Finite Group Databasebeta

← #13138050

Subgroups — #13138050

Order 1728.

659 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88649C211yesmaximal, characteristic
2C4⋊C4×#272C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#63636⋊C3C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590609—31nomaximal
9#10590607—31nomaximal
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C2×#6200—21no—
15C2×((C3⋊D3)⋊C3)⋊C4—21no—
16#88649—22no—
17#88649C414yescomplemented
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88435—22no—
24C2×#3799—31no—
25C2×#3795—31no—
26C3×D3×C4⋊C4—31no—
27C3×D3×C4⋊C4—31no—
28C3×D3×C4⋊C4—61no—
29C2×#3711—31no—
30C2×#3711—31no—
31#63637D311yescharacteristic, complemented
32#63636D311yescharacteristic, complemented
33C3×#810—31no—
34C3×#809—31no—
35#63567—31no—
36#63565—31no—
37#63564—31no—
38#63562—31no—
39C22×#272D411yescharacteristic
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084D411yescharacteristic
45C4×#272—42no—
46C4×#272C2×C412yes—
47C4×#272D412yes—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
51#634⋊C3—21no—
52#634⋊C3C2×C412yescomplemented
53#6200—24no—
54((C3⋊D3)⋊C3)⋊C4—24no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57#4966—91no—
58#4803—91no—
59#4771—91no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3—62no—
69C2×C3×Dic6—31no—
70C2×#626—61no—
71C2×#625—61no—
72C2×#624—61no—
73C2×D3×Dic3—61no—
74#3799—34no—
75#3799—62no—
76#3795—34no—
77#3795—62no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4C2×D311yescharacteristic
80C32×C4⋊C4C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×Dic3⋊C4—61no—
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—62no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91#3711—62no—
92#3711—62no—
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C2×#272—21no—
95C2×#272—21no—
96C2×#272C22⋊C412yes—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—41no—
100C4×H(3,𝔽3)—41no—
101C4×H(3,𝔽3)C22×C411yescharacteristic
102C4×H(3,𝔽3)C2×D411yescharacteristic
103#1084—41no—
104#1084C22⋊C412yes—
105(C3⋊Dic3)⋊C3—22no—
106C2×#239—91no—
107C2×#237—91no—
108C2×#235—91no—
109C2×C3×C4⋊C4—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113D3×C4⋊C4—31no—
114D3×C4⋊C4—31no—
115D3×C4⋊C4—61no—
116C3×#103—91no—
117C3×#102—91no—
118#835—91no—
119#832—91no—
120#821—91no—
121#816—91no—
122#810—31no—
123#810—91no—
124#810—91no—
125#809—31no—
126#809—91no—
127#809—91no—
128C22×C3⋊D3—21no—
129C22×C3×D3—31no—
130C22×C3×D3—31no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C3⋊Dic3—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C4×D312yescomplemented
142#634—21no—
143#634C4×D312yescomplemented
144C2×C3×Dic3—31no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C3×D12—32no—
152C3×D12—61no—
153C3×C4×D3—32no—
154C3×C4×D3—32no—
155C3×C4×D3—32no—
156C3×C4×D3—32no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×C4×D3—124no—
160C3×C4×D3—122no—
161C3×C4×D3—122no—
162C3×Dic6—32no—
163C3×Dic6—61no—
164#626—64no—
165#625—64no—
166#624—64no—
167D3×Dic3—64no—
168C3⋊(C3⋊C8)—21no—
169C3⋊(C3⋊C8)—21no—
170C3×C3⋊C8—61no—
171C3×C3⋊C8—61no—
172#417—271noSylow
173C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
174C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
175C2×H(3,𝔽3)#13711yescharacteristic
176#272—22no—
177(C3⋊D3)⋊C3—42no—
178C2×C3×Q8—91no—
179C2×C3×D4—91no—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—181no—
183C2×#67—181no—
184C22×Dic3—181no—
185#239—94no—
186#239—182no—
187#237—94no—
188#237—182no—
189#235—94no—
190#235—182no—
191C2×D12—31no—
192C2×D12—91no—
193C2×D12—181no—
194C2×C4×D3—31no—
195C2×C4×D3—31no—
196C2×C4×D3—61no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×C4×D3—62no—
200C2×C4×D3—181no—
201C2×C4×D3—181no—
202C2×Dic6—31no—
203C2×Dic6—91no—
204C2×C3×C8—91no—
205C2×C3×C8—91no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—91no—
210C3×C4⋊C4—181no—
211C3×C4⋊C4D3211yescharacteristic
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—61no—
215Dic3⋊C4—61no—
216Dic3⋊C4—61no—
217Dic3⋊C4—62no—
218#208—182no—
219C2×C3⋊C8—31no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8—91no—
222C2×C3⋊C8—91no—
223ΓL(1,𝔽25)—182no—
224ΓL(1,𝔽25)—182no—
225C22×C32—21no—
226C22×C32D3×D411yescharacteristic
227C22×C32D3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3×D3—32no—
232C2×C3×D3—32no—
233C2×C3×D3—62no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—61no—
240C2×C3×D3—62no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—42no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4C2×C4×D311yescharacteristic
249C32×C4C2×C4×D311yescharacteristic
250C32×C4D3×D411yescharacteristic
