Finite Group Databasebeta

← #13140591

Subgroups — #13140591

Order 1728.

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

normal subgroups only

This group has 598 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#11721850C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721136C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591429—31nomaximal
9#10591428—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6235—21no—
12C2×#6235—21no—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×((C3⋊D3)⋊C3)⋊C4—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#88436—22no—
23C2×#3753D311yescharacteristic, complemented
24C3×#940—31no—
25C3×#940—31no—
26C3×#940—61no—
27C2×C3×#213—31no—
28#63894—31no—
29Dic3×D12—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63563—31no—
37#63562—31no—
38C23×H(3,𝔽3)—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44#6235—42no—
45#6235—42no—
46C2×#1084D411yescharacteristic
47C2×#1084D411yescharacteristic
48C4⋊#272—21no—
49C4⋊#272—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52C4⋊((C3⋊D3)⋊C3)D412yescomplemented
53((C3⋊D3)⋊C3)⋊C4—42no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5150—91no—
57#5072—91no—
58#4786—91no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D311yescharacteristic
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C2×C3×#67—61no—
66C2×C3×#67—61no—
67C2×C3×#67—62no—
68C2×C3×#67—62no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×#624—61no—
74C2×D3×Dic3—61no—
75#3753—22no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×#213—62no—
80C3×C4⋊Dic3—31no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—31no—
83C3×C4×Dic3—61no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86#3722—61no—
87(C2×D3)⋊Dic3—61no—
88#3714—62no—
89#3713—62no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)—41no—
92C22×H(3,𝔽3)—41no—
93C22×H(3,𝔽3)C2×D411yescharacteristic
94C2×#272—41no—
95C2×#272—41no—
96C2×#272—41no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—41no—
99C2×(C3⋊D3)⋊C3—41no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)#4111yescharacteristic
102#1084—21no—
103#1084D812yescomplemented
104(C3⋊Dic3)⋊C3—41no—
105C2×C3×D8—91no—
106C3×#127—91no—
107#940—31no—
108#940—31no—
109#940—61no—
110D4×Dic3—91no—
111C2×#213—31no—
112C2×#213—91no—
113C2×#213—91no—
114C2×#213—91no—
115C2×D24—91no—
116#893—91no—
117C4×D12—91no—
118C3×#102—91no—
119#832—31no—
120#832—91no—
121#832—91no—
122#821—91no—
123Dic3⋊C8—91no—
124Dic3⋊C8—91no—
125#809—91no—
126C4⋊(C3⋊C8)—91no—
127C23×C32—22no—
128C23×C32—21no—
129C23×C32—21no—
130C22×C3⋊D3—21no—
131C22×C3×D3—61no—
132C22×C3×D3—62no—
133C22×C3×D3—61no—
134C22×C3×D3—61no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—41no—
140C32×D4—41no—
141C2×C32×C4—21no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C3⋊Dic3—21no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)#6712yescomplemented
147C3×#67—122no—
148C3×#67—122no—
149C3×#67—124no—
150C3×#67—124no—
151C3×#67—122no—
152C3×#67—122no—
153C2×C3×Dic3—31no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C3×D12—32no—
161C3×D12—61no—
162C3×D12—61no—
163C3×D12—61no—
164C3×D12—61no—
165C3×D12—61no—
166C3×D12—121no—
167C3×D12—121no—
168#624—122no—
169D3×Dic3—122no—
