Finite Group Databasebeta

← #10592750⋊C3 #13142876

Subgroups — #10592750⋊C3 #13142876

Order 1728.

473 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 106 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
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2C2×#3756⋊C3C211yesmaximal, characteristic, complemented
3#64133⋊C3C211yesmaximal, characteristic
4C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592750C311yesmaximal, characteristic, complemented
9C3×#5051—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16#3757⋊C3—21no—
17#3756⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3—22no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×C22⋊Q8—21no—
24C32×C22⋊Q8D311yescharacteristic
25C32×C22⋊Q8C2×C311yescharacteristic
26C2×#3756C2×C311yescharacteristic, complemented
27#64133C2×C311yescharacteristic
28C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
29C2×C3×#216—31no—
30C3×#924—31no—
31C22×C3×C3⋊C8—31no—
32#63662C2×C311yescharacteristic, complemented
33#63637C2×C311yescharacteristic, complemented
34#63634C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#810—31no—
37C3×#807—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)D411yescharacteristic
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53C3×#467—91no—
54#5051—31no—
55#5051—91no—
56C2×C32×Q8—21no—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C32×Q8C22×C311yescharacteristic
59C22×C32×C4—21no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C22×C311yescharacteristic
62C2×#634C22×C311yescharacteristic
63C2×C3×Dic6—31no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4—21no—
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70C32×C22⋊C4—22no—
71C32×C22⋊C4—21no—
72C32×C22⋊C4—21no—
73#3757—21no—
74#3756—22no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76#3750—21no—
77C2×C3⋊(C3⋊C8)—22no—
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—61no—
81C3×#216—62no—
82C3×C4⋊Dic3—31no—
83C3×Dic3⋊C4—61no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—62no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)Q1612yescomplemented
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)—41no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3—41no—
98C2×C3×Q16—91no—
99C22×C3×C8—91no—
100C3×C22⋊Q8—31no—
101C3×C22⋊Q8—31no—
102C3×C22⋊Q8—61no—
103C3×C22⋊Q8—91no—
104C3×C22⋊Q8C3×D311yescharacteristic
105C2×#216—31no—
106C2×#216—91no—
107#924—31no—
108#924—91no—
109C22×C3⋊C8—31no—
110C22×C3⋊C8—91no—
111C3×#107—91no—
112C3×#103—91no—
113C3×#103—91no—
114#835—31no—
115#835—91no—
116#810—31no—
117#810—91no—
118#807—31no—
119#807—91no—
120C23×C32—21no—
121C23×C32#6711yescharacteristic
122C23×C32C3×D411yescharacteristic
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—41no—
128C32×Q8—41no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4#6711yescharacteristic
144C2×C32×C4C3×D411yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C23×C311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149#634—21no—
150#634—21no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)C3×D412yes—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—61no—
159#467—91noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)C2×Q1611yescharacteristic
163C2×H(3,𝔽3)#13511yescharacteristic
164C2×C3×Q8—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169C22×C3×C4—31no—
170C22×C3×C4—31no—
171C22×C3×C4—61no—
172C22×C3×C4C2×C3×D311yescharacteristic
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×Q16—182no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C2×C3×C8—182no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—61no—
184C3×C4⋊C4—62no—
185C3×C4⋊C4—61no—
186C3×C4⋊C4—91no—
187C3×C4⋊C4—181no—
188C3×C4⋊C4C2×C3×D311yescharacteristic
189C3×C22⋊C4—21no—
190C3×C22⋊C4—61no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—62no—
193C3×C22⋊C4—181no—
194#217—61no—
195#217—181no—
196#216—62no—
197#216—182no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200Dic3⋊C4—61no—
201Dic3⋊C4—181no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—62no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—182no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32#21612yescomplemented
217C22×C32C3×Q1612yescomplemented
218C22×C32C2×#6711yescharacteristic
219C22×C32C2×C3×D411yescharacteristic
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—42no—
226C32×C4—42no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4D3×D411yescharacteristic
232C32×C4#23711yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C32×C4C3×#4111yescharacteristic
235C3⋊Dic3—41no—
236C3⋊Dic3—41no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×Q16—91no—
240C22×C8—91no—
241C22⋊Q8—91no—
242C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
243#107—91no—
244#103—91no—
245#103—91no—
246H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3C3×#6711yescharacteristic
251C3×Q8—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—121no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—121no—
282C2×C3×C4—121no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C3×#6711yescharacteristic
286C2×C3×C4C22×C3×D311yescharacteristic
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—181no—
290C2×Dic3—181no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—92no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—92no—
300C3⋊C8—181no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—42no—
307C2×C32#93911yescharacteristic
308C2×C32C2×#21611yescharacteristic
309C2×C32#95011yescharacteristic
310C2×C32C3×#12111yescharacteristic
311C2×C32C2×C3×Q1611yescharacteristic
312C2×C32C3×#13511yescharacteristic
313C2×Q8—91no—
314C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
315C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
316Q16—182no—
317C2×C8—91no—
318C2×C8—91no—
319C2×C8—182no—
320C4⋊C4—21no—
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—181no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—32no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3C3×#21612yes—
341C22×C3C2×C3×#6711yescharacteristic
342C3×C4—21no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—121no—
354C3×C4—121no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—121no—
358C3×C4—122no—
359C3×C4—122no—
360C3×C4—361no—
361C3×C4—361no—
362C3×C4C3×D3×D411yescharacteristic
363C3×C4C3×#23711yescharacteristic
364Dic3—121no—
365Dic3—121no—
366Dic3—361no—
367Dic3—361no—
368C32—21no—
369C32#505111yescharacteristic, complemented
370C32C3×#46711yescharacteristic, complemented
371C23#638⋊C311yescharacteristic
372Q8—21no—
373Q8—21no—
374Q8—181no—
375Q8—181no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4#638⋊C311yescharacteristic
383C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
384C8—92no—
385C8—181no—
386C2×C3—22no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—62no—
394C2×C3—62no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—61no—
398C2×C3—122no—
399C2×C3C3×#93911yescharacteristic
400C2×C3C2×C3×#21611yescharacteristic
401C2×C3C3×#95011yescharacteristic
402C22—21no—
403C22—21no—
404C22#3756⋊C312yescomplemented
405C22C2×#638⋊C311yescharacteristic
406C4—21no—
407C4—41no—
408C4—41no—
409C4—361no—
410C4—361no—
411C4D4×(C3⋊D3)⋊C311yescharacteristic
412C4#3830⋊C311yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3C3×#505111yescharacteristic
417C2—22no—
418C2#64148⋊C311yescharacteristic
419C2C2×#3756⋊C311yescharacteristic
420C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC23×H(3,𝔽3)D4
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#3756⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #63637⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592750C3semidirect
26. C2×#3756C2×C3semidirect
32. #63662C2×C3semidirect
33. #63637C2×C3semidirect
34. #63634C2×C3semidirect
89. C22×H(3,𝔽3)Q16semidirect
216. C22×C32#216semidirect
217. C22×C32C3×Q16semidirect
246. H(3,𝔽3)#467semidirect
369. C32#5051semidirect
370. C32C3×#467semidirect
404. C22#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
C2×#3756⋊C32
#64133⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#63662⋊C32
#63637⋊C32
#63634⋊C32
#105927503
C3×#50513