Finite Group Databasebeta

← #10592861⋊C3 #13143363

Subgroups — #10592861⋊C3 #13143363

Order 1728.

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

normal subgroups only

This group has 488 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
3D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4C2×#3754⋊C3C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63641⋊C3C211yesmaximal, characteristic, complemented
8#10592861C311yesmaximal, characteristic, complemented
9C3×#5162—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14SD16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16#3757⋊C3—21no—
17#3754⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20#3750⋊C3—21no—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C2×C32×SD16—21no—
24C2×C32×SD16D311yescharacteristic
25C2×C32×SD16C2×C311yescharacteristic
26#64153C2×C311yescharacteristic, complemented
27D4×C3⋊Dic3C2×C311yescharacteristic
28C2×#3754C2×C311yescharacteristic
29C3×#944—31no—
30C3×D4×Dic3—31no—
31C2×C3×#214—31no—
32#63662C2×C311yescharacteristic, complemented
33#63643C2×C311yescharacteristic, complemented
34#63641C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#816—31no—
37C3×Dic3⋊C8—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)D412yes—
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)⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50#634⋊C3—21no—
51#634⋊C3—21no—
52C8×H(3,𝔽3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54C3×#461—91no—
55#5162—31no—
56#5162—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
63C22×C3⋊Dic3—21no—
64C2×#634C22×C311yescharacteristic
65C22×C3×Dic3—61no—
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—21no—
74#3754—22no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76#3750—21no—
77#3750—21no—
78C4×C3⋊Dic3C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—61no—
81C3×#214—62no—
82C3×C4⋊Dic3—31no—
83C3×Dic3⋊C4—61no—
84C3×Dic3⋊C4—61no—
85C3×C4×Dic3—31no—
86C2×C3×C3⋊C8—31no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1612yescomplemented
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×C4×D4—91no—
105#944—31no—
106#944—91no—
107D4×Dic3—31no—
108D4×Dic3—91no—
109C2×#214—31no—
110C2×#214—91no—
111C3×C4⋊C8—91no—
112C3×#103—91no—
113C3×#103—91no—
114#835—31no—
115#835—91no—
116#816—31no—
117#816—91no—
118Dic3⋊C8—31no—
119Dic3⋊C8—91no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—41no—
123C32×Q8—22no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—41no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—22no—
132C32×D4—41no—
133C32×D4#6712yes—
134C32×D4C3×D412yes—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C23×C311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—42no—
144C2×C3⋊Dic3C3×D411yescharacteristic
145C2×C3⋊Dic3C3×D411yescharacteristic
146#634—21no—
147#634—21no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—122no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C32×C8—22no—
156C32×C8—21no—
157C32×C8—21no—
158C3⋊(C3⋊C8)—21no—
159C3×C3⋊C8—61no—
160#461—91noSylow
161C2×H(3,𝔽3)—42no—
162C2×H(3,𝔽3)#12111yescharacteristic
163C2×H(3,𝔽3)C2×SD1611yescharacteristic
164C2×H(3,𝔽3)#13711yescharacteristic
165C2×C3×Q8—31no—
166C2×C3×Q8—31no—
167C2×C3×Q8—61no—
168C2×C3×Q8—91no—
169C2×C3×Q8C2×C3×D311yescharacteristic
170C2×C3×D4—31no—
171C2×C3×D4—31no—
172C2×C3×D4—61no—
173C2×C3×D4C2×C3×D311yescharacteristic
174C22×C3×C4—181no—
175C22×Dic3—61no—
176C22×Dic3—181no—
177C2×Dic6—31no—
178C2×Dic6—91no—
179C3×SD16—22no—
180C3×SD16—62no—
181C3×SD16—62no—
182C3×SD16—64no—
183C3×SD16—182no—
184C2×C3×C8—31no—
185C2×C3×C8—31no—
186C2×C3×C8—61no—
187C2×C3×C8—91no—
188C2×C3×C8C2×C3×D311yescharacteristic
189C3×C4⋊C4—91no—
190C3×C4⋊C4—181no—
191C3×C4⋊C4—181no—
192C3×C22⋊C4—181no—
193C3×C42—91no—
194#217—61no—
195#217—181no—
196#214—62no—
197#214—182no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200Dic3⋊C4—61no—
201Dic3⋊C4—61no—
202Dic3⋊C4—181no—
203Dic3⋊C4—181no—
204C4×Dic3—31no—
205C4×Dic3—91no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—91no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—21no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32D3×D411yescharacteristic
219C22×C32C2×C3×D411yescharacteristic