170C3⋊(C3⋊C8)—21no—
171C3⋊(C3⋊C8)—21no—
172C3×C3⋊C8—61no—
173C3×C3⋊C8—61no—
174#459—271noSylow
175C2×H(3,𝔽3)—81no—
176C2×H(3,𝔽3)#12111yescharacteristic
177C2×H(3,𝔽3)C2×D811yescharacteristic
178C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
179#272—81no—
180(C3⋊D3)⋊C3—42no—
181C2×C3×D4—31no—
182C2×C3×D4—31no—
183C2×C3×D4—61no—
184C2×C3×D4—91no—
185C2×C3×D4—91no—
186C2×C3×D4—181no—
187C2×C3×D4—181no—
188C2×C3×D4D3211yescharacteristic
189C2×#67—61no—
190C2×#67—61no—
191C2×#67—61no—
192C2×#67—61no—
193C2×#67—62no—
194C2×#67—62no—
195C22×Dic3—181no—
196C2×D12—31no—
197C2×D12—31no—
198C2×D12—31no—
199C2×D12—61no—
200C2×D12—91no—
201C2×C4×D3—181no—
202C2×C4×D3—181no—
203C3×D8—182no—
204C2×C3×C8—91no—
205C2×C3×C8—91no—
206C3×C4⋊C4—91no—
207C3×C42—91no—
208#217—181no—
209#213—62no—
210#213—182no—
211#213—182no—
212#213—182no—
213(C2×D3)⋊C4—181no—
214(C2×D3)⋊C4—181no—
215C4⋊Dic3—31no—
216C4⋊Dic3—91no—
217C4×Dic3—31no—
218C4×Dic3—31no—
219C4×Dic3—61no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8—31no—
222C2×C3⋊C8—91no—
223C2×C3⋊C8—91no—
224D24—182no—
225C22×C32—21no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32D3×D411yescharacteristic
236C22×C32D3×D411yescharacteristic
237C2×C3⋊D3—22no—
238C2×C3⋊D3—41no—
239C2×C3⋊D3—41no—
240C2×C3×D3—62no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C2×C3×D3—122no—
244C2×C3×D3—122no—
245C2×C3×D3—122no—
246C2×C3×D3—121no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—121no—
250C2×C3×D3—121no—
251C2×C3×D3—121no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4#23511yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4#23711yescharacteristic
257C32×C4C2×#6711yescharacteristic
258C3⋊Dic3—41no—
259C3×Dic3—32no—
260C3×Dic3—32no—
261C3×Dic3—62no—
262C3×Dic3—62no—
263C3×Dic3—61no—
264C3×Dic3—61no—
265C3×Dic3—121no—
266C2×D8—91no—
267C2×D8—271no—
268#127—91no—
269C4×D4—271no—
270C4⋊C8—271no—
271#102—91no—
272#102—271no—
273H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
274C23×C3—21no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—62no—
278C23×C3—181no—
279C23×C3—181no—
280C22×D3—61no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—62no—
284C22×D3—181no—
285C3×D4—21no—
286C3×D4—21no—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4—61no—
290C3×D4—61no—
291C3×D4—92no—
292C3×D4—121no—
293C3×D4—121no—
294C3×D4—181no—
295C3×D4—181no—
296C3×D4—181no—
297C3×D4—362no—
298C3×D4—362no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—61no—
302C2×C3×C4—91no—
303C2×C3×C4—91no—
304C2×C3×C4—181no—
305C2×C3×C4C2×D3211yescharacteristic
306#67—122no—
307#67—122no—
308#67—124no—
309#67—122no—
310#67—122no—
311#67—124no—
312C2×Dic3—31no—
313C2×Dic3—31no—
314C2×Dic3—31no—
315C2×Dic3—31no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—181no—
320C2×Dic3—362no—
321D12—32no—
322D12—61no—
323D12—61no—
324D12—61no—
325D12—61no—
326D12—61no—
327D12—92no—
328D12—121no—
329D12—121no—
330D12—181no—
331C4×D3—362no—
332C4×D3—362no—
333C3×C8—181no—
334C3×C8—181no—
335C3⋊C8—61no—
336C3⋊C8—61no—
337C3⋊C8—181no—
338C3⋊C8—181no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—82no—
343C2×C32—81no—
344C2×C32—81no—
345C2×C32#88411yescharacteristic
346C2×C32D3×D811yescharacteristic
