Finite Group Databasebeta

← #13138002

Subgroups — #13138002

Order 1728.

519 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 519 conjugacy classes of subgroups — too many to draw. Only its 77 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#64104⋊C3C211yesmaximal, characteristic, complemented
2#11721156C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590509—31nomaximal
9#10590499—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×#6200—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21#88436—22no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64104D311yescharacteristic, complemented
24C3×#895—31no—
25#63916—31no—
26#63908—31no—
27C2×#3714—31no—
28C2×#3713—31no—
29C2×#3710—31no—
30C2×D3×C3⋊C8—31no—
31#63634D311yescharacteristic, complemented
32C3×#809—31no—
33C3×#809—31no—
34C3×#809—61no—
35C3×#807—31no—
36#63564—31no—
37#63563—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C4⋊#272—21no—
44C4⋊#272—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—42no—
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6200—42no—
53#6180—21no—
54(C3⋊(C3⋊C8))⋊C3D412yescomplemented
55#5038—91no—
56#4883—91no—
57#4775—91no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×D12—61no—
66C2×C3×C4×D3—31no—
67C2×#626—61no—
68C2×#625—61no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C2×D311yescharacteristic
72#3752—21no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81(C2×D3)⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83#3714—62no—
84#3713—62no—
85#3710—62no—
86D3×C3⋊C8—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—41no—
89C2×#272—41no—
90C2×#272—41no—
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3D812yescomplemented
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C2×D411yescharacteristic
99C4×H(3,𝔽3)#4111yescharacteristic
100(C3⋊Dic3)⋊C3—21no—
101(C3⋊Dic3)⋊C3—41no—
102C3×#121—91no—
103#938—91no—
104C2×#213—91no—
105C2×#213—91no—
106#930—91no—
107C22×C3⋊C8—91no—
108C2×D24—91no—
109C2×D3×C8—91no—
110C2×D3×C8—91no—
111#895—31no—
112#895—91no—
113#895—91no—
114C3×#107—91no—
115C3×#102—91no—
116#832—91no—
117#821—91no—
118#809—31no—
119#809—31no—
120#809—61no—
121#809—91no—
122#807—31no—
123#807—91no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3#6711yescharacteristic
126C22×C3×D3—31no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—121no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—41no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—42no—
142C4×C3⋊D3—21no—
143C4×C3⋊D3—21no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C3×D12—62no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—121no—
154C3×D12—122no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157#626—122no—
158#625—122no—
159C3⋊(C3⋊C8)D1212yescomplemented
160C3×C3⋊C8—32no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—121no—
164#466—271noSylow
165C2×H(3,𝔽3)#12111yescharacteristic
166C2×H(3,𝔽3)C2×D811yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168#272—81no—
169(C3⋊D3)⋊C3—22no—
170(C3⋊D3)⋊C3—81no—
171C2×C3×D4—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—91no—
175C2×#67—181no—
176C2×#67—181no—
177C2×D12—31no—
178C2×D12—31no—
179C2×D12—31no—
180C2×D12—61no—
181C2×D12—61no—
182C2×D12—91no—
183C2×D12—181no—
184C2×D12—181no—
185C2×C4×D3—31no—
186C2×C4×D3—91no—
187C2×C3×C8—91no—
188C2×C3×C8—91no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4D3211yescharacteristic
194C3×C22⋊C4—181no—
195#217—181no—
196#213—182no—
197#213—182no—
198(C2×D3)⋊C4—61no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—31no—
203C4⋊Dic3—91no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—182no—
210D24—182no—
211D3×C8—182no—
212D3×C8—182no—
213C22×C32—21no—
214C22×C32D3×D411yescharacteristic
215C22×C32C2×#6711yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3#21312yescomplemented
222C2×C3×D3—32no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C2×C3×D3—122no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—241no—
236C2×C3×D3—241no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—81no—
