Finite Group Databasebeta

← #10592874⋊C3 #13143365

Subgroups — #10592874⋊C3 #13143365

Order 1728.

565 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 108 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×SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64153⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#64149⋊C3C211yesmaximal, characteristic
5C2×#3754⋊C3C211yesmaximal, characteristic, complemented
6C2×#3744⋊C3C211yesmaximal, characteristic, complemented
7C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
8#10592874C311yesmaximal, characteristic, complemented
9C3×#5175—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#634⋊C3C2211yescharacteristic
15SD16×H(3,𝔽3)—22no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3—22no—
18#3754⋊C3—22no—
19#3750⋊C3—22no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3744⋊C3—22no—
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic, complemented
25C2×C32×SD16C2×C311yescharacteristic, complemented
26#64153C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28#64149C2×C311yescharacteristic
29C2×#3754C2×C311yescharacteristic, complemented
30C2×#3744C2×C311yescharacteristic, complemented
31C3×#944—31no—
32C2×C3×#215—31no—
33C3×#940—31no—
34C2×C3×#214—31no—
35C2×C3×#204—31no—
36C8×C3⋊Dic3C2×C311yescharacteristic, complemented
37C3×C8×Dic3—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46#638⋊C3—44no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3D412yes—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#634⋊C3—21no—
52#634⋊C3—21no—
53C8×H(3,𝔽3)D412yescomplemented
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55C3×#492—91no—
56#5175—31no—
57#5175—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C22×C311yescharacteristic
61C2×C32×D4—21no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—22no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×#634C22×C311yescharacteristic
67C2×C3×#67—62no—
68C2×C3×D12—31no—
69C2×C3×Dic6—31no—
70C32×SD16—24no—
71C32×SD16—22no—
72C32×SD16—22no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C22×C311yescharacteristic
76#3755—22no—
77#3754—22no—
78#3750—22no—
79C4×C3⋊Dic3C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81#3744—22no—
82C3×#215—62no—
83C3×#214—62no—
84C3×Dic3⋊C4—62no—
85C3×C4×Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C3×#204—62no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—42no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—42no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)C2×D411yescharacteristic
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1614yescomplemented
98C2×C3×SD16—31no—
99C2×C3×SD16—31no—
100C2×C3×SD16—61no—
101C2×C3×SD16—91no—
102C2×C3×SD16—91no—
103C2×C3×SD16—91no—
104C2×C3×SD16C3×D311yescharacteristic
105C3×C4⋊Q8—91no—
106C3×#127—91no—
107#944—31no—
108#944—91no—
109C2×#215—31no—
110C2×#215—91no—
111#940—31no—
112#940—91no—
113C2×#214—31no—
114C2×#214—91no—
115C2×#204—31no—
116C2×#204—91no—
117C3×C4×C8—91no—
118C8×Dic3—31no—
119C8×Dic3—91no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—41no—
123C22×C3⋊D3—21no—
124C22×C3×D3—61no—
125C32×Q8—22no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—41no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—41no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C23×C311yescharacteristic
143#638—44no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3C3×D412yes—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148#634—21no—
149#634—21no—
150C3×#67—124no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×Dic6—61no—
156C3×Dic6—61no—
157C32×C8—22no—
158C32×C8#6712yescomplemented
159C32×C8C3×D412yescomplemented
160C3⋊(C3⋊C8)C3×D412yescomplemented
161C3×C3⋊C8—32no—
162#492—91noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12711yescharacteristic
165C2×H(3,𝔽3)C2×SD1612yes—
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—91no—
171C2×C3×Q8C2×C3×D311yescharacteristic
172C2×C3×D4—31no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—91no—
176C2×C3×D4—182no—
177C2×C3×D4C2×C3×D311yescharacteristic
178C2×#67—62no—
179C2×#67—182no—
180C2×D12—31no—
181C2×D12—91no—
182C2×Dic6—31no—
183C2×Dic6—91no—
184C3×SD16—22no—
185C3×SD16—62no—
186C3×SD16—62no—
187C3×SD16—64no—
188C3×SD16—182no—
189C3×SD16—182no—
190C3×SD16—182no—
191C2×C3×C8—31no—
192C2×C3×C8—31no—
193C2×C3×C8—61no—
194C2×C3×C8—91no—
195C2×C3×C8C2×C3×D311yescharacteristic
196C3×C4⋊C4—182no—
197C3×C42—91no—
198#215—62no—
199#215—182no—
200#214—62no—
201#214—182no—
202Dic3⋊C4—62no—
203Dic3⋊C4—182no—
204C4×Dic3—31no—
205C4×Dic3—91no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—91no—
208#204—62no—
209#204—182no—
210C22×C32—21no—
211C22×C32—42no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32—82no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×C3×D411yescharacteristic
