Finite Group Databasebeta

← #13140906

Subgroups — #13140906

Order 1728.

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

normal subgroups only

This group has 609 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×#88722C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic, complemented
3C2×#88430C211yesmaximal, characteristic, complemented
4#11720583C211yesmaximal, characteristic, complemented
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591554—31nomaximal
9#10591550—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—22no—
12#88722C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊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×#3832D311yescharacteristic
25C2×C3×#239—31no—
26#63900—31no—
27D3×C4⋊Dic3—31no—
28C2×#3712—31no—
29C2×#3711—31no—
30#63662D311yescharacteristic, complemented
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—61no—
35#63564—31no—
36#63564—31no—
37#63563—31no—
38#63562—31no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41Q8×H(3,𝔽3)—21no—
42Q8×H(3,𝔽3)C2×C412yescomplemented
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×#1084—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—24no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
51C4×(C3⋊D3)⋊C3—24no—
52C4×(C3⋊D3)⋊C3C2×C412yes—
53C4×(C3⋊D3)⋊C3D412yes—
54((C3⋊D3)⋊C3)⋊C4—42no—
55#6180—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57#5232—91no—
58#4803—91no—
59#4803—91no—
60C2×C32×Q8—21no—
61C2×C32×Q8C2×D311yescharacteristic
62C2×C32×Q8C2×D311yescharacteristic
63#3832—22no—
64#3832Dic314yescomplemented
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C3×#239—34no—
70C3×#239—62no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
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)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—22no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3C22⋊C412yes—
98C4×H(3,𝔽3)—22no—
99C4×H(3,𝔽3)C22×C411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101#1084—41no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3C22⋊C412yes—
104C2×C3×#41—91no—
105C2×#239—31no—
106C2×#239—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×#103—91no—
114C3×#103—91no—
115#835—31no—
116#835—31no—
117#835—31no—
118#835—61no—
119#835—91no—
120#832—91no—
121#832—91no—
122#821—91no—
123#821—91no—
124#809—91no—
125#809—91no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3#6711yescharacteristic
128C22×C3×D3—31no—
129C22×C3×D3—61no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—22no—
133C32×Q8—41no—
134C32×Q8C4×D312yescomplemented
135C32×Q8C2×Dic312yescomplemented
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3#6711yescharacteristic
144C4⋊(C3⋊D3)—24no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)C2×Dic312yescomplemented
147C4×C3⋊D3—24no—
148C4×C3⋊D3C2×Dic312yes—
149C4×C3⋊D3#6712yes—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—121no—
155C3×D12—32no—
156C3×D12—64no—
157C3×D12—61no—
158C3×C4×D3—32no—
159C3×C4×D3—32no—
160C3×C4×D3—64no—
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)C2×C22⋊C411yescharacteristic
170C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
171C2×H(3,𝔽3)#13711yescharacteristic
172(C3⋊D3)⋊C3—22no—
173(C3⋊D3)⋊C3—42no—
174C3×#41—94no—
175C3×#41—182no—
176C2×C3×Q8—31no—
177C2×C3×Q8—31no—
178C2×C3×Q8—61no—
179C2×C3×Q8D3211yescharacteristic
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184C22×Dic3—181no—
185#239—34no—
186#239—62no—
187#239—94no—
188#239—182no—
189C2×D12—31no—
190C2×D12—61no—
191C2×D12—91no—
192C2×D12—181no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C2×C4×D3—91no—
196C2×C4×D3—181no—
197C2×C4×D3—181no—
198C2×C4×D3—181no—
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—21no—
220C22×C32D3×D411yescharacteristic
221C22×C32C2×#6711yescharacteristic
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3#21712yes—
228C2×C3×D3—32no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—62no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—42no—
239C32×C4C2×C4×D311yescharacteristic
240C32×C4D3×D411yescharacteristic
241C32×C4C22×Dic311yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—41no—
244C3⋊Dic3#21712yes—
245C3×Dic3—32no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C3×Dic3—241no—
250C2×#41—91no—
251C2×M16—271no—
252C2×C4⋊C4—271no—
253#103—91no—
