Finite Group Databasebeta

← #13137967

Subgroups — #13137967

Order 1728.

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

normal subgroups only

This group has 514 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
2#11721094C211yesmaximal, characteristic, complemented
3C2×#88437C211yesmaximal, characteristic, complemented
4C2×#88435C211yesmaximal, characteristic, complemented
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590588—31nomaximal
9#10590515—31nomaximal
10C2×C4⋊#272—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15#88542—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88448—21no—
21#88437—22no—
22#88435—22no—
23C3×#895—31no—
24C3×#895—31no—
25C3×#895—61no—
26#63857—31no—
27#63856—31no—
28C2×#3716—31no—
29C2×#3715—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32#63637D311yescharacteristic, complemented
33#63634D311yescharacteristic, complemented
34C3×#810—31no—
35C3×#807—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×#272D411yescharacteristic
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084D411yescharacteristic
43C4⋊#272—21no—
44C4⋊#272—21no—
45C4⋊#272—42no—
46C4×#272—21no—
47C4×#272—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3D412yescomplemented
54#5042—91no—
55#4886—91no—
56#4812—91no—
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×D12—61no—
62C2×C3×D12—61no—
63C2×C3×D12—62no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4C2×D311yescharacteristic
70C32×C4⋊C4C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×(C2×D3)⋊C4—61no—
75C3×(C2×D3)⋊C4—61no—
76C3×(C2×D3)⋊C4—62no—
77C3×C4⋊Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80#3724—61no—
81Dic3⋊Dic3—61no—
82(C2×D3)⋊Dic3—61no—
83(C2×D3)⋊Dic3—61no—
84#3716—62no—
85#3715—62no—
86#3711—62no—
87#3711—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—21no—
90C2×#272—21no—
91C2×#272—21no—
92C2×#272—41no—
93C2×#272—41no—
94C2×#272—41no—
95C4×H(3,𝔽3)—41no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98#1084—21no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3—41no—
101C3×#121—91no—
102#946—91no—
103C2×#215—91no—
104C2×#214—91no—
105#924—91no—
106C2×#208—91no—
107C2×#204—91no—
108C2×ΓL(1,𝔽25)—91no—
109C2×ΓL(1,𝔽25)—91no—
110#897—91no—
111#895—31no—
112#895—31no—
113#895—61no—
114C3×#107—91no—
115C3×#103—91no—
116#832—91no—
117#821—91no—
118#810—31no—
119#810—91no—
120#809—91no—
121#807—31no—
122#807—91no—
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C22×C3×D3—61no—
126C22×C3×D3—62no—
127C22×C3×D3—61no—
128C22×C3×D3—61no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—121no—
149C3×D12—121no—
150C3×D12—124no—
151C3×D12—122no—
152C3×D12—122no—
153C3×C4×D3—61no—
154C3×C4×D3—61no—
155C3×C4×D3—61no—
156C3×C4×D3—61no—
157C3×C4×D3—121no—
158C3×C4×D3—121no—
159C3×Dic6—61no—
160C3×Dic6—61no—
161C3⋊(C3⋊C8)—21no—
162C3⋊(C3⋊C8)D1212yescomplemented
163C3×C3⋊C8—32no—
164C3×C3⋊C8—61no—
165#468—271noSylow
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
168C2×H(3,𝔽3)#13711yescharacteristic
169#272—41no—
170#272—81no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—91no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—61no—
178C2×D12—61no—
179C2×D12—61no—
180C2×D12—62no—
181C2×C4×D3—31no—
182C2×C4×D3—31no—
183C2×C4×D3—61no—
184C2×Dic6—31no—
185C2×Dic6—91no—
186C2×C3×C8—91no—
187C2×C3×C8—91no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—91no—
192C3×C4⋊C4D3211yescharacteristic
193C3×C22⋊C4—181no—
194#217—181no—
195#215—182no—
196#214—182no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—61no—
199(C2×D3)⋊C4—62no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—31no—
203C4⋊Dic3—91no—
204C4⋊Dic3—181no—
205Dic3⋊C4—181no—
206Dic3⋊C4—181no—
207#208—182no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—91no—
211C2×C3⋊C8—91no—
212#204—182no—
213ΓL(1,𝔽25)—182no—
214ΓL(1,𝔽25)—182no—
215C22×C32—21no—
216C22×C32D3×D411yescharacteristic
217C22×C32D3×D411yescharacteristic
218C2×C3×D3—62no—
219C2×C3×D3—61no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—122no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—121no—
236C32×C4—21no—
237C32×C4—21no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4C2×D1211yescharacteristic
