Finite Group Databasebeta

← #13140620

Subgroups — #13140620

Order 1728.

488 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88546C211yesmaximal, characteristic
2D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
3C2×#3754⋊C3C211yesmaximal, characteristic
4#11721090C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591497—31nomaximal
9#10591489—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14#88546—22no—
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3754⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88448—21no—
22#88448—21no—
23D4×C3⋊Dic3D311yescharacteristic
24C2×#3754D311yescharacteristic
25C3×D4×Dic3—31no—
26C2×C3×#214—31no—
27C2×C3×#214—31no—
28C2×C3×#214—31no—
29C2×C3×#214—61no—
30#63873—31no—
31#63866—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63567—31no—
35#63566—31no—
36#63566—31no—
37#63565—31no—
38C23×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084—21no—
43#6229—21no—
44#6229—21no—
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—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5162—91no—
55#5162—91no—
56#5019—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C22×C3⋊Dic3—21no—
61C2×#634C2×D311yescharacteristic
62C22×C3×Dic3—61no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3757—21no—
68#3754—22no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#217—61no—
73C3×#214—64no—
74C3×#214—62no—
75C3×#214—62no—
76C3×#214—62no—
77C3×C4⋊Dic3—31no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83#3724—61no—
84#3724—61no—
85Dic3⋊Dic3—61no—
86Dic3⋊Dic3—61no—
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)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93#1084—41no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1612yescomplemented
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C3×C4×D4—91no—
101#944—91no—
102#944—91no—
103D4×Dic3—31no—
104D4×Dic3—91no—
105C2×#214—31no—
106C2×#214—31no—
107C2×#214—31no—
108C2×#214—61no—
109C2×#214—91no—
110#869—91no—
111#835—91no—
112#835—91no—
113#816—91no—
114#816—91no—
115Dic3⋊C8—91no—
116Dic3⋊C8—91no—
117#810—91no—
118#810—91no—
119C4⋊(C3⋊C8)—91no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—41no—
123C32×D4—21no—
124C32×D4—21no—
125C32×D4—22no—
126C32×D4—41no—
127C32×D4D1212yes—
128C32×D4#6712yes—
129C2×C32×C4—21no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—42no—
135C2×C3⋊Dic3#6711yescharacteristic
136C2×C3⋊Dic3#6711yescharacteristic
137#634—21no—
138#634—21no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C2×C3×Dic3—122no—
147C3×Dic6—62no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—121no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160#461—271noSylow
161C2×H(3,𝔽3)—42no—
162C2×H(3,𝔽3)#12111yescharacteristic
163C2×H(3,𝔽3)C2×SD1611yescharacteristic
164C2×H(3,𝔽3)#13711yescharacteristic
165C2×C3×Q8—91no—
166C2×C3×Q8—91no—
167C2×C3×D4—31no—
168C2×C3×D4—31no—
169C2×C3×D4—61no—
170C2×C3×D4D3211yescharacteristic
171C22×C3×C4—181no—
172C22×Dic3—61no—
173C22×Dic3—181no—
174C2×Dic6—31no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C2×Dic6—61no—
178C2×Dic6—91no—
179C3×SD16—182no—
180C3×SD16—182no—
181C2×C3×C8—91no—
182C2×C3×C8—91no—
183C3×C4⋊C4—91no—
184C3×C22⋊C4—181no—
185C3×C42—91no—
186#217—61no—
187#217—181no—
188#214—62no—
189#214—62no—
190#214—64no—
191#214—62no—
192#214—182no—
193C4⋊Dic3—31no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196C4⋊Dic3—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—181no—
199Dic3⋊C4—181no—
200Dic3⋊C4—181no—
201C4×Dic3—31no—
202C4×Dic3—91no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—61no—
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×#6711yescharacteristic
220C32×C4—21no—
221C32×C4—21no—
222C32×C4C2×D1211yescharacteristic
223C32×C4#23511yescharacteristic
224C32×C4#23711yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C3⋊Dic3—21no—
227C3⋊Dic3—41no—
228C3⋊Dic3—41no—
229C3⋊Dic3#21412yescomplemented
230C3×Dic3—32no—
231C3×Dic3—61no—
232C3×Dic3—122no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C2×SD16—91no—
238C2×SD16—91no—
239C4⋊Q8—271no—
240C4×D4—91no—
241C4⋊C8—271no—
242#103—271no—
243#103—271no—
244H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
245C23×C3—21no—
246C23×C3—61no—
247C23×C3—61no—
248C23×C3—121no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×Q8—181no—
253C3×D4—21no—
254C3×D4—32no—
255C3×D4—32no—
256C3×D4—61no—
257C3×D4—61no—
258C3×D4—62no—
259C3×D4—121no—
260C3×D4#62612yes—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4—362no—
270C2×C3×C4C2×D3211yescharacteristic
271C2×Dic3—31no—
272C2×Dic3—31no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—91no—
278C2×Dic3—91no—
279C2×Dic3—121no—
280C2×Dic3—122no—
281C2×Dic3—181no—
282C2×Dic3—181no—
283C2×Dic3—362no—
284Dic6—61no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—62no—
289Dic6—61no—
290Dic6—61no—
291Dic6—121no—
292Dic6—181no—
293Dic6—181no—
294C3×C8—181no—
295C3×C8—181no—
296C3⋊C8—61no—
297C3⋊C8—61no—
298C3⋊C8—62no—
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#93011yescharacteristic
310C2×C32C2×#21411yescharacteristic
311C2×C32#93911yescharacteristic
312C2×C32#95111yescharacteristic
313C2×Q8—91no—
314C2×Q8—91no—
315C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
316C22×C4—181no—
317SD16—182no—
318SD16—182no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—91no—
322C4⋊C4—541no—
323C4⋊C4—541no—
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×C3D3×#6711yescharacteristic
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—92no—
349C3×C4—181no—
350C3×C4—361no—
351C3×C4—361no—
352C3×C4—361no—
353C3×C4#380411yescharacteristic
354C3×C4C2×#62611yescharacteristic
355Dic3—32no—
356Dic3—61no—
357Dic3—92no—
358Dic3—121no—
359Dic3—121no—
360Dic3—122no—
361Dic3—121no—
362Dic3—121no—
363Dic3—181no—
364Dic3—361no—
365Dic3—361no—
366C32—21no—
367C32#501911yescharacteristic, complemented
368C32#516211yescharacteristic, complemented
369C23—21no—
370Q8—181no—
371Q8—181no—
372Q8—181no—
373Q8—181no—
374D4—21no—
375D4#620012yes—
376C2×C4—91no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—362no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—181no—
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×C3D3×#21411yescharacteristic
399C2×C3#6392911yescharacteristic
400C2×C3#6397811yescharacteristic
401C22—22no—
402C22—41no—
403C22—41no—
404C22#8867611yescharacteristic
405C4—92no—
406C4—181no—
407C4—361no—
408C4—361no—
409C4—361no—
410C4#8867111yescharacteristic
411C4C2×#620011yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059148911yescharacteristic
416C2—42no—
417C2#1172119811yescharacteristic
418C2#1172119911yescharacteristic
419C2#1172132211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
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
4. #11721090C2semidirect
5. #11720326C2semidirect
6. #11720313C2semidirect
7. #11720311C2semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
229. C3⋊Dic3#214semidirect
244. H(3,𝔽3)#461semidirect
367. C32#5019semidirect
368. C32#5162semidirect

Maximal subgroups

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

subgroupindex
C2×#885462
D4×(C3⋊Dic3)⋊C32
C2×#3754⋊C32
#117210902
#117203262
#117203132
#117203112
#105914973
#105914893