347C2×C32D3×D811yescharacteristic
348C2×C32#91111yescharacteristic
349C2×C32#91411yescharacteristic
350C2×C32#93911yescharacteristic
351C3⋊D3—42no—
352C3×D3—122no—
353C3×D3—242no—
354C3×D3—241no—
355C3×D3—241no—
356C2×D4—91no—
357C2×D4—91no—
358C2×D4—181no—
359C2×D4—181no—
360C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C22×C4—541no—
362D8—182no—
363D8—542no—
364C2×C8—91no—
365C2×C8—91no—
366C4⋊C4—91no—
367C22⋊C4—541no—
368C42—91no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—61no—
375C22×C3—121no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3—122no—
382C22×C3—122no—
383C22×C3—122no—
384C22×C3—182no—
385C22×C3—361no—
386C22×C3—361no—
387C22×C3—361no—
388C22×C3—361no—
389C22×C3—361no—
390C22×C3#381111yescharacteristic
391C2×D3—62no—
392C2×D3—121no—
393C2×D3—121no—
394C2×D3—121no—
395C2×D3—121no—
396C2×D3—121no—
397C2×D3—121no—
398C2×D3—121no—
399C2×D3—121no—
400C2×D3—122no—
401C2×D3—122no—
402C2×D3—122no—
403C2×D3—182no—
404C2×D3—361no—
405C2×D3—361no—
406C3×C4—31no—
407C3×C4—31no—
408C3×C4—31no—
409C3×C4—31no—
410C3×C4—61no—
411C3×C4—61no—
412C3×C4—92no—
413C3×C4—181no—
414C3×C4—361no—
415C3×C4#380411yescharacteristic
416C3×C4D3×#6711yescharacteristic
417Dic3—32no—
418Dic3—32no—
419Dic3—62no—
420Dic3—61no—
421Dic3—61no—
422Dic3—62no—
423Dic3—121no—
424Dic3—361no—
425C32—21no—
426C32#478611yescharacteristic, complemented
427C32#515011yescharacteristic, complemented
428C23—21no—
429C23—181no—
430C23—181no—
431D4—21no—
432D4—21no—
433D4—92no—
434D4—181no—
435D4—181no—
436D4—181no—
437D4—362no—
438D4—362no—
439C2×C4—91no—
440C2×C4—91no—
441C2×C4—181no—
442C2×C4—1082no—
443C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
444C8—181no—
445C8—181no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—61no—
453C2×C3—61no—
454C2×C3—61no—
455C2×C3—81no—
456C2×C3—241no—
457C2×C3—241no—
458C2×C3—242no—
459C2×C3—362no—
460C2×C3—721no—
461C2×C3#6392611yescharacteristic
462C2×C3#6393711yescharacteristic
463C2×C3#6397111yescharacteristic
464D3—122no—
465D3—241no—
466D3—241no—
467D3—242no—
468D3—362no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—182no—
473C22—361no—
474C22—361no—
475C22—361no—
476C22—361no—
477C22—361no—
478C22#8867711yescharacteristic
479C4—92no—
480C4—181no—
481C4—361no—
482C4#8867111yescharacteristic
483C4#8867611yescharacteristic
484C3—31no—
485C3—31no—
486C3—61no—
487C3#1059143311yescharacteristic
488C2—81no—
489C2—362no—
490C2—721no—
491C2#1172119011yescharacteristic
492C2#1172119311yescharacteristic
493C2#1172131211yescharacteristic
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)#459
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. #11721850C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721136C2semidirect
4. C2×#88436C2semidirect
5. #63659⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720304C2semidirect
23. C2×#3753D3semidirect
32. #63659D3semidirect
52. C4⋊((C3⋊D3)⋊C3)D4semidirect
103. #1084D8semidirect
146. C4⋊(C3⋊D3)#67semidirect
273. H(3,𝔽3)#459semidirect
426. C32#4786semidirect
427. C32#5150semidirect

Maximal subgroups

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

subgroupindex
#117218502
C2×#3753⋊C32
#117211362
C2×#884362
#63659⋊C32
#117203242
#117203042
#105914293
#105914283