Finite Group Databasebeta

← #13140601

Subgroups — #13140601

Order 1728.

726 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 69 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#11722664C211yesmaximal, characteristic
2#64144⋊C3C211yesmaximal, characteristic, complemented
3#11721301C212yesmaximal, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#11720335C212yesmaximal, complemented
6#10591534—31nomaximal
7#10591533—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C2×#6235—21no—
10C2×#6235C2212yes—
11#88744—22no—
12D4×#272—22no—
13#88740—21no—
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
15#3757⋊C3—21no—
16#3757⋊C3C2212yescomplemented
17#3757⋊C3C2211yescharacteristic, complemented
18#3750⋊C3—21no—
19#88451—22no—
20#88448—22no—
21C3×#1004—31no—
22C3×#1004—31no—
23C3×#1004—61no—
24#64144D311yescharacteristic, complemented
25C3×#935—31no—
26#63962—32no—
27#63961—32no—
28#63661D311yescharacteristic, complemented
29C3×#834—31no—
30#63580—32no—
31#63580—32no—
32C23×H(3,𝔽3)D411yescharacteristic
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×#272—21no—
35C22×#272D412yes—
36D4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)D411yescharacteristic
38#6235—22no—
39#6235—22no—
40#6235—22no—
41C2×#1084—21no—
42C2×#1084D412yes—
43C4⋊#272—21no—
44C4×#272—22no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—22no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50#5243—91no—
51#5053—91no—
52#4742—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C2×D311yescharacteristic
56C22×C3⋊Dic3C2×D311yescharacteristic, complemented
57C2×C3×#67—32no—
58C2×C3×#67—32no—
59C2×C3×#67—62no—
60C2×C3×#67—62no—
61C2×C3×#67—61no—
62C2×C3×#67—61no—
63C22×C3×Dic3—31no—
64C3×#237—62no—
65C3×#237—62no—
66C3×#237—64no—
67C3×D3×D4—62no—
68C3×D3×D4—62no—
69C3×D3×D4—64no—
70C3×#235—62no—
71C3×#235—61no—
72C3×#235—61no—
73#3757—21no—
74#3757C2×D311yescharacteristic, complemented
75#3757C2×D312yescomplemented
76#3750—21no—
77C3×#217—31no—
78C3×#217—32no—
79C3×#217—61no—
80C3×Dic3⋊C4—61no—
81#3724—62no—
82Dic3⋊Dic3—62no—
83(C2×D3)⋊Dic3—62no—
84(C2×D3)⋊Dic3—62no—
85C22×H(3,𝔽3)—21no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C22×H(3,𝔽3)C2×D412yes—
90C2×#272—22no—
91C2×#272—42no—
92C2×#272—81no—
93C4×H(3,𝔽3)—21no—
94#1084—22no—
95(C3⋊Dic3)⋊C3—42no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3—41no—
98C3×H(5,𝔽2)—91no—
99#1004—31no—
100#1004—31no—
101#1004—61no—
102C3×#123—91no—
103#935—31no—
104#935—91no—
105#935—92no—
106#929—92no—
107#886—92no—
108C3×#99—91no—
109#834—31no—
110#834—91no—
111#834—92no—
112#806—92no—
113#806—92no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32D1211yescharacteristic
117C23×C32#6711yescharacteristic
118C23×C32C22×D311yescharacteristic
119C23×C32C22×D311yescharacteristic
120C22×C3×D3—32no—
121C22×C3×D3—32no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C32×D4—24no—
127C32×D4—22no—
128C32×D4—22no—
129C2×C32×C4—21no—
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3—21no—
136C3×#67—64no—
137C3×#67—64no—
138C3×#67—62no—
139C3×#67—62no—
140C3×#67—64no—
141C3×#67—62no—
