Finite Group Databasebeta

← #13137968

Subgroups — #13137968

Order 1728.

549 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721763C211yesmaximal, characteristic, complemented
2#11721159C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic, complemented
4C2×#88435C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590586—31nomaximal
9#10590504—31nomaximal
10C2×C4⋊#272—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×#6200—21no—
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16#88542—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88436—22no—
22#88435—22no—
23C3×#895—31no—
24C3×#895—31no—
25C3×#895—61no—
26#63909—31no—
27#63908—31no—
28C2×#3714—31no—
29C2×#3713—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63636D311yescharacteristic, complemented
33#63635D311yescharacteristic, complemented
34C3×#809—31no—
35C3×#808—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×#272D411yescharacteristic
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084D411yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—21no—
46C4⋊#272—42no—
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#5039—91no—
56#4865—91no—
57#4782—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C3×D12—31no—
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—61no—
63C2×C3×D12—61no—
64C2×C3×D12—61no—
65C2×C3×D12—62no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—61no—
69C2×#626—61no—
70C2×#625—61no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2×D311yescharacteristic
73C32×C4⋊C4C2×D311yescharacteristic
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×(C2×D3)⋊C4—61no—
78C3×(C2×D3)⋊C4—61no—
79C3×(C2×D3)⋊C4—62no—
80C3×C4⋊Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85#3714—62no—
86#3713—62no—
87#3711—62no—
88#3711—62no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×#272—21no—
91C2×#272—21no—
92C2×#272—21no—
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)—41no—
100C4×H(3,𝔽3)C2×D411yescharacteristic
101C4×H(3,𝔽3)#4111yescharacteristic
102#1084—21no—
103(C3⋊Dic3)⋊C3—41no—
104C3×#121—91no—
105#938—91no—
106C2×#213—91no—
107C2×#213—91no—
108#930—91no—
109C2×#208—91no—
110C2×D24—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—91no—
113#895—31no—
114#895—31no—
115#895—61no—
116#895—91no—
117C3×#106—91no—
118C3×#102—91no—
119#832—91no—
120#821—91no—
121#809—31no—
122#809—91no—
123#809—91no—
124#808—31no—
125#808—91no—
126C22×C3⋊D3—21no—
127C22×C3×D3—31no—
128C22×C3×D3—31no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C2×C32×C4—22no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—121no—
154C3×D12—121no—
155C3×D12—124no—
156C3×D12—122no—
157C3×D12—122no—
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)D1212yescomplemented
168C3×C3⋊C8—32no—
169C3×C3⋊C8—61no—
170#469—271noSylow
171C2×H(3,𝔽3)#12111yescharacteristic
172C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
173C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
174#272—41no—
175#272—81no—
176(C3⋊D3)⋊C3—81no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C22×C3×C4—91no—
181C2×#67—181no—
182C2×#67—181no—
183C2×D12—31no—
184C2×D12—31no—
185C2×D12—31no—
186C2×D12—61no—
187C2×D12—61no—
188C2×D12—61no—
189C2×D12—62no—
190C2×D12—91no—
191C2×D12—181no—
192C2×C4×D3—31no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C2×C3×C8—91no—
196C2×C3×C8—91no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—91no—
201C3×C4⋊C4D3211yescharacteristic
202C3×C22⋊C4—181no—
203#217—181no—
204#213—182no—
205#213—182no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—62no—
209(C2×D3)⋊C4—181no—
210(C2×D3)⋊C4—181no—
211C4⋊Dic3—31no—
212C4⋊Dic3—91no—
213#208—182no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—91no—
217C2×C3⋊C8—91no—
218D24—182no—
219ΓL(1,𝔽25)—182no—
220ΓL(1,𝔽25)—182no—
221C22×C32—21no—
222C22×C32D3×D411yescharacteristic
223C22×C32D3×D411yescharacteristic
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3×D3—62no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
