Finite Group Databasebeta

← #13140560

Subgroups — #13140560

Order 1728.

484 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#64147⋊C3C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic, complemented
3#11721106C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591465—31nomaximal
9#10591460—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3754⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88449—21no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64147D311yescharacteristic
24C2×#3754D311yescharacteristic, complemented
25C3×#938—31no—
26C2×C3×#214—31no—
27#63871—31no—
28#63857—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—61no—
34#63574—31no—
35#63573—31no—
36#63567—31no—
37#63566—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#638⋊C3—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4×(C3⋊D3)⋊C3—21no—
49C4×(C3⋊D3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5171—91no—
55#5041—91no—
56#4778—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×#638—21no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×#634C2×D311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×C4×D3—31no—
65C2×C3×Dic6—31no—
66#3757—21no—
67#3754—22no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#217—61no—
71C3×#214—62no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80Dic3⋊Dic3—61no—
81Dic3⋊Dic3—61no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84#3710—62no—
85D3×C3⋊C8—62no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×(C3⋊D3)⋊C3—21no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3SD1612yescomplemented
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—41no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C3×#121—91no—
101#946—91no—
102#938—31no—
103#938—91no—
104C2×#214—31no—
105C2×#214—91no—
106#924—91no—
107C22×C3⋊C8—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110#897—91no—
111C3×#102—91no—
112#835—91no—
113#832—31no—
114#832—31no—
115#832—61no—
116#817—91no—
117#816—91no—
118#810—91no—
119#808—91no—
120#808—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3#6711yescharacteristic
125C22×C3×D3—31no—
126C32×D4—22no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—41no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135#638—42no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4×C3⋊D3—21no—
140C4×C3⋊D3—21no—
141#634—21no—
142#634—21no—
143C3×#67—122no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—121no—
150C3×C4×D3—61no—
151C3×C4×D3—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3⋊(C3⋊C8)#6712yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159#465—271noSylow
160C2×H(3,𝔽3)—81no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)C2×SD1611yescharacteristic
163C2×H(3,𝔽3)#13511yescharacteristic
164(C3⋊D3)⋊C3—22no—
165C2×C3×Q8—91no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4—61no—
169C2×C3×D4—181no—
170C2×C3×D4D3211yescharacteristic
171C22×C3×C4—91no—
172C2×#67—61no—
173C2×#67—181no—
174C2×C4×D3—31no—
175C2×C4×D3—91no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×SD16—182no—
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C3×C4⋊C4—91no—
182C3×C4⋊C4—91no—
183C3×C22⋊C4—181no—
184#217—61no—
185#217—181no—
186#217—181no—
187#214—62no—
188#214—182no—
189(C2×D3)⋊C4—181no—
190(C2×D3)⋊C4—181no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—61no—
195C4⋊Dic3—91no—
196C4⋊Dic3—181no—
197Dic3⋊C4—181no—
198Dic3⋊C4—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—182no—
205D3×C8—182no—
206D3×C8—182no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3#21412yescomplemented
222C2×C3×D3—32no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4D3×D411yescharacteristic
228C32×C4#23711yescharacteristic
229C32×C4#23711yescharacteristic
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—21no—
232C3⋊Dic3—41no—
233C3⋊Dic3—41no—
234C3×Dic3—61no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C3×Dic3—241no—
240C2×SD16—91no—
241C22×C8—271no—
242C22⋊Q8—271no—
243#121—91no—
244#106—271no—
245#103—271no—
246#102—91no—
247H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—91no—
252C23×C3—121no—
253C22×D3—31no—
254C22×D3—91no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—62no—
264C3×D4—121no—
265C3×D4—362no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C2×D3211yescharacteristic
276#67—122no—
277#67—362no—
278C2×Dic3—31no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—91no—
284C2×Dic3—121no—
285C2×Dic3—181no—
286C2×Dic3—181no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—181no—
290C4×D3—181no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—92no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—92no—
301C3⋊C8—121no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—82no—
306C2×C32—81no—
307C2×C32—81no—
308C2×C32D3×SD1611yescharacteristic
309C2×C32#91611yescharacteristic
310C2×C32C2×#21411yescharacteristic
311C2×C32#93811yescharacteristic
312C2×C32#93911yescharacteristic
313C2×C32#95011yescharacteristic
314C3⋊D3—22no—
315C3×D3—62no—
316C2×Q8—91no—
317C2×D4—181no—
318C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C22×C4—91no—
320SD16—182no—
321C2×C8—91no—
322C2×C8—91no—
323C2×C8—542no—
324C4⋊C4—91no—
325C4⋊C4—91no—
326C4⋊C4—541no—
327C22⋊C4—181no—
328C22⋊C4—541no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—61no—
335C22×C3—92no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—121no—
342C22×C3—122no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—241no—
346C22×C3—241no—
347C22×C3D3×#6711yescharacteristic
348C2×D3—32no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—92no—
352C2×D3—181no—
353C2×D3—181no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—181no—
361C3×C4—361no—
362C3×C4—361no—
363C3×C4—361no—
364C3×C4#380511yescharacteristic
365C3×C4D3×#6711yescharacteristic
366Dic3—61no—
367Dic3—121no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—181no—
372Dic3—241no—
373Dic3—361no—
374Dic3—361no—
375C32—21no—
376C32#504111yescharacteristic, complemented
377C32#517111yescharacteristic, complemented
378C23—21no—
379C23—91no—
380Q8—181no—
381Q8—181no—
382D4—21no—
383D4—21no—
384D4—362no—
385C2×C4—91no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C8—92no—
393C8—181no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—81no—
404C2×C3—182no—
405C2×C3—241no—
406C2×C3—241no—
407C2×C3—242no—
408C2×C3D3×#21411yescharacteristic
409C2×C3#6393511yescharacteristic
410C2×C3#6397011yescharacteristic
411D3—62no—
412D3—182no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—92no—
417C22—181no—
418C22—181no—
419C22#8867611yescharacteristic
420C4—181no—
421C4—361no—
422C4—361no—
423C4—361no—
424C4#8867211yescharacteristic
425C4#8867611yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3#1059149911yescharacteristic
430C2—81no—
431C2—182no—
432C2#1172119711yescharacteristic
433C2#1172119811yescharacteristic
434C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#64147⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
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
2. C2×#3754⋊C3C2semidirect
3. #11721106C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #11720578C2semidirect
6. #11720313C2semidirect
7. #11720307C2semidirect
24. C2×#3754D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
92. C2×(C3⋊D3)⋊C3SD16semidirect
154. C3⋊(C3⋊C8)#67semidirect
221. C2×C3⋊D3#214semidirect
247. H(3,𝔽3)#465semidirect
376. C32#5041semidirect
377. C32#5171semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
C2×#3754⋊C32
#117211062
C2×((C3⋊D3)⋊C3)⋊C82
#117205782
#117203132
#117203072
#105914653
#105914603