242C32×C4C2×D1211yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4#23711yescharacteristic
245C32×C4#23911yescharacteristic
246C3⋊Dic3—21no—
247C3⋊Dic3—41no—
248C3×Dic3—61no—
249C3×Dic3—121no—
250C2×D8—271no—
251C22×C8—271no—
252#121—91no—
253#121—271no—
254#107—91no—
255#102—91no—
256#102—271no—
257H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
258C23×C3—91no—
259C23×C3—181no—
260C23×C3—181no—
261C22×D3—31no—
262C22×D3—61no—
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—91no—
266C22×D3—121no—
267C22×D3—181no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—362no—
273C2×C3×C4—21no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—91no—
280C2×C3×C4—121no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4C2×D3211yescharacteristic
285#67—362no—
286#67—362no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—91no—
290C2×Dic3—181no—
291D12—61no—
292D12—61no—
293D12—61no—
294D12—61no—
295D12—62no—
296D12—61no—
297D12—61no—
298D12—121no—
299D12—122no—
300D12—181no—
301D12—181no—
302D12—362no—
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#89511yescharacteristic
319C2×C32D3×D811yescharacteristic
320C2×C32#91911yescharacteristic
321C2×C32C2×#21311yescharacteristic
322C2×C32#93911yescharacteristic
323C2×C32#95011yescharacteristic
324C3⋊D3—22no—
325C3⋊D3—81no—
326C3×D3—62no—
327C3×D3—242no—
328C3×D3—241no—
329C3×D3—241no—
330C3×D3—241no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—181no—
334C2×D4—541no—
335C22×C4—91no—
336D8—542no—
337C2×C8—91no—
338C2×C8—91no—
339C2×C8—542no—
340C4⋊C4—91no—
341C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
342C22⋊C4—181no—
343C22⋊C4—541no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—61no—
347C22×C3—92no—
348C22×C3—181no—
349C22×C3—181no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3—361no—
356C22×C3D3×#6711yescharacteristic
357C2×D3—32no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—92no—
361C2×D3—121no—
362C2×D3—121no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—122no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—181no—
372C2×D3—181no—
373C2×D3—241no—
374C2×D3—241no—
375C2×D3—361no—
376C2×D3—361no—
377C2×D3—361no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—31no—
382C3×C4—41no—
383C3×C4—61no—
384C3×C4—61no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—181no—
388C3×C4—241no—
389C3×C4—361no—
390C3×C4#379611yescharacteristic
391C3×C4D3×D1211yescharacteristic
392Dic3—61no—
393Dic3—121no—
394Dic3—181no—
395Dic3—361no—
396C32—21no—
397C32#488311yescharacteristic, complemented
398C32#503811yescharacteristic, complemented
399C23—91no—
400C23—181no—
401C23—181no—
402D4—181no—
403D4—181no—
404D4—181no—
405D4—181no—
406D4—362no—
407D4—1082no—
408C2×C4—21no—
409C2×C4—91no—
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—182no—
426C2×C3—721no—
427C2×C3—721no—
428C2×C3#6388611yescharacteristic
429C2×C3D3×#21311yescharacteristic
430C2×C3#6394911yescharacteristic
431D3—62no—
432D3—182no—
433D3—241no—
434D3—241no—
435D3—242no—
436D3—241no—
437D3—721no—
438C22—92no—
439C22—181no—
440C22—181no—
441C22—361no—
442C22—361no—
443C22—361no—
444C22—361no—
445C22—361no—
446C22—361no—
447C22#8867611yescharacteristic
448C4—41no—
449C4—181no—
450C4—361no—
451C4#8864911yescharacteristic
452C4#8865411yescharacteristic
453C3—31no—
454C3—31no—
455C3—61no—
456C3#1059057611yescharacteristic
457C2—182no—
458C2—721no—
459C2—721no—
460C2#1172112711yescharacteristic
461C2#1172119411yescharacteristic
462C2#1172123511yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64104⋊C3C2semidirect
2. #11721156C2semidirect
3. C2×#88436C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #11720531C2semidirect
6. #63634⋊C3C2semidirect
7. #11720318C2semidirect
23. #64104D3semidirect
31. #63634D3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
96. C2×(C3⋊D3)⋊C3D8semidirect
159. C3⋊(C3⋊C8)D12semidirect
221. C2×C3⋊D3#213semidirect
257. H(3,𝔽3)#466semidirect
397. C32#4883semidirect
398. C32#5038semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
#117211562
C2×#884362
C2×((C3⋊D3)⋊C3)⋊C82
#117205312
#63634⋊C32
#117203182
#105905093
#105904993