Finite Group Databasebeta

← #13140558

Subgroups — #13140558

Order 1728.

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

normal subgroups only

This group has 505 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#64147⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic, complemented
4C2×#88430C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10591454—31nomaximal
9#10591453—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6198—21no—
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3757⋊C3—21no—
18#3753⋊C3—22no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88430—22no—
23#64147D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic, complemented
25C3×#938—31no—
26C2×C3×#213—31no—
27#63916—31no—
28#63909—31no—
29C2×#3712—31no—
30C2×#3711—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—61no—
34#63574—31no—
35#63573—31no—
36#63564—31no—
37#63563—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45#638⋊C3—42no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6198—42no—
53#6180—21no—
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5159—91no—
56#5039—91no—
57#4782—91no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×#67—61no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—31no—
67C2×#626—61no—
68C2×#626—61no—
69#3757—21no—
70#3753—22no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×#217—61no—
74C3×#213—62no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×C4⋊Dic3—61no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85#3712—62no—
86#3711—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—21no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—41no—
97C4×H(3,𝔽3)C2×D411yescharacteristic
98C4×H(3,𝔽3)#4111yescharacteristic
99#1084—41no—
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102C2×C3×D8—91no—
103C3×#121—91no—
104#938—31no—
105#938—91no—
106#938—91no—
107C2×#213—31no—
108C2×#213—91no—
109#930—91no—
110C2×#208—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113#895—91no—
114C3×#102—91no—
115#832—31no—
116#832—31no—
117#832—61no—
118#832—91no—
119#821—91no—
120#817—91no—
121#809—91no—
122#808—91no—
123#808—91no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—41no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3#6711yescharacteristic
129C22×C3×D3—31no—
130C22×C3×D3—61no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—21no—
135C32×D4—21no—
136C32×D4—41no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140#638—42no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3#6711yescharacteristic
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—21no—
146C4×C3⋊D3—21no—
147C3×#67—122no—
148C2×C3×Dic3—31no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—121no—
153C3×D12—61no—
154C3×D12—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157#626—122no—
158#626—122no—
159C3⋊(C3⋊C8)#6712yescomplemented
160C3×C3⋊C8—32no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—121no—
164#469—271noSylow
165C2×H(3,𝔽3)—81no—
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169(C3⋊D3)⋊C3—41no—
170(C3⋊D3)⋊C3—81no—
171C2×C3×D4—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4—181no—
176C2×C3×D4D3211yescharacteristic
177C22×C3×C4—91no—
178C2×#67—61no—
179C2×#67—181no—
180C2×#67—181no—
181C2×#67—181no—
182C2×D12—31no—
183C2×D12—91no—
184C2×D12—181no—
185C2×C4×D3—31no—
186C2×C4×D3—91no—
187C3×D8—182no—
188C2×C3×C8—91no—
189C2×C3×C8—91no—
190C3×C4⋊C4—91no—
191C3×C4⋊C4—91no—
192C3×C22⋊C4—181no—
193#217—61no—
194#217—181no—
195#217—181no—
196#213—62no—
197#213—182no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200C4⋊Dic3—31no—
201C4⋊Dic3—31no—
202C4⋊Dic3—31no—
203C4⋊Dic3—61no—
204C4⋊Dic3—91no—
205#208—182no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—91no—
211ΓL(1,𝔽25)—182no—
212ΓL(1,𝔽25)—182no—
213C22×C32—21no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—81no—
222C22×C32—81no—
223C22×C32D3×D411yescharacteristic
224C22×C32C2×#6711yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—21no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—41no—
230C2×C3⋊D3—41no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4D3×D411yescharacteristic
