Finite Group Databasebeta

← #13140188

Subgroups — #13140188

Order 1728.

514 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721830C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88437C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591180—31nomaximal
9#10591176—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14#88542—21no—
15C4⋊#1084C2211yescharacteristic
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19#3748⋊C3—22no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88437—22no—
23#64133D311yescharacteristic, complemented
24C2×#3748D311yescharacteristic, complemented
25C3×#930—31no—
26C3×#930—31no—
27C3×#930—61no—
28C3×#924—31no—
29C2×C3×#208—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63576—31no—
33#63575—31no—
34#63567—31no—
35#63565—31no—
36#63563—31no—
37#63562—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×#272—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—42no—
45C2×#1084—21no—
46C4⋊#272—21no—
47C4⋊#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5134—91no—
55#5049—91no—
56#5042—91no—
57C22×C32×C4—21no—
58C22×C32×C4C2×D311yescharacteristic
59C22×C32×C4C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×#67—62no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×D12—31no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×Dic6—31no—
68#3757—21no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71#3748—22no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×#217—61no—
75C3×(C2×D3)⋊C4—62no—
76C3×(C2×D3)⋊C4—61no—
77C3×(C2×D3)⋊C4—61no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×Dic3⋊C4—61no—
83C3×#208—62no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86#3716—62no—
87#3715—62no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×#272—41no—
93C2×#272—41no—
94C2×#272—41no—
95C4×H(3,𝔽3)—21no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98#1084—41no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3—41no—
101C2×C3×M16—91no—
102C3×C22⋊Q8—91no—
103C3×#121—91no—
104C2×#215—91no—
105C2×#214—91no—
106#930—31no—
107#930—31no—
108#930—61no—
109#924—31no—
110#924—91no—
111C2×#208—31no—
112C2×#208—91no—
113C2×#204—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#818—91no—
118#816—91no—
119#810—91no—
120#809—91no—
121#807—91no—
122#807—91no—
123C23×C32—21no—
124C23×C32D1211yescharacteristic
125C23×C32#6711yescharacteristic
126C22×C3×D3—62no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4D1211yescharacteristic
138C2×C32×C4#6711yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143#634—21no—
144#634—21no—
145C3×#67—124no—
146C3×#67—122no—
147C3×#67—122no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—121no—
158C3×D12—121no—
159C3×Dic6—61no—
160C3×Dic6—61no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)#6712yescomplemented
163C3×C3⋊C8—32no—
164C3×C3⋊C8—61no—
165#468—271noSylow
166C2×H(3,𝔽3)—41no—
167C2×H(3,𝔽3)#12111yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169C2×H(3,𝔽3)#13711yescharacteristic
170#272—81no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—61no—
177C22×C3×C4D3211yescharacteristic
178C2×#67—61no—
179C2×#67—61no—
180C2×#67—62no—
181C2×D12—31no—
182C2×D12—31no—
183C2×D12—61no—
184C2×Dic6—31no—
185C2×Dic6—91no—
186C3×M16—182no—
187C2×C3×C8—91no—
188C2×C3×C8—91no—
189C3×C4⋊C4—91no—
190C3×C4⋊C4—91no—
191C3×C4⋊C4—181no—
192C3×C22⋊C4—181no—
193C3×C22⋊C4—181no—
194#217—61no—
195#217—181no—
196#215—182no—
197#214—182no—
198(C2×D3)⋊C4—61no—
199(C2×D3)⋊C4—61no—
200(C2×D3)⋊C4—62no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—61no—
205C4⋊Dic3—91no—
206Dic3⋊C4—61no—
207Dic3⋊C4—181no—
208#208—62no—
209#208—182no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—91no—
214#204—182no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32C2×D1211yescharacteristic
226C22×C32C2×#6711yescharacteristic
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4#23511yescharacteristic
242C32×C4D3×D411yescharacteristic
243C32×C4#23711yescharacteristic
244C32×C4C2×#6711yescharacteristic
245C3⋊Dic3—41no—
246C3⋊Dic3—41no—
247C3×Dic3—122no—