142C3×#67—62no—
143C3×#67—62no—
144C3×#67—62no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C3×D12—62no—
156C3×D12—61no—
157C3×D12—61no—
158C3×C4×D3—64no—
159C3×C4×D3—62no—
160C3×C4×D3—62no—
161C3×Dic6—62no—
162C3×Dic6—61no—
163C3×Dic6—61no—
164#458—271noSylow
165C2×H(3,𝔽3)—22no—
166C2×H(3,𝔽3)—21no—
167C2×H(3,𝔽3)—41no—
168C2×H(3,𝔽3)#12011yescharacteristic
169#272—42no—
170C3×#41—182no—
171C3×#41—181no—
172C2×C3×D4—31no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—92no—
176C2×C3×D4—182no—
177C2×C3×D4—181no—
178C2×C3×D4D3211yescharacteristic
179C22×C3×C4—91no—
180C2×#67—32no—
181C2×#67—32no—
182C2×#67—62no—
183C2×#67—61no—
184C2×#67—61no—
185C2×#67—62no—
186C22×Dic3—31no—
187C22×Dic3—91no—
188#237—62no—
189#237—62no—
190#237—64no—
191D3×D4—62no—
192D3×D4—62no—
193D3×D4—64no—
194#235—61no—
195#235—61no—
196#235—62no—
197C3×C4⋊C4—181no—
198C3×C22⋊C4—91no—
199C3×C22⋊C4—92no—
200C3×C22⋊C4—181no—
201#217—32no—
202#217—31no—
203#217—61no—
204#217—91no—
205#217—92no—
206#217—181no—
207#217—182no—
208(C2×D3)⋊C4—182no—
209(C2×D3)⋊C4—182no—
210C4⋊Dic3—182no—
211Dic3⋊C4—61no—
212Dic3⋊C4—181no—
213Dic3⋊C4—182no—
214Dic3⋊C4—182no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—22no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—81no—
225C22×C32—81no—
226C22×C32C2×D1211yescharacteristic
227C22×C32D3×D412yes—
228C22×C32D3×D412yes—
229C22×C32C2×#6711yescharacteristic
230C2×C3×D3—64no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—122no—
234C2×C3×D3—122no—
235C2×C3×D3—242no—
236C2×C3×D3—242no—
237C2×C3×D3—241no—
238C2×C3×D3—241no—
239C32×C4—22no—
240C32×C4—21no—
241C32×C4—21no—
242C3⋊Dic3—42no—
243C3⋊Dic3—41no—
244C3⋊Dic3—41no—
245C3×Dic3—64no—
246C3×Dic3—62no—
247C3×Dic3—62no—
248C3×Dic3—121no—
249C3×Dic3—122no—
250C3×Dic3—121no—
251H(5,𝔽2)—91no—
252#123—91no—
253#123—272no—
254#99—91no—
255#99—272no—
256H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
257C23×C3—31no—
258C23×C3—31no—
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—92no—
264C23×C3—181no—
265C23×C3#62611yescharacteristic
266C23×C3C2×D3211yescharacteristic
267C22×D3—32no—
268C22×D3—32no—
269C22×D3—62no—
270C22×D3—61no—
271C22×D3—61no—
272C22×D3—62no—
273C3×Q8—181no—
274C3×D4—22no—
275C3×D4—62no—
276C3×D4—62no—
277C3×D4—64no—
278C3×D4—181no—
279C3×D4—182no—
280C3×D4—182no—
281C3×D4—182no—
282C2×C3×C4—31no—
283C2×C3×C4—31no—
284C2×C3×C4—61no—
285C2×C3×C4—92no—
286C2×C3×C4—182no—
287C2×C3×C4—181no—
288C2×C3×C4—182no—
289C2×C3×C4—182no—
290C2×C3×C4—181no—
291C2×C3×C4—181no—
292C2×C3×C4#62611yescharacteristic
293#67—62no—
294#67—62no—
295#67—62no—
296#67—62no—
297#67—64no—
298#67—62no—
299#67—62no—
300#67—64no—
301#67—64no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—62no—
305C2×Dic3—61no—
306C2×Dic3—61no—
307C2×Dic3—62no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—182no—
313C2×Dic3—181no—
314C2×Dic3—181no—
315C2×Dic3—182no—
316D12—61no—
317D12—61no—
318D12—62no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—64no—
322Dic6—61no—
323Dic6—61no—
324Dic6—62no—
325C2×C32—21no—
326C2×C32—22no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—42no—
