Finite Group Databasebeta

← #13138962

Subgroups — #13138962

Order 1728.

566 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#88529C211yesmaximal, characteristic, complemented
2#11721158C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590753—31nomaximal
9#10590749—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C2×#6200—21no—
15C2×#6199—21no—
16C2×C8×H(3,𝔽3)C2211yescharacteristic
17#88529—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88436—22no—
23C2×C3×D24—31no—
24C2×C3×D24—31no—
25C2×C3×D24—61no—
26#63914—31no—
27#63911—31no—
28#63898—31no—
29Dic3×D12—31no—
30C2×#3714—31no—
31C2×#3713—31no—
32#63648D311yescharacteristic, complemented
33#63641D311yescharacteristic, complemented
34C3×#821—31no—
35C3×Dic3⋊C8—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×#272—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C4⋊#272—21no—
45C4⋊#272D412yescomplemented
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6200—42no—
52#6200—42no—
53#6199—42no—
54C8×H(3,𝔽3)—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5150—91no—
57#4843—91no—
58#4786—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×C3×D12—61no—
67C2×#626—61no—
68C2×#626—61no—
69C2×#625—61no—
70C2×#625—61no—
71C2×D3×Dic3—61no—
72C2×D3×Dic3—61no—
73C2×C32×C8—21no—
74C2×C32×C8C2×D311yescharacteristic
75C2×C32×C8C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4×C3⋊Dic3C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C3×C4⋊Dic3—31no—
80C3×C4×Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C3×D24—62no—
83C3×D24—62no—
84C3×D24—64no—
85(C2×D3)⋊Dic3—61no—
86(C2×D3)⋊Dic3—61no—
87#3714—62no—
88#3713—62no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×#272—22no—
91C2×#272—41no—
92C2×#272—41no—
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)C2×D411yescharacteristic
100C4×H(3,𝔽3)#4111yescharacteristic
101(C3⋊Dic3)⋊C3—21no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3D812yescomplemented
104C2×C3×D8—91no—
105#940—91no—
106#940—91no—
107D4×Dic3—91no—
108C2×#213—91no—
109C2×#213—91no—
110C2×D24—31no—
111C2×D24—31no—
112C2×D24—61no—
113C2×D24—91no—
114#893—91no—
115#874—91no—
116C4×D12—91no—
117C3×C4⋊C8—91no—
118C3×#102—91no—
119#832—91no—
120#821—31no—
121#821—91no—
122#821—91no—
123Dic3⋊C8—31no—
124Dic3⋊C8—91no—
125#809—91no—
126C22×C3⋊D3—21no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—121no—
133C22×C3×D3—121no—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3D1211yescharacteristic
139C2×C3⋊Dic3D1211yescharacteristic
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C3×D12—32no—
146C3×D12—32no—
147C3×D12—62no—
148C3×D12—62no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—61no—
156C3×D12—61no—
157C3×D12—121no—
158C3×D12—121no—
159#626—122no—
160#626—122no—
161#625—122no—
162#625—122no—
163D3×Dic3—122no—
164D3×Dic3—122no—
165C32×C8—22no—
166C32×C8—21no—
167C32×C8—21no—
168C3⋊(C3⋊C8)—21no—
169C3×C3⋊C8—61no—
170#459—271noSylow
171C2×H(3,𝔽3)#12111yescharacteristic
172C2×H(3,𝔽3)C2×D811yescharacteristic
173C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
174#272—42no—
175#272—81no—
176(C3⋊D3)⋊C3—81no—
177C2×C3×D4—91no—
178C2×C3×D4—91no—
179C2×C3×D4—91no—
180C2×#67—181no—
181C2×#67—181no—
182C2×#67—181no—
183C2×#67—181no—
184C22×Dic3—181no—
185C2×D12—31no—
186C2×D12—31no—
187C2×D12—31no—
188C2×D12—31no—
189C2×D12—31no—
190C2×D12—61no—
191C2×D12—61no—
192C2×D12—91no—
193C2×D12—181no—
194C2×D12—181no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C3×D8—182no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—61no—
201C2×C3×C8—91no—
202C2×C3×C8D3211yescharacteristic
203C3×C4⋊C4—91no—
204C3×C42—91no—
205#217—181no—
206#213—182no—
207#213—182no—
208(C2×D3)⋊C4—181no—
209(C2×D3)⋊C4—181no—
210C4⋊Dic3—31no—
211C4⋊Dic3—91no—
212C4×Dic3—31no—
213C4×Dic3—91no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8—91no—
216D24—62no—
217D24—62no—
218D24—64no—
219D24—182no—
220C22×C32—21no—
221C22×C32C2×D1211yescharacteristic
222C22×C32D3×D411yescharacteristic
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—122no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C2×C3×D3—241no—
244C2×C3×D3—241no—
245C2×C3×D3—241no—
246C2×C3×D3—241no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4#23511yescharacteristic
250C32×C4D3×D411yescharacteristic
251C32×C4D3×D411yescharacteristic
252C32×C4#23911yescharacteristic
253C3⋊Dic3—21no—
254C3⋊Dic3—41no—
255C3⋊Dic3D2412yescomplemented
256C3×Dic3—32no—
257C3×Dic3—61no—
258C3×Dic3—121no—
259C2×D8—91no—
260C2×D8—271no—
