Finite Group Databasebeta

← #13140571

Subgroups — #13140571

Order 1728.

563 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 73 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#11721848C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic, complemented
4C2×#88434C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591498—31nomaximal
9#10591449—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×C4×#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#6200—21no—
15#88552—21no—
16C4⋊#1084C2211yescharacteristic
17#3753⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88434—22no—
23C2×#3753D311yescharacteristic, complemented
24C3×#938—31no—
25C3×#938—31no—
26C3×#938—61no—
27C2×C3×#213—31no—
28#63909—31no—
29#63908—31no—
30C2×D3×C3⋊C8—31no—
31C2×D3×C3⋊C8—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63576—31no—
35#63575—31no—
36#63564—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272D411yescharacteristic
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—42no—
45C2×#1084—21no—
46C2×#1084D411yescharacteristic
47C4×#272—21no—
48C4×#272—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52#6200—42no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5157—91no—
56#5038—91no—
57#4775—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×#67—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—61no—
69C2×#626—61no—
70C2×#625—61no—
71#3753—22no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×#217—61no—
76C3×#217—61no—
77C3×#217—62no—
78C3×#213—62no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—61no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—31no—
85#3722—61no—
86(C2×D3)⋊Dic3—61no—
87D3×C3⋊C8—62no—
88D3×C3⋊C8—62no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C2×#272—21no—
94C2×#272—21no—
95C2×#272D812yescomplemented
96C2×(C3⋊D3)⋊C3—41no—
97C2×(C3⋊D3)⋊C3—41no—
98C2×(C3⋊D3)⋊C3—41no—
99C4×H(3,𝔽3)C2×D411yescharacteristic
100C4×H(3,𝔽3)#4111yescharacteristic
101#1084—21no—
102#1084—41no—
103(C3⋊Dic3)⋊C3—41no—
104C2×C3×D8—91no—
105C3×#121—91no—
106#938—31no—
107#938—31no—
108#938—61no—
109#938—91no—
110C2×#213—31no—
111C2×#213—91no—
112#930—91no—
113C22×C3⋊C8—91no—
114C2×D3×C8—91no—
115C2×D3×C8—91no—
116#895—91no—
117C3×#102—91no—
118#832—31no—
119#832—91no—
120#832—91no—
121#821—91no—
122#818—91no—
123#809—91no—
124#807—91no—
125#807—91no—
126C23×C32—22no—
127C23×C32—21no—
128C23×C32—21no—
129C22×C3⋊D3—21no—
130C22×C3×D3—31no—
131C22×C3×D3—31no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—41no—
139C32×D4—41no—
140C2×C32×C4—21no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—21no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146C3×#67—122no—
147C3×#67—122no—
148C3×#67—124no—
149C2×C3×Dic3—31no—
150C2×C3×Dic3—31no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C3×D12—61no—
157C3×D12—61no—
158C3×C4×D3—61no—
159C3×C4×D3—61no—
160C3×C4×D3—61no—
161C3×C4×D3—61no—
162C3×C4×D3—121no—
163C3×C4×D3—121no—
164#626—122no—
165#625—122no—
166C3⋊(C3⋊C8)—21no—
167C3⋊(C3⋊C8)#6712yescomplemented
168C3×C3⋊C8—32no—
169C3×C3⋊C8—61no—
170#466—271noSylow
171C2×H(3,𝔽3)—81no—
172C2×H(3,𝔽3)#12111yescharacteristic
173C2×H(3,𝔽3)C2×D811yescharacteristic
174C2×H(3,𝔽3)#13511yescharacteristic
175#272—22no—
176(C3⋊D3)⋊C3—81no—
177C2×C3×D4—31no—
178C2×C3×D4—31no—
179C2×C3×D4—61no—
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C2×C3×D4D3211yescharacteristic
183C22×C3×C4—91no—
184C2×#67—61no—
185C2×#67—61no—
186C2×#67—62no—
187C2×#67—181no—
188C2×#67—181no—
189C2×D12—31no—
190C2×D12—91no—
191C2×D12—181no—
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C3×D8—182no—
196C2×C3×C8—91no—
197C2×C3×C8—91no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—91no—
200C3×C22⋊C4—181no—
201#217—61no—
202#217—61no—
203#217—62no—
204#217—181no—
205#213—62no—
206#213—182no—
207(C2×D3)⋊C4—181no—
208(C2×D3)⋊C4—181no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—31no—
212C4⋊Dic3—61no—
213C4⋊Dic3—91no—
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×C32—42no—
223C22×C32—42no—
224C22×C32—42no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32D3×D411yescharacteristic
232C22×C32D3×D411yescharacteristic
233C2×C3⋊D3—41no—
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3×D3—32no—
237C2×C3×D3—32no—
238C2×C3×D3—62no—
239C2×C3×D3—61no—
240C2×C3×D3—61no—
241C2×C3×D3—61no—
242C2×C3×D3—61no—
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#23511yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4#23711yescharacteristic