331C2×C32—41no—
332C2×C32—42no—
333C2×C32—41no—
334C2×C32—41no—
335C2×C32#88211yescharacteristic
336C2×C32#93711yescharacteristic
337C3×D3—124no—
338C3×D3—122no—
339C3×D3—122no—
340#41—182no—
341#41—181no—
342C2×D4—92no—
343C2×D4—182no—
344C2×D4—181no—
345C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C22×C4—91no—
347C4⋊C4—181no—
348C4⋊C4—542no—
349C22⋊C4—92no—
350C22⋊C4—91no—
351C22⋊C4—181no—
352C22⋊C4—542no—
353C22×C3—21no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—32no—
357C22×C3—32no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—62no—
363C22×C3—61no—
364C22×C3—62no—
365C22×C3—121no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3—182no—
370C22×C3—241no—
371C22×C3—241no—
372C22×C3—362no—
373C22×C3—721no—
374C22×C3C2×#62611yescharacteristic
375C22×C3#381112yes—
376C2×D3—62no—
377C2×D3—62no—
378C2×D3—64no—
379C2×D3—122no—
380C2×D3—122no—
381C2×D3—242no—
382C2×D3—241no—
383C2×D3—241no—
384C2×D3—242no—
385C3×C4—21no—
386C3×C4—61no—
387C3×C4—61no—
388C3×C4—62no—
389C3×C4—182no—
390C3×C4—361no—
391C3×C4—362no—
392C3×C4—361no—
393Dic3—62no—
394Dic3—62no—
395Dic3—64no—
396Dic3—121no—
397Dic3—122no—
398Dic3—121no—
399Dic3—361no—
400Dic3—362no—
401Dic3—361no—
402C32—21no—
403C32#474211yescharacteristic, complemented
404C32#505311yescharacteristic, complemented
405C23—92no—
406C23—181no—
407C23#620011yescharacteristic
408C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
409Q8—181no—
410D4—22no—
411D4—181no—
412D4—182no—
413D4—182no—
414D4—182no—
415C2×C4—92no—
416C2×C4—182no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4#620011yescharacteristic
423C2×C3—21no—
424C2×C3—22no—
425C2×C3—31no—
426C2×C3—31no—
427C2×C3—41no—
428C2×C3—62no—
429C2×C3—61no—
430C2×C3—62no—
431C2×C3—61no—
432C2×C3—61no—
433C2×C3—122no—
434C2×C3—121no—
435C2×C3—121no—
436C2×C3—121no—
437C2×C3—122no—
438C2×C3—362no—
439C2×C3#6398111yescharacteristic
440D3—122no—
441D3—122no—
442D3—124no—
443C22—21no—
444C22—41no—
445C22—41no—
446C22—182no—
447C22—362no—
448C22—721no—
449C22C2×#620011yescharacteristic
450C22#8867712yes—
451C4—21no—
452C4—182no—
453C4—361no—
454C4—362no—
455C4—361no—
456C3—31no—
457C3—31no—
458C3—61no—
459C3#1059153511yescharacteristic
460C2—21no—
461C2—22no—
462C2—41no—
463C2—362no—
464C2#1172132311yescharacteristic
subgroupG/H
centerC2#11721323
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3#63981
prime radical#11722664C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#458
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #64144⋊C3C2semidirect
3. #11721301C2semidirect
4. #63661⋊C3C2semidirect
5. #11720335C2semidirect
14. C22×(C3⋊Dic3)⋊C3C22semidirect
16. #3757⋊C3C22semidirect
17. #3757⋊C3C22semidirect
24. #64144D3semidirect
28. #63661D3semidirect
56. C22×C3⋊Dic3C2×D3semidirect
74. #3757C2×D3semidirect
75. #3757C2×D3semidirect
256. H(3,𝔽3)#458semidirect
403. C32#4742semidirect
404. C32#5053semidirect

Maximal subgroups

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

subgroupindex
#117226642
#64144⋊C32
#117213012
#63661⋊C32
#117203352
#105915343
#105915333