251C32×C4D3×D411yescharacteristic
252C3⋊Dic3—22no—
253C3×Dic3—32no—
254C3×Dic3—32no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C3×Dic3—122no—
258C3×Dic3—121no—
259C3×Dic3—121no—
260C2×#41—271no—
261C2×M16—271no—
262C2×C4⋊C4—91no—
263#103—91no—
264#103—271no—
265#102—91no—
266#102—271no—
267H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
268C23×C3—91no—
269C23×C3—181no—
270C22×D3—31no—
271C22×D3—31no—
272C22×D3—61no—
273C22×D3—61no—
274C22×D3—181no—
275C3×Q8—92no—
276C3×Q8—181no—
277C3×D4—92no—
278C3×D4—181no—
279C2×C3×C4—21no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—62no—
285C2×C3×C4—61no—
286C2×C3×C4—91no—
287C2×C3×C4—92no—
288C2×C3×C4—92no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—362no—
292C2×C3×C4C2×D3211yescharacteristic
293#67—184no—
294#67—184no—
295C2×Dic3—31no—
296C2×Dic3—31no—
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—61no—
301C2×Dic3—62no—
302C2×Dic3—181no—
303C2×Dic3—184no—
304D12—32no—
305D12—61no—
306D12—92no—
307D12—181no—
308D12—184no—
309C4×D3—32no—
310C4×D3—32no—
311C4×D3—32no—
312C4×D3—32no—
313C4×D3—62no—
314C4×D3—62no—
315C4×D3—122no—
316C4×D3—122no—
317C4×D3—124no—
318C4×D3—184no—
319C4×D3—184no—
320Dic6—32no—
321Dic6—61no—
322Dic6—92no—
323Dic6—181no—
324C3×C8—181no—
325C3×C8—181no—
326C3⋊C8—61no—
327C3⋊C8—61no—
328C3⋊C8—181no—
329C3⋊C8—181no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32D3×C22⋊C411yescharacteristic
334C2×C32D3×C22⋊C411yescharacteristic
335C2×C32#91111yescharacteristic
336C2×C32#91411yescharacteristic
337C2×C32#91511yescharacteristic
338C2×C32#91811yescharacteristic
339C3⋊D3—42no—
340C3×D3—64no—
341C3×D3—62no—
342C3×D3—62no—
343C3×D3—122no—
344#41—274no—
345#41—542no—
346C2×Q8—91no—
347C2×D4—91no—
348C2×D4—541no—
349C22×C4—91no—
350C22×C4—181no—
351C22×C4—541no—
352M16—542no—
353C2×C8—91no—
354C2×C8—91no—
355C4⋊C4—91no—
356C4⋊C4—181no—
357C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
358C22×C3—31no—
359C22×C3—31no—
360C22×C3—61no—
361C22×C3—92no—
362C22×C3—182no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3—361no—
366C22×C3—361no—
367C22×C3#381111yescharacteristic
368C2×D3—32no—
369C2×D3—32no—
370C2×D3—62no—
371C2×D3—61no—
372C2×D3—61no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—62no—
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—182no—
381C2×D3—361no—
382C2×D3—361no—
383C3×C4—31no—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—31no—
387C3×C4—41no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—92no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—122no—
394C3×C4—182no—
395C3×C4—361no—
396C3×C4C4×D3211yescharacteristic
397C3×C4#380211yescharacteristic
398Dic3—32no—
399Dic3—32no—
400Dic3—62no—
401Dic3—62no—
402Dic3—121no—
403Dic3—121no—
404Dic3—122no—
405Dic3—182no—
406C32—21no—
407C32#477111yescharacteristic, complemented
408C32#480311yescharacteristic, complemented
409C23—91no—
410C23—181no—
411Q8—92no—
412Q8—181no—
413D4—92no—
414D4—181no—
415D4—544no—
416C2×C4—21no—
417C2×C4—91no—
418C2×C4—92no—
419C2×C4—92no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—362no—
423C2×C4—544no—
424C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C8—181no—
426C8—181no—
427C2×C3—31no—
428C2×C3—31no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—61no—
434C2×C3—61no—
435C2×C3—61no—
436C2×C3—182no—
437C2×C3—362no—
438C2×C3#6392111yescharacteristic
439C2×C3#6393711yescharacteristic
440C2×C3#6395111yescharacteristic
441D3—62no—
442D3—62no—
443D3—64no—
444D3—122no—
445D3—362no—
446C22—92no—
447C22—181no—
448C22—181no—
449C22—182no—
450C22—361no—
451C22—361no—
452C22#8867711yescharacteristic
453C4—41no—
454C4—92no—
455C4—182no—
456C4—361no—
457C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
458C4#8865511yescharacteristic
459C3—31no—
460C3—31no—
461C3—61no—
462C3#1059061211yescharacteristic
463C2—182no—
464C2—362no—
465C2#1172116111yescharacteristic
466C2#1172119311yescharacteristic
467C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88649C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#417
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×#272C2semidirect
3. C2×#88435C2semidirect
4. #63636⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
17. #88649C4semidirect
31. #63637D3semidirect
32. #63636D3semidirect
50. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
52. #634⋊C3C2×C4semidirect
141. C4⋊(C3⋊D3)C4×D3semidirect
143. #634C4×D3semidirect
267. H(3,𝔽3)#417semidirect
407. C32#4771semidirect
408. C32#4803semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C4⋊C4×#2722
C2×#884352
#63636⋊C32
#63637⋊C32
#117203092
#117203082
#105906093
#105906073