219C2×C3⋊D3—41no—
220C2×C3⋊D3—42no—
221C2×C3×D3—122no—
222C2×C3×D3—121no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—42no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4D3×D411yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C3⋊Dic3—41no—
233C3⋊Dic3C3×SD1614yescomplemented
234C3×Dic3—34no—
235C3×Dic3—121no—
236C2×SD16—91no—
237C2×SD16—91no—
238C2×SD16—91no—
239C2×SD16(C3⋊D3)⋊C311yescharacteristic, complemented
240C4⋊Q8—91no—
241#127—91no—
242C4×C8—91no—
243H(3,𝔽3)#49211yescharacteristic, complemented, Sylow
244C23×C3—21no—
245C23×C3—61no—
246C23×C3—61no—
247C23×C3—121no—
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—181no—
251C3×Q8—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—62no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—21no—
262C3×D4—21no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—62no—
268C3×D4—121no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—364no—
272C2×C3×C4—21no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—92no—
279C2×C3×C4—121no—
280C2×C3×C4—181no—
281C2×C3×C4C22×C3×D311yescharacteristic
282#67—124no—
283#67—364no—
284C2×Dic3—32no—
285C2×Dic3—61no—
286C2×Dic3—92no—
287C2×Dic3—181no—
288D12—61no—
289D12—61no—
290D12—181no—
291D12—181no—
292Dic6—61no—
293Dic6—61no—
294Dic6—181no—
295Dic6—181no—
296C3×C8—32no—
297C3×C8—32no—
298C3×C8—62no—
299C3×C8—92no—
300C3×C8C3×#6712yes—
301C3⋊C8—32no—
302C3⋊C8—92no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—82no—
306C2×C32—81no—
307C2×C32—81no—
308C2×C32D3×SD1612yes—
309C2×C32#94011yescharacteristic
310C2×C32C3×#12711yescharacteristic
311C2×C32C2×C3×SD1612yes—
312C3⋊D3—81no—
313C3×D3—241no—
314C2×Q8—91no—
315C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
316C2×D4—91no—
317C2×D4—182no—
318C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
319SD16—22no—
320SD16—182no—
321SD16—182no—
322SD16—182no—
323C2×C8—91no—
324C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
325C4⋊C4—182no—
326C42—91no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—61no—
332C22×C3—122no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—122no—
336C22×C3—122no—
337C22×C3—242no—
338C22×C3—361no—
339C22×C3—362no—
340C22×C3C3×D3×D411yescharacteristic
341C2×D3—122no—
342C2×D3—121no—
343C2×D3—361no—
344C2×D3—362no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—31no—
349C3×C4—41no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—94no—
353C3×C4—121no—
354C3×C4—121no—
355C3×C4—122no—
356C3×C4—361no—
357C3×C4C3×D3×D411yescharacteristic
358C3×C4C2×C3×#6711yescharacteristic
359Dic3—34no—
360Dic3—94no—
361Dic3—121no—
362Dic3—361no—
363C32—21no—
364C32#517511yescharacteristic, complemented
365C32C3×#49211yescharacteristic, complemented
366C23—21no—
367C23—181no—
368Q8—21no—
369Q8—21no—
370Q8—181no—
371Q8—181no—
372D4—21no—
373D4—21no—
374D4—181no—
375D4—181no—
376D4—364no—
377C2×C4—21no—
378C2×C4—92no—
379C2×C4—181no—
380C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
381C8—92no—
382C8#638⋊C312yescomplemented
383C2×C3—32no—
384C2×C3—32no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—62no—
388C2×C3—61no—
389C2×C3—81no—
390C2×C3—241no—
391C2×C3—241no—
392C2×C3—242no—
393C2×C3—721no—
394C2×C3C3×D3×SD1612yes—
395C2×C3C3×#94011yescharacteristic
396D3—241no—
397D3—721no—
398C22—42no—
399C22—41no—
400C22—362no—
401C22—361no—
402C22D4×(C3⋊D3)⋊C311yescharacteristic
403C4—41no—
404C4—94no—
405C4—361no—
406C4D4×(C3⋊D3)⋊C311yescharacteristic
407C4C2×#638⋊C311yescharacteristic
408C3—31no—
409C3—31no—
410C3—61no—
411C3C3×#517511yescharacteristic
412C2—81no—
413C2—721no—
414C2SD16×(C3⋊D3)⋊C312yes—
415C2#64149⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
prime radical#64149⋊C3C2
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#492
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. C2×SD16×H(3,𝔽3)C2semidirect
2. #64153⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
5. C2×#3754⋊C3C2semidirect
6. C2×#3744⋊C3C2semidirect
7. C8×(C3⋊Dic3)⋊C3C2semidirect
8. #10592874C3semidirect
24. C2×C32×SD16D3semidirect
25. C2×C32×SD16C2×C3semidirect
26. #64153C2×C3semidirect
27. C2×#3755C2×C3semidirect
29. C2×#3754C2×C3semidirect
30. C2×#3744C2×C3semidirect
36. C8×C3⋊Dic3C2×C3semidirect
53. C8×H(3,𝔽3)D4semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
158. C32×C8#67semidirect
159. C32×C8C3×D4semidirect
160. C3⋊(C3⋊C8)C3×D4semidirect
233. C3⋊Dic3C3×SD16semidirect
239. C2×SD16(C3⋊D3)⋊C3semidirect
243. H(3,𝔽3)#492semidirect
364. C32#5175semidirect
365. C32C3×#492semidirect
382. C8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×H(3,𝔽3)2
#64153⋊C32
C2×#3755⋊C32
#64149⋊C32
C2×#3754⋊C32
C2×#3744⋊C32
C8×(C3⋊Dic3)⋊C32
#105928743
C3×#51753