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4#23711yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C32×C4C2×C3×D411yescharacteristic
228C32×C4C3×#4111yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3—41no—
231C3⋊Dic3—41no—
232C3⋊Dic3C3×SD1612yescomplemented
233C3×Dic3—32no—
234C3×Dic3—61no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C2×SD16—91no—
238C2×SD16(C3⋊D3)⋊C311yescharacteristic
239C4⋊Q8—91no—
240C4×D4—91no—
241C4⋊C8—91no—
242#103—91no—
243#103—91no—
244H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
245C23×C3—21no—
246C23×C3—61no—
247C23×C3—61no—
248C23×C3—121no—
249C3×Q8—21no—
250C3×Q8—21no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—62no—
256C3×Q8—121no—
257C3×Q8—181no—
258C3×Q8—181no—
259C3×D4—21no—
260C3×D4—32no—
261C3×D4—32no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—62no—
265C3×D4—121no—
266C3×D4C3×#6712yes—
267C2×C3×C4—21no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—121no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—362no—
279C2×C3×C4C22×C3×D311yescharacteristic
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—91no—
285C2×Dic3—91no—
286C2×Dic3—122no—
287C2×Dic3—181no—
288C2×Dic3—181no—
289C2×Dic3—362no—
290Dic6—61no—
291Dic6—61no—
292Dic6—181no—
293Dic6—181no—
294C3×C8—21no—
295C3×C8—61no—
296C3×C8—61no—
297C3×C8—62no—
298C3×C8—181no—
299C3⋊C8—61no—
300C3⋊C8—181no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—42no—
305C2×C32—42no—
306C2×C32—82no—
307C2×C32D3×SD1611yescharacteristic
308C2×C32#91511yescharacteristic
309C2×C32#93911yescharacteristic
310C2×C32C3×#12111yescharacteristic
311C2×C32C2×C3×SD1611yescharacteristic
312C2×C32C3×#13711yescharacteristic
313C2×Q8—91no—
314C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
315C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
316C22×C4—181no—
317SD16—22no—
318SD16—182no—
319C2×C8—91no—
320C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4—181no—
324C22⋊C4—181no—
325C42—91no—
326C22×C3—22no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—62no—
332C22×C3—62no—
333C22×C3—61no—
334C22×C3—122no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—241no—
340C22×C3—241no—
341C22×C3C3×D3×D411yescharacteristic
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—41no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—92no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4—122no—
353C3×C4—181no—
354C3×C4—361no—
355C3×C4—361no—
356C3×C4C3×#23711yescharacteristic
357C3×C4C2×C3×#6711yescharacteristic
358Dic3—32no—
359Dic3—61no—
360Dic3—92no—
361Dic3—121no—
362Dic3—121no—
363Dic3—181no—
364Dic3—361no—
365Dic3—361no—
366C32—21no—
367C32#516211yescharacteristic, complemented
368C32C3×#46111yescharacteristic, complemented
369C23—21no—
370Q8—21no—
371Q8—21no—
372Q8—181no—
373Q8—181no—
374D4—21no—
375D4#638⋊C312yes—
376C2×C4—21no—
377C2×C4—91no—
378C2×C4—91no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—362no—
382C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
383C8—21no—
384C8—181no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—42no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—61no—
395C2×C3—122no—
396C2×C3—122no—
397C2×C3—242no—
398C2×C3C3×D3×SD1611yescharacteristic
399C2×C3C3×#91511yescharacteristic
400C2×C3C3×#93911yescharacteristic
401C22—22no—
402C22—41no—
403C22—41no—
404C22D4×(C3⋊D3)⋊C311yescharacteristic
405C4—41no—
406C4—92no—
407C4—181no—
408C4—361no—
409C4—361no—
410C4#3830⋊C311yescharacteristic
411C4C2×#638⋊C311yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3C3×#516211yescharacteristic
416C2—42no—
417C2SD16×(C3⋊D3)⋊C311yescharacteristic
418C2#64124⋊C311yescharacteristic
419C2#64148⋊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×#461
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
5. #63662⋊C3C2semidirect
6. #63643⋊C3C2semidirect
7. #63641⋊C3C2semidirect
8. #10592861C3semidirect
26. #64153C2×C3semidirect
32. #63662C2×C3semidirect
33. #63643C2×C3semidirect
34. #63641C2×C3semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
232. C3⋊Dic3C3×SD16semidirect
244. H(3,𝔽3)#461semidirect
367. C32#5162semidirect
368. C32C3×#461semidirect

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
D4×(C3⋊Dic3)⋊C32
C2×#3754⋊C32
#63662⋊C32
#63643⋊C32
#63641⋊C32
#105928613
C3×#51623