Finite Group Databasebeta

← #13138049

Subgroups — #13138049

Order 1728.

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

normal subgroups only

This group has 629 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
2#11721760C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63636⋊C3C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10590610—31nomaximal
9#10590608—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C2×((C3⋊D3)⋊C3)⋊C4—21no—
15#88649—22no—
16#88649C414yescomplemented
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18#88542—21no—
19C4⋊#1084C2211yescharacteristic
20C4×#1084—21no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#88434—22no—
24C2×#3799—31no—
25C2×#3795—31no—
26C3×#892—31no—
27C3×#892—31no—
28C3×#892—61no—
29C2×D3×C3⋊C8—31no—
30C2×D3×C3⋊C8—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×#272C2×C412yes—
46C4×#272D412yes—
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2×C412yes—
50#634⋊C3—21no—
51#634⋊C3C2×C412yes—
52#6200—24no—
53((C3⋊D3)⋊C3)⋊C4—24no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5002—91no—
57#4804—91no—
58#4772—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C2×#626—61no—
67C2×#625—61no—
68C2×#624—61no—
69C2×D3×Dic3—61no—
70#3799—34no—
71#3799—62no—
72#3795—34no—
73#3795—62no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×(C2×D3)⋊C4—61no—
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—62no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4×Dic3—61no—
86C3×C4×Dic3—61no—
87C3×C4×Dic3—62no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90D3×C3⋊C8—62no—
91D3×C3⋊C8—62no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—21no—
94C2×#272—21no—
95C2×#272C22⋊C412yes—
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—41no—
99C4×H(3,𝔽3)—41no—
100C4×H(3,𝔽3)C22×C411yescharacteristic
101C4×H(3,𝔽3)C2×D411yescharacteristic
102#1084—41no—
103#1084C22⋊C412yes—
104(C3⋊Dic3)⋊C3—22no—
105C2×#239—91no—
106C2×#237—91no—
107C2×#235—91no—
108C3×#117—91no—
109C22×C3⋊C8—91no—
110C2×D3×C8—91no—
111C2×D3×C8—91no—
112#892—31no—
113#892—31no—
114#892—61no—
115C3×#103—91no—
116C3×#102—91no—
117#835—91no—
118#832—91no—
119#821—91no—
120#816—91no—
121#810—31no—
122#810—91no—
123#810—91no—
124#809—31no—
125#809—91no—
126#809—91no—
127C22×C3⋊D3—21no—
128C22×C3×D3—31no—
129C22×C3×D3—31no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C2×C32×C4—22no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—21no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C4×D312yes—
141#634—21no—
142#634C4×D312yes—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C3×D12—32no—
151C3×D12—61no—
152C3×C4×D3—32no—
153C3×C4×D3—32no—
154C3×C4×D3—32no—
155C3×C4×D3—32no—
156C3×C4×D3—62no—
157C3×C4×D3—62no—
158C3×Dic6—32no—
159C3×Dic6—61no—
160#626—64no—
161#625—64no—
162#624—64no—
163D3×Dic3—64no—
164C3⋊(C3⋊C8)—21no—
165C3⋊(C3⋊C8)—21no—
166C3×C3⋊C8—61no—
167C3×C3⋊C8—61no—
168#416—271noSylow
169C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
170C2×H(3,𝔽3)#13511yescharacteristic
171C2×H(3,𝔽3)#13511yescharacteristic
172#272—22no—
173(C3⋊D3)⋊C3—42no—
174C2×C3×Q8—91no—
175C2×C3×D4—91no—
176C22×C3×C4—91no—
177C2×#67—181no—
178C2×#67—181no—
179C22×Dic3—181no—
180#239—94no—
181#239—182no—
182#237—94no—
183#237—182no—
184#235—94no—
185#235—182no—
186C2×D12—31no—
187C2×D12—91no—
188C2×D12—181no—
189C2×C4×D3—31no—
190C2×C4×D3—31no—
191C2×C4×D3—61no—
192C2×C4×D3—181no—
193C2×C4×D3—181no—
194C2×Dic6—31no—
195C2×Dic6—91no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—91no—
202C3×C4⋊C4D3211yescharacteristic
203C3×C22⋊C4—181no—
204C3×C42—181no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—62no—
208C4⋊Dic3—31no—
209C4⋊Dic3—31no—
210C4⋊Dic3—61no—
211C4×Dic3—61no—
212C4×Dic3—61no—
213C4×Dic3—62no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218C2×C3⋊C8—182no—
219D3×C8—182no—
220D3×C8—182no—
221C22×C32—21no—
222C22×C32D3×D411yescharacteristic
223C22×C32D3×D411yescharacteristic
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3×D3—32no—
228C2×C3×D3—32no—
229C2×C3×D3—62no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—62no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—42no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4C2×C4×D311yescharacteristic