261#127—271no—
262C4×D4—271no—
263C4⋊C8—91no—
264#102—91no—
265#102—271no—
266H(3,𝔽3)#45911yescharacteristic, complemented, Sylow
267C23×C3—181no—
268C23×C3—181no—
269C23×C3—181no—
270C22×D3—61no—
271C22×D3—61no—
272C22×D3—61no—
273C22×D3—61no—
274C22×D3—61no—
275C22×D3—121no—
276C22×D3—121no—
277C22×D3—181no—
278C3×D4—92no—
279C3×D4—181no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4—181no—
283C3×D4—181no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—61no—
287C2×C3×C4—91no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4C2×D3211yescharacteristic
291#67—362no—
292#67—362no—
293#67—362no—
294#67—362no—
295C2×Dic3—31no—
296C2×Dic3—31no—
297C2×Dic3—61no—
298C2×Dic3—91no—
299C2×Dic3—91no—
300C2×Dic3—181no—
301C2×Dic3—362no—
302D12—32no—
303D12—32no—
304D12—62no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—61no—
309D12—61no—
310D12—61no—
311D12—62no—
312D12—61no—
313D12—61no—
314D12—121no—
315D12—121no—
316D12—181no—
317D12—181no—
318D12—362no—
319D12—362no—
320C4×D3—362no—
321C4×D3—362no—
322C3×C8—21no—
323C3×C8—61no—
324C3×C8—61no—
325C3×C8—62no—
326C3×C8—181no—
327C3⋊C8—61no—
328C3⋊C8—181no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#88411yescharacteristic
333C2×C32#89511yescharacteristic
334C2×C32C2×D2411yescharacteristic
335C2×C32#90811yescharacteristic
336C2×C32D3×D811yescharacteristic
337C2×C32#91411yescharacteristic
338C3⋊D3—81no—
339C3×D3—122no—
340C3×D3—122no—
341C3×D3—242no—
342C3×D3—242no—
343C3×D3—241no—
344C3×D3—241no—
345C3×D3—241no—
346C2×D4—91no—
347C2×D4—91no—
348C2×D4—91no—
349C2×D4—541no—
350C2×D4—541no—
351C22×C4—541no—
352D8—182no—
353D8—542no—
354C2×C8—91no—
355C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C4⋊C4—91no—
357C22⋊C4—541no—
358C42—91no—
359C22×C3—31no—
360C22×C3—31no—
361C22×C3—61no—
362C22×C3—182no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3—361no—
366C22×C3—361no—
367C22×C3—361no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3—361no—
371C22×C3D3×D1211yescharacteristic
372C2×D3—62no—
373C2×D3—62no—
374C2×D3—122no—
375C2×D3—121no—
376C2×D3—121no—
377C2×D3—121no—
378C2×D3—121no—
379C2×D3—121no—
380C2×D3—121no—
381C2×D3—121no—
382C2×D3—121no—
383C2×D3—121no—
384C2×D3—121no—
385C2×D3—122no—
386C2×D3—121no—
387C2×D3—121no—
388C2×D3—121no—
389C2×D3—241no—
390C2×D3—241no—
391C2×D3—241no—
392C2×D3—241no—
393C2×D3—361no—
394C2×D3—361no—
395C2×D3—361no—
396C3×C4—31no—
397C3×C4—31no—
398C3×C4—31no—
399C3×C4—31no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—92no—
403C3×C4—181no—
404C3×C4—361no—
405C3×C4#379911yescharacteristic
406C3×C4#381111yescharacteristic
407Dic3—32no—
408Dic3—61no—
409Dic3—92no—
410Dic3—121no—
411Dic3—181no—
412Dic3—361no—
413C32—21no—
414C32#478611yescharacteristic, complemented
415C32#484311yescharacteristic, complemented
416C23—181no—
417C23—181no—
418C23—181no—
419D4—92no—
420D4—181no—
421D4—181no—
422D4—181no—
423D4—181no—
424D4—181no—
425D4—1082no—
426D4—1082no—
427C2×C4—91no—
428C2×C4—91no—
429C2×C4—181no—
430C2×C4—1082no—
431C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C8—21no—
433C8—181no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—31no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—61no—
443C2×C3—362no—
444C2×C3—721no—
445C2×C3—721no—
446C2×C3D3×D2411yescharacteristic
447C2×C3#6379411yescharacteristic
448C2×C3#6391011yescharacteristic
449D3—122no—
450D3—122no—
451D3—242no—
452D3—241no—
453D3—241no—
454D3—242no—
455D3—241no—
456D3—721no—
457C22—182no—
458C22—361no—
459C22—361no—
460C22—361no—
461C22—361no—
462C22—361no—
463C22—361no—
464C22—361no—
465C22—361no—
466C22#8865411yescharacteristic
467C4—92no—
468C4—181no—
469C4—361no—
470C4#8865211yescharacteristic
471C4#8867711yescharacteristic
472C3—31no—
473C3—31no—
474C3—61no—
475C3#1059076511yescharacteristic
476C2—362no—
477C2—721no—
478C2—721no—
479C2#1172084711yescharacteristic
480C2#1172084811yescharacteristic
481C2#1172115711yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88529C2semidirect
2. #11721158C2semidirect
3. #11721149C2semidirect
4. C2×#88436C2semidirect
5. #63648⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. #11720324C2semidirect
32. #63648D3semidirect
33. #63641D3semidirect
45. C4⋊#272D4semidirect
103. (C3⋊Dic3)⋊C3D8semidirect
255. C3⋊Dic3D24semidirect
266. H(3,𝔽3)#459semidirect
414. C32#4786semidirect
415. C32#4843semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#117211582
#117211492
C2×#884362
#63648⋊C32
#63641⋊C32
#117203242
#105907533
#105907493