242C32×C4#23511yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4#23911yescharacteristic
245C3⋊Dic3—41no—
246C3⋊Dic3—41no—
247C3×Dic3—61no—
248C3×Dic3—61no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C3×Dic3—121no—
252C2×SD16—271no—
253C2×M16—271no—
254C22⋊Q8—271no—
255#121—91no—
256#107—91no—
257#103—91no—
258#102—271no—
259H(3,𝔽3)#46811yescharacteristic, complemented, Sylow
260C23×C3—91no—
261C23×C3—181no—
262C22×D3—31no—
263C22×D3—31no—
264C22×D3—61no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—62no—
268C3×Q8—181no—
269C3×Q8—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—62no—
279C2×C3×C4—61no—
280C2×C3×C4—91no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C2×D3211yescharacteristic
286C2×Dic3—31no—
287C2×Dic3—31no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293D12—61no—
294D12—61no—
295D12—61no—
296D12—61no—
297D12—121no—
298D12—121no—
299D12—122no—
300D12—122no—
301D12—124no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—121no—
307C4×D3—121no—
308Dic6—61no—
309Dic6—61no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—92no—
313C3×C8—181no—
314C3⋊C8—32no—
315C3⋊C8—61no—
316C3⋊C8—92no—
317C3⋊C8—181no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32#88411yescharacteristic
322C2×C32#89511yescharacteristic
323C2×C32#91111yescharacteristic
324C2×C32#91411yescharacteristic
325C2×C32#91811yescharacteristic
326C2×C32#91811yescharacteristic
327C3×D3—122no—
328C3×D3—121no—
329C3×D3—121no—
330C3×D3—242no—
331C3×D3—241no—
332C3×D3—241no—
333C2×Q8—91no—
334C2×D4—91no—
335C2×D4—181no—
336C22×C4—91no—
337SD16—542no—
338M16—542no—
339C2×C8—91no—
340C2×C8—91no—
341C4⋊C4—91no—
342C4⋊C4—541no—
343C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C22⋊C4—181no—
345C22⋊C4—541no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—61no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—181no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3—361no—
355C22×C3#381111yescharacteristic
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—62no—
363C2×D3—121no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—122no—
372C2×D3—122no—
373C2×D3—122no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—41no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—121no—
382C3×C4—121no—
383C3×C4—122no—
384C3×C4—181no—
385C3×C4—361no—
386C3×C4—361no—
387C3×C4#379911yescharacteristic
388C3×C4D3×D1211yescharacteristic
389Dic3—61no—
390Dic3—61no—
391Dic3—121no—
392Dic3—121no—
393Dic3—121no—
394Dic3—361no—
395Dic3—361no—
396C32—21no—
397C32#481211yescharacteristic, complemented
398C32#488611yescharacteristic, complemented
399C23—91no—
400C23—181no—
401Q8—181no—
402Q8—181no—
403D4—181no—
404D4—181no—
405D4—362no—
406C2×C4—21no—
407C2×C4—91no—
408C2×C4—181no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
413C8—92no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—361no—
425C2×C3—721no—
426C2×C3#6391011yescharacteristic
427C2×C3#6393711yescharacteristic
428C2×C3#6394511yescharacteristic
429D3—121no—
430D3—121no—
431D3—122no—
432D3—241no—
433D3—241no—
434D3—242no—
435C22—181no—
436C22—181no—
437C22—181no—
438C22—361no—
439C22—361no—
440C22—361no—
441C22#8867711yescharacteristic
442C4—41no—
443C4—181no—
444C4—361no—
445C4—361no—
446C4#8865211yescharacteristic
447C4#8865411yescharacteristic
448C3—31no—
449C3—31no—
450C3—61no—
451C3#1059052211yescharacteristic
452C2—361no—
453C2—721no—
454C2#1172115711yescharacteristic
455C2#1172119311yescharacteristic
456C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721763C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#468
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
2. #11721094C2semidirect
3. C2×#88437C2semidirect
4. C2×#88435C2semidirect
5. #63637⋊C3C2semidirect
6. #63634⋊C3C2semidirect
7. #11720324C2semidirect
32. #63637D3semidirect
33. #63634D3semidirect
53. (C3⋊(C3⋊C8))⋊C3D4semidirect
162. C3⋊(C3⋊C8)D12semidirect
259. H(3,𝔽3)#468semidirect
397. C32#4812semidirect
398. C32#4886semidirect

Maximal subgroups

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

subgroupindex
#117217632
#117210942
C2×#884372
C2×#884352
#63637⋊C32
#63634⋊C32
#117203242
#105905883
#105905153