Finite Group Databasebeta

← #13140546

Subgroups — #13140546

Order 1728.

617 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3830⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic, complemented
3C2×#88430C211yesmaximal, characteristic, complemented
4#11720578C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591418—31nomaximal
9#10591414—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C22×(C3⋊Dic3)⋊C3—21no—
13#3830⋊C3—22no—
14#3830⋊C3C414yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×((C3⋊D3)⋊C3)⋊C4—21no—
18C4⋊#1084C2211yescharacteristic
19C2×#6180C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88448—21no—
23#88430—22no—
24C2×#3830D311yescharacteristic
25C2×C3×#237—31no—
26#63900—31no—
27D3×C4⋊Dic3—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63659D311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—31no—
34C3×#832—61no—
35#63567—31no—
36#63567—31no—
37#63566—31no—
38#63565—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C412yescomplemented
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45#638⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3D412yes—
52#634⋊C3—21no—
53#634⋊C3C2×C412yescomplemented
54((C3⋊D3)⋊C3)⋊C4—42no—
55#6180—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57#5232—91no—
58#4771—91no—
59#4771—91no—
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C2×D311yescharacteristic
63C2×#638—21no—
64C22×C3⋊Dic3—21no—
65#3830—22no—
66#3830Dic314yescomplemented
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634C2×D311yescharacteristic
69C2×C3×#67—61no—
70C22×C3×Dic3—61no—
71C3×#237—34no—
72C3×#237—62no—
73C2×C3×C4×D3—31no—
74C2×C3×Dic6—31no—
75C2×#624—61no—
76C2×D3×Dic3—61no—
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—61no—
87#3724—61no—
88Dic3⋊Dic3—61no—
89#3712—62no—
90#3711—62no—
91C22×H(3,𝔽3)—22no—
92C22×H(3,𝔽3)—41no—
93C22×H(3,𝔽3)—41no—
94C22×H(3,𝔽3)C2×D411yescharacteristic
95C2×(C3⋊D3)⋊C3—21no—
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3C22⋊C412yes—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084—41no—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3C22⋊C412yes—
104C2×C3×#41—91no—
105C2×#237—31no—
106C2×#237—91no—
107C2×C4⋊Dic3—91no—
108C2×#208—91no—
109C2×ΓL(1,𝔽25)—91no—
110C2×ΓL(1,𝔽25)—91no—
111D3×C4⋊C4—91no—
112D3×C4⋊C4—91no—
113C3×#102—91no—
114C3×#102—91no—
115#835—91no—
116#835—91no—
117#832—31no—
118#832—31no—
119#832—31no—
120#832—61no—
121#832—91no—
122#816—91no—
123#816—91no—
124#810—91no—
125#810—91no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—41no—
129C22×C3⋊D3#6711yescharacteristic
130C22×C3×D3—31no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—22no—
134C32×D4—41no—
135C32×D4C4×D312yescomplemented
136C32×D4C2×Dic312yescomplemented
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140#638—24no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—24no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3#6711yescharacteristic
145C4×C3⋊D3C2×Dic312yes—
146C4×C3⋊D3#6712yes—
147#634—21no—
148#634C2×Dic312yescomplemented
149C3×#67—64no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—121no—
157C3×C4×D3—32no—
158C3×C4×D3—32no—
159C3×Dic6—32no—
160C3×Dic6—61no—
161#624—122no—
162D3×Dic3—122no—
163C3⋊(C3⋊C8)—21no—
164C3×C3⋊C8—61no—
165C3×C3⋊C8—61no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—121no—
168#417—271noSylow
169C2×H(3,𝔽3)—42no—
170C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
171C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
172C2×H(3,𝔽3)#13711yescharacteristic
173(C3⋊D3)⋊C3—22no—
174C3×#41—94no—
175C3×#41—182no—
176C2×C3×Q8—91no—
177C2×C3×D4—31no—
178C2×C3×D4—31no—
179C2×C3×D4—61no—
180C2×C3×D4—181no—
181C2×C3×D4D3211yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184C2×#67—61no—
185C2×#67—181no—
186C22×Dic3—61no—
187C22×Dic3—181no—
188C22×Dic3—181no—
189#237—34no—
190#237—62no—
191#237—94no—
192#237—182no—
193C2×C4×D3—31no—
194C2×C4×D3—91no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×Dic6—31no—
198C2×Dic6—91no—
199C2×C3×C8—91no—
200C2×C3×C8—91no—
201C3×C4⋊C4—91no—
202C3×C4⋊C4—91no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—61no—
207C4⋊Dic3—91no—
208C4⋊Dic3—181no—
209Dic3⋊C4—181no—
210Dic3⋊C4—181no—
211#208—182no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—31no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—61no—
216C2×C3⋊C8—91no—
217ΓL(1,𝔽25)—182no—
218ΓL(1,𝔽25)—182no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—81no—
228C22×C32—81no—
229C22×C32D3×D411yescharacteristic
230C22×C32C2×#6711yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—21no—
233C2×C3⋊D3#21712yes—
234C2×C3×D3—32no—
