Finite Group Databasebeta

← #13140631

Subgroups — #13140631

Order 1728.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88546C211yesmaximal, characteristic, complemented
2D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3753⋊C3C211yesmaximal, characteristic
4#11721110C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591476—31nomaximal
9#10591471—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14#88546—22no—
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3753⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88450—21no—
23D4×C3⋊Dic3D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic
25C3×D4×Dic3—31no—
26C2×C3×#214—31no—
27C2×C3×#214—31no—
28C2×C3×#214—61no—
29C2×C3×#213—31no—
30#63864—31no—
31#63863—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63566—31no—
35#63565—31no—
36#63564—31no—
37#63563—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)D412yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—42no—
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5165—91no—
55#5153—91no—
56#5016—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C22×C3⋊Dic3—21no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C22×C3×Dic3—61no—
63C2×C3×D12—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3757—21no—
68#3753—22no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#217—61no—
73C3×#214—64no—
74C3×#214—62no—
75C3×#214—62no—
76C3×#213—62no—
77C3×C4⋊Dic3—31no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83#3722—61no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86(C2×D3)⋊Dic3—61no—
87C22×H(3,𝔽3)—22no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C2×C3×D8—91no—
102C3×C4×D4—91no—
103#947—91no—
104#936—91no—
105D4×Dic3—31no—
106D4×Dic3—91no—
107C2×#214—31no—
108C2×#214—31no—
109C2×#214—61no—
110C2×#213—31no—
111C2×#213—91no—
112#875—91no—
113#835—91no—
114#832—91no—
115#821—91no—
116#816—91no—
117Dic3⋊C8—91no—
118Dic3⋊C8—91no—
119#810—91no—
120#809—91no—
121C4⋊(C3⋊C8)—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3—21no—
126C22×C3×D3—61no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—22no—
130C32×D4—41no—
131C32×D4D1212yes—
132C32×D4#6712yes—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—42no—
138C2×C3⋊Dic3#6711yescharacteristic
139C2×C3⋊Dic3#6711yescharacteristic
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)—21no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—121no—
148C2×C3×Dic3—122no—
149C3×D12—61no—
150C3×D12—61no—
151C3×Dic6—62no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—121no—
157C3⋊(C3⋊C8)—21no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162#463—271noSylow
163C2×H(3,𝔽3)—42no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)#13511yescharacteristic
166C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
167(C3⋊D3)⋊C3—81no—
168C2×C3×Q8—91no—
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4D3211yescharacteristic
174C22×C3×C4—181no—
175C22×Dic3—61no—
176C22×Dic3—181no—
177C2×D12—31no—
178C2×D12—91no—
179C2×Dic6—31no—
180C2×Dic6—31no—
181C2×Dic6—61no—
182C3×SD16—182no—
183C3×D8—182no—
184C2×C3×C8—91no—
185C2×C3×C8—91no—
186C3×C4⋊C4—91no—
187C3×C22⋊C4—181no—
188C3×C42—91no—
189#217—61no—
190#217—181no—
191#217—181no—
192#217—181no—
193#214—62no—
194#214—62no—
195#214—64no—
196#213—62no—
197#213—182no—
198(C2×D3)⋊C4—181no—
199(C2×D3)⋊C4—181no—
200(C2×D3)⋊C4—181no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—31no—
203C4⋊Dic3—91no—
204C4×Dic3—31no—
205C4×Dic3—91no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—91no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—42no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—81no—
220C22×C32—81no—
221C22×C32D3×D411yescharacteristic
222C22×C32C2×#6711yescharacteristic
223C2×C3⋊D3—41no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4C2×D1211yescharacteristic
232C32×C4#23511yescharacteristic
233C32×C4#23711yescharacteristic
234C32×C4C2×#6711yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—21no—
