Finite Group Databasebeta

← #13140539

Subgroups — #13140539

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#64149⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic, complemented
6#11720310C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591487—31nomaximal
9#10591402—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×#638⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#6196—21no—
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#64149D311yescharacteristic
22C3×#940—31no—
23#63919—31no—
24#63918—31no—
25#63659D311yescharacteristic, complemented
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63564—31no—
31#63564—31no—
32#63563—31no—
33#63562—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C23×H(3,𝔽3)—21no—
37C22×(C3⋊D3)⋊C3—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#638⋊C3—42no—
43#638⋊C3—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#6196—42no—
50#6180—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52#5070—91no—
53#4760—91no—
54#4760—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—21no—
59C2×#638—21no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×#67—61no—
62C2×C3×#67—61no—
63C2×C3×D12—31no—
64C2×#627—61no—
65C2×#627—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—61no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78#3724—61no—
79Dic3⋊Dic3—61no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)—41no—
82C22×H(3,𝔽3)—41no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
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—
93C3×#127—91no—
94#940—31no—
95#940—91no—
96#888—91no—
97#888—91no—
98#869—91no—
99C3×#102—91no—
100C3×#102—91no—
101#832—31no—
102#832—31no—
103#832—31no—
104#832—61no—
105#832—91no—
106#832—91no—
107#832—91no—
108#821—91no—
109#821—91no—
110C8×Dic3—91no—
111C8×Dic3—91no—
112#809—91no—
113#809—91no—
114C4×C3⋊C8—91no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C22×C3⋊D3—21no—
119C22×C3×D3—61no—
120C32×D4—22no—
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—41no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129#638—42no—
130#638—42no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3#6711yescharacteristic
133C2×C3⋊Dic3#6711yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C3×#67—122no—
137C3×#67—122no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C3×D12—61no—
145C3×D12—61no—
146#627—122no—
147#627—122no—
148C3⋊(C3⋊C8)—21no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—121no—
153#486—271noSylow
154C2×H(3,𝔽3)—81no—
155C2×H(3,𝔽3)#12411yescharacteristic
156C2×H(3,𝔽3)C2×D811yescharacteristic
157C2×H(3,𝔽3)C2×SD1611yescharacteristic
158(C3⋊D3)⋊C3—81no—
159C2×C3×D4—31no—
160C2×C3×D4—31no—
161C2×C3×D4—61no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C2×C3×D4—181no—
165C2×C3×D4D3211yescharacteristic
166C2×#67—61no—
167C2×#67—61no—
168C2×#67—181no—
169C2×#67—181no—
170C2×D12—31no—
171C2×D12—91no—
172C2×Dic6—181no—
173C2×Dic6—181no—
174C2×Dic6—181no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—91no—
179C3×C42—91no—
180C4⋊Dic3—31no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—61no—
184C4⋊Dic3—91no—
185C4⋊Dic3—181no—
186Dic3⋊C4—181no—
187Dic3⋊C4—181no—
188C4×Dic3—31no—
189C4×Dic3—91no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—31no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—91no—
195C22×C32—21no—
196C22×C32—42no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—81no—
204C22×C32—81no—
205C22×C32D3×D411yescharacteristic
206C22×C32C2×#6711yescharacteristic
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—121no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4#23511yescharacteristic
216C32×C4#23711yescharacteristic
217C32×C4#23711yescharacteristic
218C32×C4#23711yescharacteristic
219C3⋊Dic3—41no—
220C3⋊Dic3#21312yes—
221C3⋊Dic3#21412yes—
222C3×Dic3—32no—
223C3×Dic3—32no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226C3×Dic3—121no—
227C3×Dic3—241no—
228C4⋊Q8—271no—
229#127—91no—
230#102—91no—
231#102—91no—
232#102—271no—
233#102—271no—
234C4×C8—271no—
235H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—61no—
238C23×C3—61no—
239C23×C3—121no—
240C23×C3—181no—
241C22×D3—61no—
242C22×D3—181no—
243C3×D4—21no—
244C3×D4—21no—
245C3×D4—61no—
246C3×D4—61no—
247C3×D4—61no—
248C3×D4—61no—
249C3×D4—62no—
250C3×D4—121no—
251C3×D4—181no—
252C3×D4—181no—
253C3×D4—362no—
254C3×D4—362no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—61no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3211yescharacteristic
263#67—122no—
264#67—122no—
265#67—362no—
266#67—362no—
267C2×Dic3—31no—
268C2×Dic3—31no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—121no—
275C2×Dic3—181no—
276D12—61no—
277D12—61no—
278D12—181no—
279D12—181no—
280Dic6—362no—
281Dic6—362no—
282Dic6—362no—
283C3×C8—181no—
284C3×C8—181no—
285C3⋊C8—61no—
286C3⋊C8—61no—
287C3⋊C8—61no—
288C3⋊C8—121no—
289C3⋊C8—181no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—82no—
294C2×C32—81no—
295C2×C32—81no—
296C2×C32#88511yescharacteristic
297C2×C32D3×D811yescharacteristic
298C2×C32D3×SD1611yescharacteristic
299C2×C32C2×#21311yescharacteristic
300C2×C32C2×#21411yescharacteristic
301C2×C32#93611yescharacteristic
302C3⋊D3—81no—
303C3×D3—241no—
304C2×Q8—541no—
305C2×D4—91no—
306C2×D4—181no—
307C2×D4—181no—
308C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
309C2×C8—91no—
310C2×C8—91no—
311C4⋊C4—91no—
312C4⋊C4—91no—
313C4⋊C4—541no—
314C42—91no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—61no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—122no—
328C22×C3—241no—
329C22×C3—241no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3—361no—
333C22×C3D3×#6711yescharacteristic
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—121no—
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#380411yescharacteristic
351C3×C4#380511yescharacteristic
352Dic3—32no—
353Dic3—32no—
354Dic3—92no—
355Dic3—92no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359Dic3—241no—
360Dic3—361no—
361C32—21no—
362C32#476011yescharacteristic, complemented
363C32#507011yescharacteristic, complemented
364C23—21no—
365C23—181no—
366Q8—1082no—
367D4—21no—
368D4—21no—
369D4—181no—
370D4—181no—
371D4—362no—
372D4—362no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C8—181no—
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—81no—
390C2×C3—241no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—721no—
394C2×C3D3×#21311yescharacteristic
395C2×C3D3×#21411yescharacteristic
396C2×C3#6395911yescharacteristic
397D3—241no—
398D3—721no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22#8867611yescharacteristic
406C4—92no—
407C4—92no—
408C4—361no—
409C4—361no—
410C4#8867111yescharacteristic
411C4#8867211yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059148711yescharacteristic
416C2—81no—
417C2—721no—
418C2#1172119411yescharacteristic
419C2#1172119811yescharacteristic
420C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64149⋊C3C2
hypercenterC2×D4((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×D4

Complements

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

NH ≅ G/Ntype
2. #11721116C2semidirect
3. #11720578C2semidirect
4. #63659⋊C3C2semidirect
5. #11720318C2semidirect
6. #11720310C2semidirect
7. #11720308C2semidirect
25. #63659D3semidirect
235. H(3,𝔽3)#486semidirect
362. C32#4760semidirect
363. C32#5070semidirect

Maximal subgroups

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

subgroupindex
#64149⋊C32
#117211162
#117205782
#63659⋊C32
#117203182
#117203102
#117203082
#105914873
#105914023