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—42no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4C2×D1211yescharacteristic
254C32×C4#23511yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4#23911yescharacteristic
257C3⋊Dic3—41no—
258C3×Dic3—61no—
259C3×Dic3—61no—
260C3×Dic3—121no—
261C3×Dic3—121no—
262C2×D8—271no—
263C2×M16—271no—
264#121—91no—
265#121—271no—
266#106—91no—
267#102—91no—
268#102—271no—
269H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
270C23×C3—91no—
271C23×C3—181no—
272C23×C3—181no—
273C22×D3—31no—
274C22×D3—31no—
275C22×D3—61no—
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—
286C2×C3×C4—21no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—62no—
292C2×C3×C4—61no—
293C2×C3×C4—91no—
294C2×C3×C4—181no—
295C2×C3×C4—181no—
296C2×C3×C4—181no—
297C2×C3×C4C2×D3211yescharacteristic
298#67—362no—
299#67—362no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—181no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—61no—
309D12—61no—
310D12—61no—
311D12—121no—
312D12—121no—
313D12—122no—
314D12—122no—
315D12—124no—
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#88411yescharacteristic
335C2×C32#89511yescharacteristic
336C2×C32#91111yescharacteristic
337C2×C32#91411yescharacteristic
338C2×C32#91411yescharacteristic
339C2×C32#91411yescharacteristic
340C3⋊D3—81no—
341C3×D3—122no—
342C3×D3—121no—
343C3×D3—121no—
344C3×D3—242no—
345C3×D3—241no—
346C3×D3—241no—
347C3×D3—241no—
348C2×D4—91no—
349C2×D4—91no—
350C2×D4—181no—
351C2×D4—541no—
352C22×C4—91no—
353D8—542no—
354M16—542no—
355C2×C8—91no—
356C2×C8—91no—
357C4⋊C4—91no—
358C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
359C22⋊C4—181no—
360C22⋊C4—541no—
361C22×C3—31no—
362C22×C3—31no—
363C22×C3—61no—
364C22×C3—181no—
365C22×C3—181no—
366C22×C3—181no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3—361no—
373C22×C3#381111yescharacteristic
374C2×D3—61no—
375C2×D3—61no—
376C2×D3—61no—
377C2×D3—61no—
378C2×D3—61no—
379C2×D3—61no—
380C2×D3—62no—
381C2×D3—121no—
382C2×D3—121no—
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—31no—
399C3×C4—31no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—41no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—122no—
408C3×C4—181no—
409C3×C4—361no—
410C3×C4#379911yescharacteristic
411C3×C4D3×D1211yescharacteristic
412Dic3—61no—
413Dic3—61no—
414Dic3—121no—
415Dic3—121no—
416Dic3—361no—
417C32—21no—
418C32#478211yescharacteristic, complemented
419C32#486511yescharacteristic, complemented
420C23—91no—
421C23—181no—
422C23—181no—
423D4—181no—
424D4—181no—
425D4—181no—
426D4—181no—
427D4—362no—
428D4—1082no—
429C2×C4—21no—
430C2×C4—91no—
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—361no—
447C2×C3—721no—
448C2×C3—721no—
449C2×C3#6391011yescharacteristic
450C2×C3#6393711yescharacteristic
451C2×C3#6394111yescharacteristic
452D3—121no—
453D3—121no—
454D3—122no—
455D3—241no—
456D3—241no—
457D3—242no—
458D3—241no—
459D3—721no—
460C22—181no—
461C22—181no—
462C22—181no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22—361no—
467C22—361no—
468C22—361no—
469C22#8867711yescharacteristic
470C4—41no—
471C4—181no—
472C4—361no—
473C4#8865211yescharacteristic
474C4#8865411yescharacteristic
475C3—31no—
476C3—31no—
477C3—61no—
478C3#1059050811yescharacteristic
479C2—361no—
480C2—721no—
481C2—721no—
482C2#1172115711yescharacteristic
483C2#1172119311yescharacteristic
484C2#1172122911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721763C2semidirect
2. #11721159C2semidirect
3. C2×#88436C2semidirect
4. C2×#88435C2semidirect
5. #63636⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. #11720324C2semidirect
32. #63636D3semidirect
33. #63635D3semidirect
54. (C3⋊(C3⋊C8))⋊C3D4semidirect
167. C3⋊(C3⋊C8)D12semidirect
269. H(3,𝔽3)#469semidirect
418. C32#4782semidirect
419. C32#4865semidirect

Maximal subgroups

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

subgroupindex
#117217632
#117211592
C2×#884362
C2×#884352
#63636⋊C32
#63635⋊C32
#117203242
#105905863
#105905043