Finite Group Databasebeta

← #13137997

Subgroups — #13137997

Order 1728.

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

normal subgroups only

This group has 484 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#64102⋊C3C211yesmaximal, characteristic, complemented
2#11721114C211yesmaximal, characteristic, complemented
3C2×#88436C211yesmaximal, characteristic
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590540—31nomaximal
9#10590530—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88451—21no—
21#88436—22no—
22#88431—22no—
23#64102D311yescharacteristic, complemented
24C3×#893—31no—
25#63863—31no—
26#63860—31no—
27C2×#3717—31no—
28C2×#3716—31no—
29C2×#3714—31no—
30C2×#3713—31no—
31#63637D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#809—31no—
34C3×#809—31no—
35C3×#809—61no—
36#63572—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—21no—
43C4⋊#272—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49C4×(C3⋊D3)⋊C3—42no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5016—91no—
55#4819—91no—
56#4787—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—31no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—21no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80(C2×D3)⋊Dic3—61no—
81(C2×D3)⋊Dic3—61no—
82#3717—62no—
83#3716—62no—
84#3714—62no—
85#3713—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×#272—41no—
88C2×#272—41no—
89C2×#272—41no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C3×C4×D4—91no—
100#947—91no—
101C2×#215—91no—
102#936—91no—
103C2×#214—91no—
104C2×#213—91no—
105C2×#213—91no—
106C2×D24—91no—
107C2×#204—91no—
108#893—31no—
109#893—91no—
110#875—91no—
111C3×#103—91no—
112C3×#102—91no—
113Dic3⋊C8—91no—
114Dic3⋊C8—91no—
115#810—31no—
116#810—91no—
117#809—31no—
118#809—31no—
119#809—61no—
120C4⋊(C3⋊C8)—91no—
121C22×C3⋊D3—21no—
122C22×C3×D3—61no—
123C22×C3×D3—61no—
124C22×C3×D3—61no—
125C22×C3×D3—121no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3#6711yescharacteristic
134C2×C3⋊Dic3#6711yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)D1212yes—
137C4×C3⋊D3—42no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—61no—
143C3×D12—32no—
144C3×D12—62no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—121no—
151C3×C4×D3—122no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—61no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—121no—
159#463—271noSylow
160C2×H(3,𝔽3)#12111yescharacteristic
161C2×H(3,𝔽3)#13511yescharacteristic
162C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
163#272—81no—
164(C3⋊D3)⋊C3—42no—
165C2×C3×Q8—91no—
166C2×C3×D4—91no—
167C2×C3×D4—91no—
168C22×C3×C4—181no—
169C2×D12—31no—
170C2×D12—31no—
171C2×D12—31no—
172C2×D12—61no—
173C2×D12—91no—
174C2×C4×D3—61no—
175C2×C4×D3—181no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4D3211yescharacteristic
184C3×C22⋊C4—181no—
185C3×C42—91no—
186#217—181no—
187#217—181no—
188#215—182no—
189#214—182no—
190#213—182no—
191#213—182no—
192(C2×D3)⋊C4—61no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197(C2×D3)⋊C4—181no—
198C4×Dic3—31no—
199C4×Dic3—91no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—91no—
205D24—182no—
206#204—182no—
207C22×C32—21no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3×D3—62no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—122no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—241no—
224C2×C3×D3—241no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—81no—
230C32×C4C2×D1211yescharacteristic
231C32×C4#23511yescharacteristic
232C32×C4#23511yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3⋊Dic3—21no—
235C3⋊Dic3—21no—
236C3⋊Dic3—41no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—121no—
240C2×SD16—271no—
241C2×D8—271no—
242#124—271no—
243C4×D4—91no—
244C4⋊C8—271no—
245#103—91no—
246#102—91no—
247H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C23×C3—181no—
250C22×D3—61no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—121no—
254C22×D3—181no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—92no—
258C3×D4—181no—
259C3×D4—181no—
260C3×D4—181no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—121no—
270C2×C3×C4—181no—
271C2×C3×C4—362no—
272C2×C3×C4C2×D3211yescharacteristic
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—61no—
276C2×Dic3—91no—
277C2×Dic3—91no—
278C2×Dic3—181no—
279D12—32no—
280D12—61no—
281D12—61no—
282D12—61no—
283D12—61no—
284D12—62no—
285D12—61no—
286D12—92no—
287D12—121no—
288D12—181no—
289C4×D3—122no—
290C4×D3—362no—
291Dic6—61no—
292Dic6—61no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—181no—
296C3×C8—181no—
297C3⋊C8—61no—
298C3⋊C8—61no—
299C3⋊C8—61no—
300C3⋊C8—121no—
301C3⋊C8—181no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32#88411yescharacteristic
306C2×C32#91411yescharacteristic
307C2×C32#91911yescharacteristic
308C2×C32#93011yescharacteristic
309C2×C32#93211yescharacteristic
310C2×C32#95011yescharacteristic
311C3⋊D3—42no—
312C3×D3—122no—
313C3×D3—242no—
314C3×D3—241no—
315C3×D3—241no—
316C2×Q8—91no—
317C2×D4—91no—
318C2×D4—91no—
319C22×C4—181no—
320SD16—542no—
321D8—542no—
322C2×C8—91no—
323C2×C8—91no—
324C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C22⋊C4—181no—
326C22⋊C4—541no—
327C22⋊C4—541no—
328C42—91no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—61no—
332C22×C3—182no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3D3×#6711yescharacteristic
339C2×D3—62no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—122no—
349C2×D3—182no—
350C2×D3—241no—
351C2×D3—241no—
352C2×D3—361no—
353C2×D3—361no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—41no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—181no—
364C3×C4—181no—
365C3×C4—241no—
366C3×C4—361no—
367C3×C4#379811yescharacteristic
368C3×C4D3×D1211yescharacteristic
369Dic3—61no—
370Dic3—61no—
371Dic3—121no—
372Dic3—181no—
373Dic3—181no—
374Dic3—361no—
375C32—21no—
376C32#481911yescharacteristic, complemented
377C32#501611yescharacteristic, complemented
378C23—181no—
379C23—181no—
380Q8—181no—
381Q8—181no—
382D4—92no—
383D4—181no—
384D4—181no—
385D4—181no—
386C2×C4—21no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—181no—
390C2×C4—362no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C8—181no—
393C8—181no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—362no—
404C2×C3—721no—
405C2×C3#6390611yescharacteristic
406C2×C3#6393411yescharacteristic
407C2×C3#6394911yescharacteristic
408D3—122no—
409D3—241no—
410D3—241no—
411D3—242no—
412D3—362no—
413C22—182no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22—361no—
419C22#8867611yescharacteristic
420C4—41no—
421C4—181no—
422C4—181no—
423C4—361no—
424C4#8865011yescharacteristic
425C4#8865411yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3#1059053411yescharacteristic
430C2—362no—
431C2—721no—
432C2#1172114011yescharacteristic
433C2#1172119611yescharacteristic
434C2#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
prime radicalC2×#88436C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#463
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. #64102⋊C3C2semidirect
2. #11721114C2semidirect
4. C2×#88431C2semidirect
5. #11720531C2semidirect
6. #63637⋊C3C2semidirect
7. #11720311C2semidirect
23. #64102D3semidirect
31. #63637D3semidirect
247. H(3,𝔽3)#463semidirect
376. C32#4819semidirect
377. C32#5016semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#117211142
C2×#884362
C2×#884312
#117205312
#63637⋊C32
#117203112
#105905403
#105905303