245C32×C4C2×C4×D311yescharacteristic
246C32×C4D3×D411yescharacteristic
247C32×C4D3×D411yescharacteristic
248C3⋊Dic3—22no—
249C3×Dic3—32no—
250C3×Dic3—32no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—122no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C2×#41—271no—
257C22×C8—271no—
258#117—91no—
259#103—91no—
260#103—271no—
261#102—91no—
262#102—271no—
263H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
264C23×C3—91no—
265C23×C3—181no—
266C22×D3—31no—
267C22×D3—31no—
268C22×D3—61no—
269C22×D3—61no—
270C22×D3—181no—
271C3×Q8—92no—
272C3×Q8—181no—
273C3×D4—92no—
274C3×D4—181no—
275C2×C3×C4—21no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—92no—
284C2×C3×C4—92no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4C2×D3211yescharacteristic
288#67—184no—
289#67—184no—
290C2×Dic3—31no—
291C2×Dic3—31no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—62no—
297C2×Dic3—181no—
298C2×Dic3—184no—
299D12—32no—
300D12—61no—
301D12—92no—
302D12—181no—
303D12—184no—
304C4×D3—32no—
305C4×D3—32no—
306C4×D3—32no—
307C4×D3—32no—
308C4×D3—62no—
309C4×D3—62no—
310C4×D3—184no—
311C4×D3—184no—
312Dic6—32no—
313Dic6—61no—
314Dic6—92no—
315Dic6—181no—
316C3×C8—181no—
317C3×C8—181no—
318C3⋊C8—61no—
319C3⋊C8—61no—
320C3⋊C8—181no—
321C3⋊C8—181no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32D3×C22⋊C411yescharacteristic
326C2×C32D3×C22⋊C411yescharacteristic
327C2×C32#91211yescharacteristic
328C2×C32#91611yescharacteristic
329C2×C32#91611yescharacteristic
330C2×C32#91911yescharacteristic
331C3⋊D3—42no—
332C3×D3—62no—
333C3×D3—62no—
334C3×D3—122no—
335C3×D3—122no—
336#41—274no—
337#41—542no—
338C2×Q8—91no—
339C2×D4—91no—
340C2×D4—541no—
341C22×C4—91no—
342C22×C4—541no—
343C2×C8—91no—
344C2×C8—91no—
345C2×C8—542no—
346C4⋊C4—91no—
347C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
348C22⋊C4—181no—
349C42—181no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—92no—
354C22×C3—182no—
355C22×C3—181no—
356C22×C3—181no—
357C22×C3—361no—
358C22×C3—361no—
359C22×C3#381111yescharacteristic
360C2×D3—32no—
361C2×D3—32no—
362C2×D3—62no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—62no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—182no—
373C2×D3—361no—
374C2×D3—361no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—41no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—92no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—122no—
386C3×C4—182no—
387C3×C4—361no—
388C3×C4C4×D3211yescharacteristic
389C3×C4#380211yescharacteristic
390Dic3—32no—
391Dic3—32no—
392Dic3—62no—
393Dic3—62no—
394Dic3—121no—
395Dic3—121no—
396Dic3—122no—
397Dic3—182no—
398C32—21no—
399C32#477211yescharacteristic, complemented
400C32#480411yescharacteristic, complemented
401C23—91no—
402C23—181no—
403Q8—92no—
404Q8—181no—
405D4—92no—
406D4—181no—
407D4—544no—
408C2×C4—21no—
409C2×C4—91no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—544no—
415C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
416C8—181no—
417C8—181no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—31no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—182no—
428C2×C3—362no—
429C2×C3#6392111yescharacteristic
430C2×C3#6393611yescharacteristic
431C2×C3#6395011yescharacteristic
432D3—62no—
433D3—62no—
434D3—122no—
435D3—122no—
436D3—362no—
437C22—92no—
438C22—181no—
439C22—181no—
440C22—182no—
441C22—361no—
442C22—361no—
443C22#8867711yescharacteristic
444C4—41no—
445C4—92no—
446C4—182no—
447C4—361no—
448C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
449C4#8865511yescharacteristic
450C3—31no—
451C3—31no—
452C3—61no—
453C3#1059061111yescharacteristic
454C2—182no—
455C2—362no—
456C2#1172116111yescharacteristic
457C2#1172119211yescharacteristic
458C2#1172123011yescharacteristic
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)#416
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. #11721760C2semidirect
3. C2×#88434C2semidirect
4. #63636⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720308C2semidirect
16. #88649C4semidirect
31. #63637D3semidirect
32. #63636D3semidirect
263. H(3,𝔽3)#416semidirect
399. C32#4772semidirect
400. C32#4804semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
#117217602
C2×#884342
#63636⋊C32
#63637⋊C32
#117203092
#117203082
#105906103
#105906083