Finite Group Databasebeta

← #13140184

Subgroups — #13140184

Order 1728.

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

normal subgroups only

This group has 549 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, complemented
2#64139⋊C3C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591167—31nomaximal
9#10591165—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4⋊#272C2211yescharacteristic
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15#88542—21no—
16C4⋊#1084C2211yescharacteristic
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3748⋊C3—22no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88436—22no—
23#64139D311yescharacteristic, complemented
24C2×#3748D311yescharacteristic, complemented
25C3×#930—31no—
26C3×#930—31no—
27C3×#930—31no—
28C3×#930—61no—
29C2×C3×#208—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32#63574—31no—
33#63573—31no—
34#63564—31no—
35#63563—31no—
36#63562—31no—
37#63562—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#6235—42no—
46C2×#1084—21no—
47C4⋊#272—21no—
48C4⋊#272—21no—
49#638⋊C3—42no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5135—91no—
56#5039—91no—
57#5039—91no—
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60C22×C32×C4C2×D311yescharacteristic
61C2×#638—21no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×#67—62no—
64C2×C3×#67—61no—
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—31no—
70C2×C3×D12—61no—
71#3752—21no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73#3748—22no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×(C2×D3)⋊C4—62no—
77C3×(C2×D3)⋊C4—61no—
78C3×(C2×D3)⋊C4—61no—
79C3×(C2×D3)⋊C4—61no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C3×#208—62no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87#3714—62no—
88#3713—62no—
89C22×H(3,𝔽3)—21no—
90C22×H(3,𝔽3)—21no—
91C22×H(3,𝔽3)—21no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—41no—
94C2×#272—41no—
95C2×#272—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—41no—
99C4×H(3,𝔽3)—21no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)#4111yescharacteristic
102#1084—41no—
103(C3⋊Dic3)⋊C3—41no—
104C2×C3×M16—91no—
105C3×#121—91no—
106C3×#121—91no—
107C2×#213—91no—
108C2×#213—91no—
109#930—31no—
110#930—31no—
111#930—31no—
112#930—61no—
113#930—91no—
114C2×#208—31no—
115C2×#208—91no—
116C2×D24—91no—
117#832—91no—
118#832—91no—
119#821—91no—
120#821—91no—
121#817—91no—
122#809—91no—
123#809—91no—
124#808—91no—
125#808—91no—
126C23×C32—21no—
127C23×C32D1211yescharacteristic
128C23×C32#6711yescharacteristic
129C22×C3⋊D3—21no—
130C22×C3×D3—62no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4D1211yescharacteristic
143C2×C32×C4#6711yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146#638—42no—
147C2×C3⋊Dic3—21no—
148C4⋊(C3⋊D3)—21no—
149C4⋊(C3⋊D3)—21no—
150C3×#67—122no—
151C3×#67—124no—
152C3×#67—122no—
153C3×#67—122no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C3×D12—61no—
159C3×D12—61no—
160C3×D12—61no—
161C3×D12—61no—
162C3×D12—61no—
163C3×D12—61no—
164C3×D12—121no—
165C3×D12—121no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)#6712yescomplemented
168C3×C3⋊C8—32no—
169C3×C3⋊C8—61no—
170#469—271noSylow
171C2×H(3,𝔽3)—41no—
172C2×H(3,𝔽3)#12111yescharacteristic
173C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
174C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
175#272—81no—
176(C3⋊D3)⋊C3—81no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C2×C3×D4—181no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—61no—
184C22×C3×C4D3211yescharacteristic
185C2×#67—61no—
186C2×#67—61no—
187C2×#67—62no—
188C2×#67—61no—
189C2×#67—181no—
190C2×D12—31no—
191C2×D12—31no—
192C2×D12—31no—
193C2×D12—61no—
194C2×D12—91no—
195C3×M16—182no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—91no—
200C3×C22⋊C4—181no—
201C3×C22⋊C4—181no—
202#213—182no—
203#213—182no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—62no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—31no—
212C4⋊Dic3—61no—
213C4⋊Dic3—91no—
214#208—62no—
215#208—182no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8—91no—
219C2×C3⋊C8—91no—
220D24—182no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32C2×D1211yescharacteristic
232C22×C32C2×#6711yescharacteristic
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3×D3—122no—
237C2×C3×D3—122no—
238C2×C3×D3—122no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—41no—
253C32×C4#23511yescharacteristic
254C32×C4D3×D411yescharacteristic
255C32×C4#23711yescharacteristic
256C32×C4C2×#6711yescharacteristic