253C32×C4C2×#6711yescharacteristic
254C3⋊Dic3—41no—
255C3×Dic3—61no—
256C3×Dic3—61no—
257C3×Dic3—121no—
258C3×Dic3—122no—
259C3×Dic3—121no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C2×D8—91no—
263C22×C8—271no—
264#121—91no—
265#121—271no—
266#107—271no—
267#102—91no—
268#102—271no—
269H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
270C23×C3—21no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—62no—
274C23×C3—91no—
275C23×C3—181no—
276C22×D3—31no—
277C22×D3—31no—
278C22×D3—61no—
279C22×D3—61no—
280C22×D3—181no—
281C3×D4—21no—
282C3×D4—21no—
283C3×D4—61no—
284C3×D4—61no—
285C3×D4—61no—
286C3×D4—61no—
287C3×D4—121no—
288C3×D4—121no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—362no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—61no—
295C2×C3×C4—91no—
296C2×C3×C4—181no—
297C2×C3×C4—181no—
298C2×C3×C4—181no—
299C2×C3×C4—181no—
300C2×C3×C4C2×D3211yescharacteristic
301#67—122no—
302#67—122no—
303#67—124no—
304#67—362no—
305#67—362no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—62no—
312C2×Dic3—61no—
313C2×Dic3—181no—
314D12—61no—
315D12—61no—
316D12—181no—
317D12—181no—
318D12—362no—
319C4×D3—61no—
320C4×D3—61no—
321C4×D3—61no—
322C4×D3—61no—
323C4×D3—121no—
324C4×D3—121no—
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—82no—
335C2×C32—81no—
336C2×C32—81no—
337C2×C32#88411yescharacteristic
338C2×C32D3×D811yescharacteristic
339C2×C32D3×D811yescharacteristic
340C2×C32#91211yescharacteristic
341C2×C32#91611yescharacteristic
342C2×C32#93811yescharacteristic
343C3⋊D3—81no—
344C3×D3—62no—
345C3×D3—62no—
346C3×D3—122no—
347C3×D3—241no—
348C2×D4—91no—
349C2×D4—181no—
350C2×D4—541no—
351C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C22×C4—91no—
353D8—182no—
354C2×C8—91no—
355C2×C8—91no—
356C2×C8—542no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C22⋊C4—181no—
360C22⋊C4—541no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—61no—
367C22×C3—92no—
368C22×C3—121no—
369C22×C3—121no—
370C22×C3—121no—
371C22×C3—121no—
372C22×C3—121no—
373C22×C3—121no—
374C22×C3—122no—
375C22×C3—122no—
376C22×C3—122no—
377C22×C3—181no—
378C22×C3—181no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3—361no—
382C22×C3#381111yescharacteristic
383C2×D3—32no—
384C2×D3—32no—
385C2×D3—62no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—121no—
391C2×D3—121no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—361no—
396C2×D3—361no—
397C2×D3—361no—
398C3×C4—31no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—61no—
403C3×C4—61no—
404C3×C4—181no—
405C3×C4—361no—
406C3×C4—361no—
407C3×C4#380411yescharacteristic
408C3×C4D3×#6711yescharacteristic
409Dic3—61no—
410Dic3—61no—
411Dic3—121no—
412Dic3—121no—
413Dic3—121no—
414Dic3—122no—
415Dic3—121no—
416Dic3—361no—
417C32—21no—
418C32#477511yescharacteristic, complemented
419C32#515711yescharacteristic, complemented
420C23—21no—
421C23—91no—
422C23—181no—
423D4—21no—
424D4—21no—
425D4—181no—
426D4—181no—
427D4—362no—
428D4—1082no—
429C2×C4—91no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4—181no—
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—61no—
444C2×C3—61no—
445C2×C3—61no—
446C2×C3—81no—
447C2×C3—182no—
448C2×C3—241no—
449C2×C3—241no—
450C2×C3—242no—
451C2×C3—721no—
452C2×C3#6392611yescharacteristic
453C2×C3#6393611yescharacteristic
454C2×C3#6395811yescharacteristic
455D3—62no—
456D3—62no—
457D3—122no—
458D3—241no—
459D3—721no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—92no—
464C22—181no—
465C22—181no—
466C22—361no—
467C22—361no—
468C22—361no—
469C22#8867711yescharacteristic
470C4—181no—
471C4—361no—
472C4—361no—
473C4#8867111yescharacteristic
474C4#8867611yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3#1059145511yescharacteristic
479C2—81no—
480C2—182no—
481C2—721no—
482C2#1172119011yescharacteristic
483C2#1172119211yescharacteristic
484C2#1172130311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721848C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721159C2semidirect
4. C2×#88434C2semidirect
5. #63659⋊C3C2semidirect
6. #11720308C2semidirect
7. #11720306C2semidirect
23. C2×#3753D3semidirect
32. #63659D3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
95. C2×#272D8semidirect
167. C3⋊(C3⋊C8)#67semidirect
269. H(3,𝔽3)#466semidirect
418. C32#4775semidirect
419. C32#5157semidirect

Maximal subgroups

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

subgroupindex
#117218482
C2×#3753⋊C32
#117211592
C2×#884342
#63659⋊C32
#117203082
#117203062
#105914983
#105914493