Finite Group Databasebeta

← #13138868

Subgroups — #13138868

Order 1728.

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

normal subgroups only

This group has 486 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
1C2×#3744⋊C3C211yesmaximal, characteristic, complemented
2#11721144C211yesmaximal, characteristic
3#11721084C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720547C211yesmaximal, characteristic, complemented
6#63648⋊C3C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590879—31nomaximal
9#10590799—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6198—21no—
14C2×#6198—21no—
15C2×C8×H(3,𝔽3)C2211yescharacteristic
16C4×#1084C2211yescharacteristic
17C2×#6180C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3744⋊C3—22no—
20#88448—21no—
21#88447—21no—
22#88431—22no—
23C2×#3744D311yescharacteristic, complemented
24C2×C3×#204—31no—
25#63911—31no—
26#63911—31no—
27#63844—31no—
28Dic3×Dic6—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31#63648D311yescharacteristic, complemented
32C3×#821—31no—
33C3×Dic3⋊C8—31no—
34C3×Dic3⋊C8—31no—
35C3×Dic3⋊C8—61no—
36#63564—31no—
37#63563—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084D411yescharacteristic
42C2×#1084D411yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49#634⋊C3—21no—
50#634⋊C3D412yescomplemented
51#6198—42no—
52#6198—42no—
53C8×H(3,𝔽3)—21no—
54#6180—21no—
55#5163—91no—
56#4848—91no—
57#4762—91no—
58C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×Dic6—31no—
64C2×#626—61no—
65C2×#626—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×C32×C8—21no—
69C2×C32×C8C2×D311yescharacteristic
70C2×C32×C8C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72#3744—22no—
73C3×C4⋊Dic3—31no—
74C3×C4×Dic3—31no—
75C3×C4×Dic3—31no—
76C3×C4×Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—61no—
80C3×#204—62no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83Dic32—61no—
84Dic32—61no—
85#3717—62no—
86#3716—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
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—21no—
97#1084SD1612yescomplemented
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101Q8×Dic3—91no—
102C2×#215—91no—
103#940—91no—
104#940—91no—
105C2×#214—91no—
106C2×#204—31no—
107C2×#204—91no—
108C2×#204—91no—
109#887—91no—
110#874—91no—
111C4×Dic6—91no—
112C3×C4⋊C8—91no—
113C3×#102—91no—
114#832—91no—
115#821—31no—
116#821—91no—
117#821—91no—
118Dic3⋊C8—31no—
119Dic3⋊C8—31no—
120Dic3⋊C8—61no—
121#809—91no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—21no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C2×C32×C4—21no—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134C4⋊(C3⋊D3)—21no—
135#634—21no—
136#634#6712yescomplemented
137C2×C3×Dic3—31no—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×D12—61no—
149C3×Dic6—32no—
150C3×Dic6—61no—
151#626—122no—
152#626—122no—
153#626—122no—
154#626—122no—
155C32×C8—21no—
156C32×C8—21no—
157C32×C8—41no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160C3×C3⋊C8—121no—
161#460—271noSylow
162C2×H(3,𝔽3)#12111yescharacteristic
163C2×H(3,𝔽3)C2×SD1611yescharacteristic
164C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
165(C3⋊D3)⋊C3—81no—
166(C3⋊D3)⋊C3—81no—
167C2×C3×Q8—91no—
168C2×C3×D4—91no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C2×#67—181no—
172C2×#67—181no—
173C2×#67—181no—
174C2×D12—31no—
175C2×D12—31no—
176C2×D12—91no—
177C2×D12—91no—
178C2×D12—181no—
179C2×D12—181no—
180C2×Dic6—31no—
181C2×Dic6—91no—
182C3×SD16—182no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—61no—
186C2×C3×C8—91no—
187C2×C3×C8D3211yescharacteristic
188C3×C4⋊C4—91no—
189C3×C42—91no—
190#215—182no—
191#214—182no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195Dic3⋊C4—181no—
196Dic3⋊C4—181no—
197C4×Dic3—31no—
198C4×Dic3—31no—
199C4×Dic3—61no—
200C4×Dic3—181no—
201C4×Dic3—181no—
202C4×Dic3—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—61no—
206#204—62no—
207#204—182no—
208#204—182no—
209C22×C32—21no—
210C22×C32D3×D411yescharacteristic
211C22×C32D3×D411yescharacteristic
