#884362
#117205542
#63648⋊C32
#63642⋊C32
#117203072
#105907463
#105907413