Finite Group Databasebeta

← #10592866⋊C3 #13143370

Subgroups — #10592866⋊C3 #13143370

Order 1728.

502 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 100 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
2#64153⋊C3C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4#64145⋊C3C211yesmaximal, characteristic
5C2×#3754⋊C3C211yesmaximal, characteristic
6C2×#3746⋊C3C211yesmaximal, characteristic, complemented
7#63642⋊C3C211yesmaximal, characteristic, complemented
8#10592866C311yesmaximal, characteristic, complemented
9C3×#5167—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—22no—
17#3756⋊C3—22no—
18#3754⋊C3—22no—
19#3750⋊C3—22no—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#3746⋊C3—22no—
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic
25C2×C32×SD16C2×C311yescharacteristic
26#64153C2×C311yescharacteristic, complemented
27C2×#3756C2×C311yescharacteristic, complemented
28#64145C2×C311yescharacteristic
29C2×#3754C2×C311yescharacteristic
30C2×#3746C2×C311yescharacteristic, complemented
31C3×#944—31no—
32C2×C3×#216—31no—
33C3×#936—31no—
34C2×C3×#214—31no—
35C2×C3×Dic12—31no—
36#63642C2×C311yescharacteristic, complemented
37C3×#815—31no—
38C23×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54C3×#497—91no—
55#5167—31no—
56#5167—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C2×#634C22×C311yescharacteristic
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C32×SD16—24no—
68C32×SD16—22no—
69C32×SD16—22no—
70C2×C32×C8—21no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C22×C311yescharacteristic
73#3757—22no—
74#3756—22no—
75#3754—22no—
76#3750—22no—
77C4×C3⋊Dic3C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79#3746—22no—
80C3×#217—62no—
81C3×#216—62no—
82C3×#214—62no—
83C3×Dic3⋊C4—62no—
84C3×C4×Dic3—31no—
85C2×C3×C3⋊C8—31no—
86C3×Dic12—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—42no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)C2×D411yescharacteristic
93(C3⋊Dic3)⋊C3—22no—
94(C3⋊Dic3)⋊C3—41no—
95(C3⋊Dic3)⋊C3—41no—
96C2×C3×Q16—91no—
97C2×C3×Q16—91no—
98C2×C3×SD16—31no—
99C2×C3×SD16—31no—
100C2×C3×SD16—61no—
101C2×C3×SD16—91no—
102C2×C3×SD16C3×D311yescharacteristic
103C3×C4⋊Q8—91no—
104C3×#124—91no—
105#944—31no—
106#944—91no—
107C2×#216—31no—
108C2×#216—91no—
109#936—31no—
110#936—91no—
111C2×#214—31no—
112C2×#214—91no—
113C2×Dic12—31no—
114C2×Dic12—91no—
115C3×#97—91no—
116#815—31no—
117#815—91no—
118C23×C32—21no—
119C23×C32—21no—
120C23×C32—41no—
121C32×Q8—22no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—41no—
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—41no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3C3×D412yes—
142#634—21no—
143#634—21no—
144#634—21no—
145#634—21no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C32×C8—22no—
154C32×C8#6712yes—
155C32×C8C3×D412yes—
156C3⋊(C3⋊C8)C3×D412yes—
157C3×C3⋊C8—32no—
158#497—91noSylow
159C2×H(3,𝔽3)—81no—
160C2×H(3,𝔽3)#12711yescharacteristic
161C2×H(3,𝔽3)#13712yes—
162C2×C3×Q8—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8—91no—
167C2×C3×Q8C2×C3×D311yescharacteristic
168C2×C3×D4—31no—
169C2×C3×D4—31no—
170C2×C3×D4—61no—
171C2×C3×D4C2×C3×D311yescharacteristic
172C2×Dic6—31no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C2×Dic6—91no—
176C3×Q16—182no—
177C3×Q16—182no—
178C3×SD16—22no—
179C3×SD16—62no—
180C3×SD16—62no—
181C3×SD16—64no—
182C3×SD16—182no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—61no—
186C2×C3×C8—91no—
187C2×C3×C8C2×C3×D311yescharacteristic
188C3×C4⋊C4—182no—
189C3×C22⋊C4—182no—
190C3×C42—91no—
191#217—62no—
192#217—182no—
193#216—62no—
194#216—182no—
195#214—62no—
196#214—182no—
197Dic3⋊C4—62no—
198Dic3⋊C4—182no—
199C4×Dic3—31no—
200C4×Dic3—91no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203Dic12—62no—
204Dic12—182no—
205C22×C32—21no—
206C22×C32—42no—
207C22×C32—42no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—42no—
211C22×C32—82no—
212C22×C32D3×D411yescharacteristic
213C22×C32C2×C3×D411yescharacteristic