240C32×C4#23711yescharacteristic
241C32×C4#23711yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—21no—
244C3⋊Dic3—41no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C3×Dic3—121no—
248C3×Dic3—121no—
249C3×Dic3—241no—
250C2×D8—91no—
251C2×M16—271no—
252#121—91no—
253#121—271no—
254#106—271no—
255#102—91no—
256#102—271no—
257H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—91no—
262C23×C3—121no—
263C23×C3—181no—
264C22×D3—31no—
265C22×D3—61no—
266C22×D3—91no—
267C22×D3—181no—
268C3×D4—21no—
269C3×D4—21no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—62no—
275C3×D4—121no—
276C3×D4—181no—
277C3×D4—181no—
278C3×D4—362no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4C2×D3211yescharacteristic
288#67—122no—
289#67—362no—
290#67—362no—
291#67—362no—
292C2×Dic3—31no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—91no—
297C2×Dic3—121no—
298C2×Dic3—181no—
299D12—61no—
300D12—61no—
301D12—181no—
302D12—181no—
303D12—362no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—181no—
307C4×D3—181no—
308C3×C8—92no—
309C3×C8—181no—
310C3⋊C8—32no—
311C3⋊C8—61no—
312C3⋊C8—61no—
313C3⋊C8—92no—
314C3⋊C8—121no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—82no—
319C2×C32—81no—
320C2×C32—81no—
321C2×C32#91111yescharacteristic
322C2×C32#91111yescharacteristic
323C2×C32#93211yescharacteristic
324C2×C32#93811yescharacteristic
325C2×C32#93911yescharacteristic
326C2×C32#94911yescharacteristic
327C3⋊D3—41no—
328C3⋊D3—81no—
329C3×D3—121no—
330C3×D3—241no—
331C2×D4—91no—
332C2×D4—181no—
333C2×D4—541no—
334C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C22×C4—91no—
336D8—182no—
337M16—542no—
338C2×C8—91no—
339C2×C8—91no—
340C4⋊C4—91no—
341C4⋊C4—91no—
342C22⋊C4—181no—
343C22⋊C4—541no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—61no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—122no—
357C22×C3—181no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—241no—
361C22×C3—241no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3D3×#6711yescharacteristic
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—181no—
373C2×D3—181no—
374C2×D3—181no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—361no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—181no—
385C3×C4—361no—
386C3×C4—361no—
387C3×C4#380511yescharacteristic
388C3×C4D3×#6711yescharacteristic
389Dic3—61no—
390Dic3—121no—
391Dic3—121no—
392Dic3—121no—
393Dic3—181no—
394Dic3—241no—
395Dic3—361no—
396C32—21no—
397C32#503911yescharacteristic, complemented
398C32#515911yescharacteristic, complemented
399C23—21no—
400C23—91no—
401C23—181no—
402D4—21no—
403D4—21no—
404D4—181no—
405D4—181no—
406D4—362no—
407D4—1082no—
408C2×C4—91no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
414C8—92no—
415C8—181no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—81no—
426C2×C3—241no—
427C2×C3—241no—
428C2×C3—242no—
429C2×C3—361no—
430C2×C3—721no—
431C2×C3#6392511yescharacteristic
432C2×C3#6393211yescharacteristic
433C2×C3#6397011yescharacteristic
434D3—121no—
435D3—241no—
436D3—361no—
437D3—721no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—181no—
442C22—181no—
443C22—181no—
444C22—361no—
445C22—361no—
446C22—361no—
447C22#8867611yescharacteristic
448C4—181no—
449C4—361no—
450C4—361no—
451C4#8867211yescharacteristic
452C4#8867611yescharacteristic
453C3—31no—
454C3—31no—
455C3—61no—
456C3#1059145111yescharacteristic
457C2—81no—
458C2—361no—
459C2—721no—
460C2#1172119511yescharacteristic
461C2#1172120011yescharacteristic
462C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#469
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. #64147⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721141C2semidirect
4. C2×#88430C2semidirect
5. #11720578C2semidirect
6. #11720318C2semidirect
7. #11720307C2semidirect
23. #64147D3semidirect
24. C2×#3753D3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
159. C3⋊(C3⋊C8)#67semidirect
257. H(3,𝔽3)#469semidirect
397. C32#5039semidirect
398. C32#5159semidirect

Maximal subgroups

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

subgroupindex
#64147⋊C32
C2×#3753⋊C32
#117211412
C2×#884302
#117205782
#117203182
#117203072
#105914543
#105914533