254#103—91no—
255#102—271no—
256#102—271no—
257H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
258C23×C3—91no—
259C23×C3—181no—
260C22×D3—31no—
261C22×D3—61no—
262C22×D3—91no—
263C22×D3—181no—
264C3×Q8—21no—
265C3×Q8—32no—
266C3×Q8—32no—
267C3×Q8—61no—
268C3×Q8—61no—
269C3×Q8—62no—
270C3×Q8—121no—
271C3×Q8D3×Dic312yes—
272C3×D4—92no—
273C3×D4—181no—
274C3×D4—184no—
275C2×C3×C4—21no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—92no—
282C2×C3×C4—92no—
283C2×C3×C4—91no—
284C2×C3×C4—121no—
285C2×C3×C4—184no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4C2×D3211yescharacteristic
289C2×Dic3—31no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—91no—
294C2×Dic3—121no—
295C2×Dic3—181no—
296C2×Dic3—362no—
297D12—32no—
298D12—64no—
299D12—61no—
300D12—92no—
301D12—181no—
302D12—184no—
303C4×D3—32no—
304C4×D3—32no—
305C4×D3—64no—
306C4×D3—92no—
307C4×D3—92no—
308C4×D3—184no—
309C4×D3—362no—
310C4×D3—362no—
311C3×C8—181no—
312C3×C8—181no—
313C3⋊C8—61no—
314C3⋊C8—61no—
315C3⋊C8—61no—
316C3⋊C8—121no—
317C3⋊C8—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32D3×C22⋊C411yescharacteristic
322C2×C32#91411yescharacteristic
323C2×C32#91811yescharacteristic
324C2×C32#94911yescharacteristic
325C2×C32#95111yescharacteristic
326C2×C32C2×#21711yescharacteristic
327C3⋊D3—22no—
328C3⋊D3—42no—
329C3×D3—62no—
330C3×D3—122no—
331#41—94no—
332#41—182no—
333C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
334C2×D4—91no—
335C2×D4—181no—
336C22×C4—91no—
337C22×C4—181no—
338C22×C4—541no—
339M16—542no—
340C2×C8—91no—
341C2×C8—91no—
342C4⋊C4—91no—
343C4⋊C4—91no—
344C4⋊C4—541no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—61no—
348C22×C3—92no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—182no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3D3×#6711yescharacteristic
355C2×D3—32no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—62no—
359C2×D3—92no—
360C2×D3—121no—
361C2×D3—121no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—182no—
365C2×D3—361no—
366C2×D3—361no—
367C3×C4—22no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—62no—
373C3×C4—62no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—92no—
377C3×C4—122no—
378C3×C4—361no—
379C3×C4—361no—
380C3×C4C2×D3×Dic311yescharacteristic
381C3×C4D3×#6711yescharacteristic
382Dic3—32no—
383Dic3—92no—
384Dic3—121no—
385Dic3—121no—
386Dic3—121no—
387Dic3—241no—
388Dic3—361no—
389C32—21no—
390C32#480311yescharacteristic, complemented
391C32#523211yescharacteristic, complemented
392C23—91no—
393C23—181no—
394Q8—21no—
395Q8((C3⋊D3)⋊C3)⋊C412yescomplemented
396D4—92no—
397D4—184no—
398D4—181no—
399C2×C4—21no—
400C2×C4—91no—
401C2×C4—92no—
402C2×C4—92no—
403C2×C4—184no—
404C2×C4—181no—
405C2×C4—181no—
406C2×C4—1082no—
407C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
408C8—181no—
409C8—181no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—31no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—182no—
420C2×C3—362no—
421C2×C3#6393911yescharacteristic
422C2×C3#6394311yescharacteristic
423C2×C3D3×#21711yescharacteristic
424D3—62no—
425D3—122no—
426D3—182no—
427D3—362no—
428C22—92no—
429C22—181no—
430C22—181no—
431C22—182no—
432C22—361no—
433C22—361no—
434C22#8867611yescharacteristic
435C4—22no—
436C4—92no—
437C4—361no—
438C4—361no—
439C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
440C4#8867611yescharacteristic
441C3—31no—
442C3—31no—
443C3—61no—
444C3#1059155011yescharacteristic
445C2—182no—
446C2—362no—
447C2#1172123311yescharacteristic
448C2#1172123711yescharacteristic
449C2#1172131011yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88722C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
2. #11721129C2semidirect
3. C2×#88430C2semidirect
4. #11720583C2semidirect
5. #63662⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720308C2semidirect
12. #88722C4semidirect
30. #63662D3semidirect
42. Q8×H(3,𝔽3)C2×C4semidirect
50. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
64. #3832Dic3semidirect
134. C32×Q8C4×D3semidirect
135. C32×Q8C2×Dic3semidirect
146. C4⋊(C3⋊D3)C2×Dic3semidirect
257. H(3,𝔽3)#417semidirect
390. C32#4803semidirect
391. C32#5232semidirect
395. Q8((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×#887222
#117211292
C2×#884302
#117205832
#63662⋊C32
#117203182
#117203082
#105915543
#105915503