237C3⋊Dic3—41no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—122no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C2×SD16—91no—
245C2×D8—91no—
246#124—271no—
247C4×D4—91no—
248C4⋊C8—271no—
249#103—271no—
250#102—271no—
251H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—121no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—181no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—21no—
262C3×D4—32no—
263C3×D4—32no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—62no—
267C3×D4—121no—
268C3×D4—181no—
269C3×D4—181no—
270C3×D4#62612yes—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—61no—
274C2×C3×C4—91no—
275C2×C3×C4—91no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—362no—
279C2×C3×C4C2×D3211yescharacteristic
280C2×Dic3—31no—
281C2×Dic3—31no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—91no—
286C2×Dic3—91no—
287C2×Dic3—121no—
288C2×Dic3—122no—
289C2×Dic3—181no—
290C2×Dic3—362no—
291D12—61no—
292D12—61no—
293D12—181no—
294D12—181no—
295Dic6—61no—
296Dic6—61no—
297Dic6—61no—
298Dic6—61no—
299Dic6—62no—
300Dic6—121no—
301C3×C8—181no—
302C3×C8—181no—
303C3⋊C8—61no—
304C3⋊C8—61no—
305C3⋊C8—62no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—42no—
312C2×C32—42no—
313C2×C32—82no—
314C2×C32#91111yescharacteristic
315C2×C32#91211yescharacteristic
316C2×C32#93011yescharacteristic
317C2×C32#93211yescharacteristic
318C2×C32#93911yescharacteristic
319C2×C32#95011yescharacteristic
320C3⋊D3—81no—
321C3×D3—241no—
322C2×Q8—91no—
323C2×D4—91no—
324C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C22×C4—181no—
326SD16—182no—
327D8—182no—
328C2×C8—91no—
329C2×C8—91no—
330C4⋊C4—91no—
331C22⋊C4—181no—
332C22⋊C4—541no—
333C22⋊C4—541no—
334C42—91no—
335C22×C3—22no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—62no—
342C22×C3—61no—
343C22×C3—122no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—241no—
349C22×C3—241no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3—361no—
353C22×C3D3×#6711yescharacteristic
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—181no—
367C3×C4—181no—
368C3×C4—361no—
369C3×C4—361no—
370C3×C4#380411yescharacteristic
371C3×C4C2×#62611yescharacteristic
372Dic3—61no—
373Dic3—61no—
374Dic3—121no—
375Dic3—121no—
376Dic3—122no—
377Dic3—121no—
378Dic3—181no—
379Dic3—181no—
380Dic3—361no—
381C32—21no—
382C32#501611yescharacteristic, complemented
383C32#515311yescharacteristic, complemented
384C23—21no—
385C23—181no—
386Q8—181no—
387Q8—181no—
388D4—21no—
389D4—181no—
390D4—181no—
391D4#620012yes—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—362no—
397C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C8—181no—
399C8—181no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—42no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—122no—
411C2×C3—122no—
412C2×C3—242no—
413C2×C3—721no—
414C2×C3#6393211yescharacteristic
415C2×C3#6393311yescharacteristic
416C2×C3#6397811yescharacteristic
417D3—241no—
418D3—721no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22#8867611yescharacteristic
426C4—181no—
427C4—181no—
428C4—361no—
429C4—361no—
430C4#8867111yescharacteristic
431C4C2×#620011yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3#1059146811yescharacteristic
436C2—42no—
437C2—721no—
438C2#1172120011yescharacteristic
439C2#1172120111yescharacteristic
440C2#1172132211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#3753⋊C3C2
hypercenterC2×D4((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 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88546C2semidirect
2. D4×(C3⋊Dic3)⋊C3C2semidirect
4. #11721110C2semidirect
5. #11720326C2semidirect
6. #11720318C2semidirect
7. #11720311C2semidirect
23. D4×C3⋊Dic3D3semidirect
251. H(3,𝔽3)#463semidirect
382. C32#5016semidirect
383. C32#5153semidirect

Maximal subgroups

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

subgroupindex
C2×#885462
D4×(C3⋊Dic3)⋊C32
C2×#3753⋊C32
#117211102
#117203262
#117203182
#117203112
#105914763
#105914713