257C3⋊Dic3—41no—
258C3×Dic3—122no—
259C3×Dic3—121no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C2×D8—271no—
263C2×M16—91no—
264#121—91no—
265#121—91no—
266#106—271no—
267#102—271no—
268#102—271no—
269H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
270C23×C3—31no—
271C23×C3—31no—
272C23×C3—61no—
273C23×C3—181no—
274C23×C3—181no—
275C23×C3#62611yescharacteristic
276C22×D3—61no—
277C22×D3—61no—
278C22×D3—62no—
279C22×D3—61no—
280C22×D3—181no—
281C3×D4—181no—
282C3×D4—181no—
283C3×D4—181no—
284C3×D4—181no—
285C3×D4—362no—
286C3×D4—362no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4#62611yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305#67—122no—
306#67—122no—
307#67—122no—
308#67—124no—
309#67—362no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—62no—
313C2×Dic3—61no—
314C2×Dic3—181no—
315D12—61no—
316D12—61no—
317D12—61no—
318D12—61no—
319D12—61no—
320D12—61no—
321D12—121no—
322D12—121no—
323D12—181no—
324D12—181no—
325C3×C8—92no—
326C3×C8—181no—
327C3⋊C8—32no—
328C3⋊C8—61no—
329C3⋊C8—92no—
330C3⋊C8—181no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—42no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32#90811yescharacteristic
338C2×C32#90811yescharacteristic
339C2×C32#93011yescharacteristic
340C2×C32#93211yescharacteristic
341C2×C32#93911yescharacteristic
342C2×C32#94911yescharacteristic
343C3⋊D3—81no—
344C3×D3—242no—
345C3×D3—241no—
346C3×D3—241no—
347C3×D3—241no—
348C2×D4—91no—
349C2×D4—91no—
350C2×D4—181no—
351C2×D4—181no—
352C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
353D8—542no—
354M16—182no—
355C2×C8—91no—
356C2×C8—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C22⋊C4—181no—
360C22⋊C4—181no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—61no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—62no—
373C22×C3—61no—
374C22×C3—121no—
375C22×C3—121no—
376C22×C3—361no—
377C22×C3—361no—
378C22×C3—361no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3C2×#62611yescharacteristic
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—121no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—122no—
390C2×D3—122no—
391C2×D3—122no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—361no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—21no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—31no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—121no—
408C3×C4—361no—
409C3×C4—361no—
410C3×C4#380411yescharacteristic
411C3×C4D3×#6711yescharacteristic
412Dic3—121no—
413Dic3—121no—
414Dic3—122no—
415Dic3—121no—
416Dic3—361no—
417C32—21no—
418C32#503911yescharacteristic, complemented
419C32#513511yescharacteristic, complemented
420C23—181no—
421C23—181no—
422C23#620011yescharacteristic
423D4—181no—
424D4—181no—
425D4—181no—
426D4—181no—
427D4—362no—
428D4—362no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4#620011yescharacteristic
434C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
435C8—92no—
436C8—181no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—31no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—41no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—61no—
447C2×C3—121no—
448C2×C3—121no—
449C2×C3—122no—
450C2×C3—721no—
451C2×C3—721no—
452C2×C3#6382511yescharacteristic
453C2×C3#6383011yescharacteristic
454C2×C3#6397811yescharacteristic
455D3—241no—
456D3—241no—
457D3—242no—
458D3—241no—
459D3—721no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22—361no—
467C22—361no—
468C22—361no—
469C22C2×#620011yescharacteristic
470C4—21no—
471C4—361no—
472C4—361no—
473C4#8867111yescharacteristic
474C4#8867611yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3#1059116511yescharacteristic
479C2—41no—
480C2—721no—
481C2—721no—
482C2#1172106811yescharacteristic
483C2#1172107111yescharacteristic
484C2#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
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#469
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
1. #11721830C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#3748⋊C3C2semidirect
4. C2×#88436C2semidirect
5. #11720324C2semidirect
6. #11720308C2semidirect
7. #11720307C2semidirect
23. #64139D3semidirect
24. C2×#3748D3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
167. C3⋊(C3⋊C8)#67semidirect
269. H(3,𝔽3)#469semidirect
418. C32#5039semidirect
419. C32#5135semidirect

Maximal subgroups

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

subgroupindex
#117218302
#64139⋊C32
C2×#3748⋊C32
C2×#884362
#117203242
#117203082
#117203072
#105911673
#105911653