235C2×C3×D3—61no—
236C2×C3×D3—61no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4C2×C4×D311yescharacteristic
240C32×C4D3×D411yescharacteristic
241C32×C4C22×Dic311yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—22no—
244C3⋊Dic3—41no—
245C3⋊Dic3#21712yes—
246C3×Dic3—32no—
247C3×Dic3—62no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—241no—
252C2×#41—91no—
253C2×M16—271no—
254C2×C4⋊C4—271no—
255#103—271no—
256#103—271no—
257#102—91no—
258#102—91no—
259H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—91no—
264C23×C3—121no—
265C22×D3—31no—
266C22×D3—91no—
267C3×Q8—92no—
268C3×Q8—181no—
269C3×D4—21no—
270C3×D4—32no—
271C3×D4—32no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—62no—
275C3×D4—121no—
276C3×D4—184no—
277C3×D4D3×Dic312yes—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—61no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—91no—
284C2×C3×C4—184no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4C2×D3211yescharacteristic
289#67—64no—
290#67—184no—
291C2×Dic3—31no—
292C2×Dic3—64no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—91no—
298C2×Dic3—121no—
299C2×Dic3—184no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302C2×Dic3—362no—
303C4×D3—32no—
304C4×D3—32no—
305C4×D3—92no—
306C4×D3—92no—
307C4×D3—362no—
308C4×D3—362no—
309Dic6—32no—
310Dic6—61no—
311Dic6—92no—
312Dic6—181no—
313C3×C8—181no—
314C3×C8—181no—
315C3⋊C8—61no—
316C3⋊C8—61no—
317C3⋊C8—61no—
318C3⋊C8—121no—
319C3⋊C8—181no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—42no—
324C2×C32—42no—
325C2×C32—82no—
326C2×C32D3×C22⋊C411yescharacteristic
327C2×C32#91111yescharacteristic
328C2×C32#91511yescharacteristic
329C2×C32#94911yescharacteristic
330C2×C32#95111yescharacteristic
331C2×C32C2×#21711yescharacteristic
332C3⋊D3—22no—
333C3×D3—62no—
334#41—94no—
335#41—182no—
336C2×Q8—91no—
337C2×D4—181no—
338C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
339C22×C4—91no—
340C22×C4—181no—
341C22×C4—541no—
342M16—542no—
343C2×C8—91no—
344C2×C8—91no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C4⋊C4—541no—
348C22×C3—22no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—92no—
357C22×C3—122no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3—241no—
365C22×C3—241no—
366C22×C3D3×#6711yescharacteristic
367C2×D3—32no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—92no—
371C2×D3—181no—
372C2×D3—181no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—61no—
378C3×C4—61no—
379C3×C4—92no—
380C3×C4—182no—
381C3×C4—361no—
382C3×C4—361no—
383C3×C4C2×D3×Dic311yescharacteristic
384C3×C4D3×#6711yescharacteristic
385Dic3—32no—
386Dic3—62no—
387Dic3—92no—
388Dic3—121no—
389Dic3—121no—
390Dic3—121no—
391Dic3—182no—
392Dic3—241no—
393Dic3—361no—
394C32—21no—
395C32#477111yescharacteristic, complemented
396C32#523211yescharacteristic, complemented
397C23—21no—
398C23—91no—
399Q8—92no—
400Q8—181no—
401D4—21no—
402D4—184no—
403D4((C3⋊D3)⋊C3)⋊C412yescomplemented
404C2×C4—91no—
405C2×C4—92no—
406C2×C4—92no—
407C2×C4—184no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—1082no—
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C8—181no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—42no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—122no—
426C2×C3—122no—
427C2×C3—182no—
428C2×C3—242no—
429C2×C3#6392511yescharacteristic
430C2×C3#6392911yescharacteristic
431C2×C3D3×#21711yescharacteristic
432D3—62no—
433D3—182no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—92no—
438C22—181no—
439C22—181no—
440C22#8867611yescharacteristic
441C4—92no—
442C4—182no—
443C4—361no—
444C4—361no—
445C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
446C4#8867611yescharacteristic
447C3—31no—
448C3—31no—
449C3—61no—
450C3#1059141411yescharacteristic
451C2—42no—
452C2—182no—
453C2#1172119511yescharacteristic
454C2#1172119911yescharacteristic
455C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#3830⋊C3C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#417
LCSG ▷ 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. #11721129C2semidirect
3. C2×#88430C2semidirect
4. #11720578C2semidirect
5. #63659⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720309C2semidirect
14. #3830⋊C3C4semidirect
30. #63659D3semidirect
42. D4×H(3,𝔽3)C2×C4semidirect
53. #634⋊C3C2×C4semidirect
66. #3830Dic3semidirect
135. C32×D4C4×D3semidirect
136. C32×D4C2×Dic3semidirect
148. #634C2×Dic3semidirect
259. H(3,𝔽3)#417semidirect
395. C32#4771semidirect
396. C32#5232semidirect
403. D4((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
#117211292
C2×#884302
#117205782
#63659⋊C32
#117203132
#117203092
#105914183
#105914143