Finite Group Databasebeta

← #13138980

Subgroups — #13138980

Order 1728.

463 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 77 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#11721158C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720554C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590885—31nomaximal
9#10590878—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6200—21no—
13C2×#6200—21no—
14C2×#6196—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#63919—31no—
22#63918—31no—
23#63914—31no—
24#63911—31no—
25#63648D311yescharacteristic, complemented
26C8×C3⋊Dic3D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#821—31no—
29C3×#821—31no—
30C3×#821—61no—
31C3×C8×Dic3—31no—
32#63564—31no—
33#63563—31no—
34#63562—31no—
35#63562—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C4⋊#272—21no—
41C4⋊#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6200—42no—
48#6200—42no—
49#6196—42no—
50C8×H(3,𝔽3)—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52#4760—91no—
53#4760—91no—
54#4694—91no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×D12—31no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×#627—61no—
61C2×#627—61no—
62C2×#626—61no—
63C2×#626—61no—
64C2×#625—61no—
65C2×#625—61no—
66C2×C32×C8—21no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C3×C4×Dic3—31no—
77C2×C3×C3⋊C8—31no—
78#3724—61no—
79Dic3⋊Dic3—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×#272—41no—
82C2×#272—41no—
83C2×#272—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89#1084—41no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3D812yes—
92(C3⋊Dic3)⋊C3SD1612yes—
93#940—91no—
94#940—91no—
95#888—91no—
96#888—91no—
97#874—91no—
98#869—91no—
99C3×#102—91no—
100C3×#102—91no—
101C3×C4×C8—91no—
102#832—91no—
103#832—91no—
104#821—31no—
105#821—31no—
106#821—31no—
107#821—61no—
108#821—91no—
109#821—91no—
110#821—91no—
111C8×Dic3—31no—
112C8×Dic3—91no—
113#809—91no—
114#809—91no—
115C22×C3⋊D3—21no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C22×C3×D3—121no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3D1211yescharacteristic
125C2×C3⋊Dic3D1211yescharacteristic
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—121no—
134C3×D12—62no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×D12—61no—
140C3×D12—61no—
141C3×D12—121no—
142#627—122no—
143#627—122no—
144#626—122no—
145#626—122no—
146#625—122no—
147#625—122no—
148C32×C8—21no—
149C32×C8—21no—
150C32×C8—41no—
151C3⋊(C3⋊C8)—21no—
152C3×C3⋊C8—61no—
153#486—271noSylow
154C2×H(3,𝔽3)#12411yescharacteristic
155C2×H(3,𝔽3)C2×D811yescharacteristic
156C2×H(3,𝔽3)C2×SD1611yescharacteristic
157#272—81no—
158(C3⋊D3)⋊C3—81no—
159C2×C3×D4—91no—
160C2×C3×D4—91no—
161C2×#67—181no—
162C2×#67—181no—
163C2×#67—181no—
164C2×#67—181no—
165C2×D12—31no—
166C2×D12—31no—
167C2×D12—31no—
168C2×D12—61no—
169C2×D12—91no—
170C2×D12—181no—
171C2×D12—181no—
172C2×Dic6—181no—
173C2×Dic6—181no—
174C2×Dic6—181no—
175C2×C3×C8—31no—
176C2×C3×C8—31no—
177C2×C3×C8—61no—
178C2×C3×C8—91no—
179C2×C3×C8D3211yescharacteristic
180C3×C4⋊C4—91no—
181C3×C4⋊C4—91no—
182C3×C42—91no—
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—61no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189Dic3⋊C4—181no—
190Dic3⋊C4—181no—
191C4×Dic3—31no—
192C4×Dic3—91no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—91no—
195C22×C32—21no—
196C22×C32C2×D1211yescharacteristic
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—241no—
212C2×C3×D3—241no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4#23511yescharacteristic
216C32×C4#23511yescharacteristic
217C32×C4#23511yescharacteristic
218C32×C4#23711yescharacteristic
219C3⋊Dic3—41no—
220C3⋊Dic3#20412yes—
221C3⋊Dic3D2412yes—
222C3×Dic3—32no—