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—42no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4D3×D411yescharacteristic
220C32×C4C2×#6711yescharacteristic
221C32×C4C2×C3×D411yescharacteristic
222C32×C4C2×C3×D411yescharacteristic
223C3⋊Dic3—22no—
224C3⋊Dic3—41no—
225C3⋊Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C2×Q16—91no—
230C2×Q16—91no—
231C2×SD16—91no—
232C2×SD16(C3⋊D3)⋊C311yescharacteristic
233C4⋊Q8—91no—
234#124—91no—
235#97—91no—
236H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
237C23×C3—21no—
238C23×C3—61no—
239C23×C3—61no—
240C23×C3—121no—
241C3×Q8—21no—
242C3×Q8—21no—
243C3×Q8—61no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8—62no—
248C3×Q8—121no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×D4—21no—
254C3×D4—21no—
255C3×D4—61no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—62no—
260C3×D4—121no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—92no—
268C2×C3×C4—121no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C22×C3×D311yescharacteristic
272C2×Dic3—32no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—92no—
276C2×Dic3—181no—
277C2×Dic3—181no—
278Dic6—61no—
279Dic6—61no—
280Dic6—61no—
281Dic6—61no—
282Dic6—181no—
283Dic6—181no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—32no—
287C3×C8—32no—
288C3×C8—62no—
289C3×C8—92no—
290C3×C8C3×#6712yes—
291C3⋊C8—32no—
292C3⋊C8—92no—
293C2×C32—22no—
294C2×C32—21no—
295C2×C32—82no—
296C2×C32—81no—
297C2×C32—81no—
298C2×C32#91512yes—
299C2×C32#94011yescharacteristic
300C2×C32C3×#12711yescharacteristic
301C2×C32C3×#13712yes—
302C2×Q8—91no—
303C2×Q8—91no—
304C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
305C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
306Q16—182no—
307Q16—182no—
308SD16—22no—
309SD16—182no—
310C2×C8—91no—
311C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
312C4⋊C4—182no—
313C22⋊C4—182no—
314C42—91no—
315C22×C3—31no—
316C22×C3—31no—
317C22×C3—42no—
318C22×C3—41no—
319C22×C3—61no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—122no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—242no—
326C22×C3C3×D3×D411yescharacteristic
327C3×C4—31no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—41no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—122no—
337C3×C4—182no—
338C3×C4—361no—
339C3×C4—361no—
340C3×C4C3×D3×D411yescharacteristic
341C3×C4C2×C3×#6711yescharacteristic
342Dic3—62no—
343Dic3—121no—
344Dic3—121no—
345Dic3—182no—
346Dic3—361no—
347Dic3—361no—
348C32—21no—
349C32#516711yescharacteristic, complemented
350C32C3×#49711yescharacteristic, complemented
351C23—21no—
352Q8—21no—
353Q8—21no—
354Q8—181no—
355Q8—181no—
356Q8—181no—
357Q8—181no—
358D4—21no—
359D4—21no—
360C2×C4—21no—
361C2×C4—92no—
362C2×C4—181no—
363C2×C4—181no—
364C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
365C8—92no—
366C8#638⋊C312yes—
367C2×C3—32no—
368C2×C3—32no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—62no—
372C2×C3—61no—
373C2×C3—81no—
374C2×C3—241no—
375C2×C3—241no—
376C2×C3—242no—
377C2×C3C3×#91512yes—
378C2×C3C3×#94011yescharacteristic
379C22—42no—
380C22—41no—
381C22D4×(C3⋊D3)⋊C311yescharacteristic
382C4—41no—
383C4—182no—
384C4—361no—
385C4—361no—
386C4D4×(C3⋊D3)⋊C311yescharacteristic
387C4C2×#638⋊C311yescharacteristic
388C3—31no—
389C3—31no—
390C3—61no—
391C3C3×#516711yescharacteristic
392C2—81no—
393C2#64124⋊C312yes—
394C2#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 radicalC2×D4×H(3,𝔽3)C22
hypercenterC2×SD16(C3⋊D3)⋊C3
hypocenterC32C3×#497
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
2. #64153⋊C3C2semidirect
3. C2×#3756⋊C3C2semidirect
6. C2×#3746⋊C3C2semidirect
7. #63642⋊C3C2semidirect
8. #10592866C3semidirect
26. #64153C2×C3semidirect
27. C2×#3756C2×C3semidirect
30. C2×#3746C2×C3semidirect
36. #63642C2×C3semidirect
236. H(3,𝔽3)#497semidirect
349. C32#5167semidirect
350. C32C3×#497semidirect

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×#3756⋊C32
#64145⋊C32
C2×#3754⋊C32
C2×#3746⋊C32
#63642⋊C32
#105928663
C3×#51673