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3—41no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4#23511yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4#23711yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—22no—
231C3⋊Dic3—41no—
232C3×Dic3—32no—
233C3×Dic3—32no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C2×SD16—91no—
241C2×SD16—271no—
242#127—271no—
243C4×Q8—271no—
244C4⋊C8—91no—
245#102—91no—
246#102—271no—
247H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C23×C3—181no—
250C22×D3—61no—
251C22×D3—61no—
252C22×D3—181no—
253C22×D3—181no—
254C3×Q8—92no—
255C3×Q8—181no—
256C3×D4—181no—
257C3×D4—181no—
258C3×D4—181no—
259C3×D4—181no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—61no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—181no—
266C2×C3×C4—181no—
267C2×C3×C4C2×D3211yescharacteristic
268#67—362no—
269#67—362no—
270#67—362no—
271#67—362no—
272C2×Dic3—31no—
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—31no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—181no—
281C2×Dic3—181no—
282D12—61no—
283D12—61no—
284D12—61no—
285D12—61no—
286D12—181no—
287D12—181no—
288D12—181no—
289D12—181no—
290D12—362no—
291D12—362no—
292Dic6—32no—
293Dic6—61no—
294Dic6—92no—
295Dic6—181no—
296C3×C8—21no—
297C3×C8—61no—
298C3×C8—61no—
299C3×C8—121no—
300C3×C8—181no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—121no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32#88411yescharacteristic
308C2×C32#91111yescharacteristic
309C2×C32D3×SD1611yescharacteristic
310C2×C32D3×SD1611yescharacteristic
311C2×C32#91411yescharacteristic
312C2×C32#93911yescharacteristic
313C3⋊D3—81no—
314C3⋊D3—81no—
315C3×D3—241no—
316C3×D3—241no—
317C2×Q8—91no—
318C2×D4—91no—
319C2×D4—91no—
320C2×D4—541no—
321C2×D4—541no—
322SD16—182no—
323SD16—542no—
324C2×C8—91no—
325C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
326C4⋊C4—91no—
327C4⋊C4—541no—
328C42—91no—
329C42—541no—
330C22×C3—31no—
331C22×C3—31no—
332C22×C3—61no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3#380211yescharacteristic
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—361no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—361no—
350C2×D3—361no—
351C2×D3—361no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—61no—
357C3×C4—61no—
358C3×C4—92no—
359C3×C4—182no—
360C3×C4—181no—
361C3×C4—361no—
362C3×C4#379511yescharacteristic
363C3×C4D3×#6711yescharacteristic
364Dic3—32no—
365Dic3—32no—
366Dic3—62no—
367Dic3—61no—
368Dic3—61no—
369Dic3—62no—
370Dic3—121no—
371Dic3—121no—
372Dic3—182no—
373Dic3—361no—
374C32—21no—
375C32#476211yescharacteristic, complemented
376C32#516311yescharacteristic, complemented
377C23—181no—
378C23—181no—
379Q8—92no—
380Q8—181no—
381D4—181no—
382D4—181no—
383D4—181no—
384D4—181no—
385D4—1082no—
386D4—1082no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C8—21no—
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—721no—
404C2×C3—721no—
405C2×C3#6379611yescharacteristic
406C2×C3#6379811yescharacteristic
407C2×C3#6388511yescharacteristic
408D3—241no—
409D3—241no—
410D3—721no—
411D3—721no—
412C22—361no—
413C22—361no—
414C22—361no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22#8865511yescharacteristic
419C4—92no—
420C4—181no—
421C4—182no—
422C4—361no—
423C4#8864811yescharacteristic
424C4#8867611yescharacteristic
425C3—31no—
426C3—31no—
427C3—61no—
428C3#1059087611yescharacteristic
429C2—721no—
430C2—721no—
431C2#1172083711yescharacteristic
432C2#1172083911yescharacteristic
433C2#1172112611yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721144C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#460
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×#3744⋊C3C2semidirect
3. #11721084C2semidirect
4. C2×#88431C2semidirect
5. #11720547C2semidirect
6. #63648⋊C3C2semidirect
7. #11720318C2semidirect
23. C2×#3744D3semidirect
31. #63648D3semidirect
50. #634⋊C3D4semidirect
97. #1084SD16semidirect
136. #634#67semidirect
247. H(3,𝔽3)#460semidirect
375. C32#4762semidirect
376. C32#5163semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
#117211442
#117210842
C2×#884312
#117205472
#63648⋊C32
#117203182
#105908793
#105907993