223C3×Dic3—32no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C4⋊Q8—271no—
229#127—271no—
230#102—91no—
231#102—91no—
232#102—271no—
233#102—271no—
234C4×C8—91no—
235H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
236C23×C3—181no—
237C23×C3—181no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—121no—
242C22×D3—181no—
243C3×D4—181no—
244C3×D4—181no—
245C3×D4—181no—
246C3×D4—181no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3211yescharacteristic
255#67—362no—
256#67—362no—
257#67—362no—
258#67—362no—
259C2×Dic3—31no—
260C2×Dic3—31no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—91no—
265C2×Dic3—91no—
266C2×Dic3—121no—
267C2×Dic3—181no—
268D12—61no—
269D12—61no—
270D12—61no—
271D12—61no—
272D12—62no—
273D12—61no—
274D12—61no—
275D12—121no—
276D12—181no—
277D12—181no—
278D12—362no—
279D12—362no—
280Dic6—362no—
281Dic6—362no—
282Dic6—362no—
283C3×C8—21no—
284C3×C8—61no—
285C3×C8—61no—
286C3×C8—121no—
287C3×C8—181no—
288C3⋊C8—61no—
289C3⋊C8—181no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32#87511yescharacteristic
294C2×C32#88511yescharacteristic
295C2×C32C2×#20411yescharacteristic
296C2×C32C2×D2411yescharacteristic
297C2×C32D3×D811yescharacteristic
298C2×C32D3×SD1611yescharacteristic
299C3⋊D3—81no—
300C3×D3—242no—
301C3×D3—241no—
302C3×D3—241no—
303C3×D3—241no—
304C2×Q8—541no—
305C2×D4—91no—
306C2×D4—91no—
307C2×D4—541no—
308C2×D4—541no—
309C2×C8—91no—
310C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—541no—
314C42—91no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—61no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3D3×D1211yescharacteristic
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—122no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—241no—
336C2×D3—241no—
337C2×D3—361no—
338C2×D3—361no—
339C2×D3—361no—
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—92no—
347C3×C4—92no—
348C3×C4—361no—
349C3×C4—361no—
350C3×C4#379811yescharacteristic
351C3×C4#380411yescharacteristic
352Dic3—32no—
353Dic3—32no—
354Dic3—92no—
355Dic3—92no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359Dic3—241no—
360Dic3—361no—
361C32—21no—
362C32#469411yescharacteristic, complemented
363C32#476011yescharacteristic, complemented
364C23—181no—
365C23—181no—
366Q8—1082no—
367D4—181no—
368D4—181no—
369D4—181no—
370D4—181no—
371D4—1082no—
372D4—1082no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C8—21no—
379C8—181no—
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—721no—
390C2×C3—721no—
391C2×C3D3×#20411yescharacteristic
392C2×C3D3×D2411yescharacteristic
393C2×C3#6385211yescharacteristic
394D3—241no—
395D3—241no—
396D3—242no—
397D3—241no—
398D3—721no—
399C22—361no—
400C22—361no—
401C22—361no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#8865411yescharacteristic
406C4—92no—
407C4—92no—
408C4—361no—
409C4—361no—
410C4#8865011yescharacteristic
411C4#8867111yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059087811yescharacteristic
416C2—721no—
417C2—721no—
418C2#1172084611yescharacteristic
419C2#1172084711yescharacteristic
420C2#1172110311yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radical#11721158C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#486
LCSG ▷ C2×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
2. #11721116C2semidirect
3. #11720554C2semidirect
4. #63648⋊C3C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720308C2semidirect
25. #63648D3semidirect
26. C8×C3⋊Dic3D3semidirect
235. H(3,𝔽3)#486semidirect
362. C32#4694semidirect
363. C32#4760semidirect

Maximal subgroups

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

subgroupindex
#117211582
#117211162
#117205542
#63648⋊C32
C8×(C3⋊Dic3)⋊C32
#117203242
#117203082
#105908853
#105908783