248C3×Dic3—121no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C2×SD16—271no—
253C2×M16—91no—
254C22⋊Q8—91no—
255#121—91no—
256#107—271no—
257#103—271no—
258#102—271no—
259H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
260C23×C3—31no—
261C23×C3—31no—
262C23×C3—61no—
263C23×C3—181no—
264C23×C3#62611yescharacteristic
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—62no—
268C3×Q8—181no—
269C3×Q8—181no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—362no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—121no—
286C2×C3×C4—121no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4#62611yescharacteristic
291C2×C3×C4C2×D3211yescharacteristic
292#67—122no—
293#67—122no—
294#67—124no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C2×Dic3—62no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302D12—61no—
303D12—61no—
304D12—61no—
305D12—61no—
306D12—121no—
307D12—121no—
308Dic6—61no—
309Dic6—61no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—92no—
313C3×C8—181no—
314C3⋊C8—32no—
315C3⋊C8—61no—
316C3⋊C8—92no—
317C3⋊C8—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—42no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32#90811yescharacteristic
325C2×C32#90911yescharacteristic
326C2×C32#93011yescharacteristic
327C2×C32#93911yescharacteristic
328C2×C32#94211yescharacteristic
329C2×C32#94911yescharacteristic
330C3×D3—242no—
331C3×D3—241no—
332C3×D3—241no—
333C2×Q8—91no—
334C2×D4—91no—
335C2×D4—181no—
336C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
337SD16—542no—
338M16—182no—
339C2×C8—91no—
340C2×C8—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4—181no—
344C22⋊C4—181no—
345C22⋊C4—181no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—61no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—62no—
358C22×C3—61no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—361no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3C2×#62611yescharacteristic
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—122no—
372C2×D3—122no—
373C2×D3—122no—
374C3×C4—21no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—121no—
384C3×C4—361no—
385C3×C4—361no—
386C3×C4—361no—
387C3×C4#380411yescharacteristic
388C3×C4D3×#6711yescharacteristic
389Dic3—121no—
390Dic3—121no—
391Dic3—122no—
392Dic3—121no—
393Dic3—121no—
394Dic3—361no—
395Dic3—361no—
396C32—21no—
397C32#504911yescharacteristic, complemented
398C32#513411yescharacteristic, complemented
399C23—181no—
400C23#620011yescharacteristic
401Q8—181no—
402Q8—181no—
403D4—181no—
404D4—181no—
405D4—362no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4#620011yescharacteristic
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C8—92no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—41no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—121no—
426C2×C3—121no—
427C2×C3—122no—
428C2×C3—721no—
429C2×C3#6382511yescharacteristic
430C2×C3#6383311yescharacteristic
431C2×C3#6397811yescharacteristic
432D3—241no—
433D3—241no—
434D3—242no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—361no—
439C22—361no—
440C22—361no—
441C22C2×#620011yescharacteristic
442C4—21no—
443C4—361no—
444C4—361no—
445C4—361no—
446C4#8867111yescharacteristic
447C4#8867611yescharacteristic
448C3—31no—
449C3—31no—
450C3—61no—
451C3#1059117811yescharacteristic
452C2—41no—
453C2—721no—
454C2#1172106811yescharacteristic
455C2#1172106911yescharacteristic
456C2#1172132211yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721830C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#468
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #64133⋊C3C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88437C2semidirect
5. #11720324C2semidirect
6. #11720309C2semidirect
7. #11720306C2semidirect
23. #64133D3semidirect
24. C2×#3748D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
162. C3⋊(C3⋊C8)#67semidirect
259. H(3,𝔽3)#468semidirect
397. C32#5049semidirect
398. C32#5134semidirect

Maximal subgroups

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

subgroupindex
#117218302
#64133⋊C32
C2×#3748⋊C32
C2×#884372
#117203242
#117203092
#117203062
#105911803
#105911763