Finite Group Databasebeta

← #13140554

Subgroups — #13140554

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#64145⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3#11721132C211yesmaximal, characteristic, complemented
4C2×#88431C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591434—31nomaximal
9#10591432—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17#3757⋊C3—21no—
18#3753⋊C3—22no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88431—22no—
23#64145D311yescharacteristic, complemented
24C2×#3753D311yescharacteristic
25C3×#936—31no—
26C2×C3×#213—31no—
27#63898—31no—
28#63894—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#832—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63565—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49#634⋊C3—21no—
50#634⋊C3—21no—
51((C3⋊D3)⋊C3)⋊C4—42no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5153—91no—
55#5057—91no—
56#4787—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×Dic6—31no—
64C2×#624—61no—
65C2×D3×Dic3—61no—
66#3757—21no—
67#3757—21no—
68#3753—22no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#217—61no—
72C3×#217—61no—
73C3×#213—62no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—61no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—61no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84#3717—62no—
85#3716—62no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—21no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×D8—91no—
100C3×#124—91no—
101C2×#215—91no—
102#936—31no—
103#936—91no—
104D4×Dic3—91no—
105C2×#214—91no—
106C2×#213—31no—
107C2×#213—91no—
108C2×#204—91no—
109#893—91no—
110C4×D12—91no—
111C3×#102—91no—
112#835—91no—
113#832—31no—
114#832—31no—
115#832—61no—
116#816—91no—
117Dic3⋊C8—91no—
118Dic3⋊C8—91no—
119#810—91no—
120C4⋊(C3⋊C8)—91no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C22×C3⋊D3—21no—
125C22×C3×D3—61no—
126C32×D4—22no—
127C32×D4—21no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—41no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3#6711yescharacteristic
137C2×C3⋊Dic3#6711yescharacteristic
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)#6712yes—
140#634—21no—
141#634—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—
148C3×D12—32no—
149C3×D12—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152#624—122no—
153D3×Dic3—122no—
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)—81no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)#13511yescharacteristic
163C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
164(C3⋊D3)⋊C3—42no—
165C2×C3×Q8—91no—
166C2×C3×D4—31no—
167C2×C3×D4—31no—
168C2×C3×D4—61no—
169C2×C3×D4—91no—
170C2×C3×D4D3211yescharacteristic
171C22×Dic3—181no—
172C2×D12—31no—
173C2×D12—91no—
174C2×C4×D3—181no—
175C2×C4×D3—181no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×D8—182no—
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C3×C4⋊C4—91no—
182C3×C22⋊C4—181no—
183C3×C22⋊C4—181no—
184C3×C42—91no—
185#217—61no—
186#217—61no—
187#217—181no—
188#217—181no—
189#217—181no—
190#215—182no—
191#214—182no—
192#213—62no—
193#213—182no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—61no—
199C4×Dic3—31no—
200C4×Dic3—91no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8—91no—
206#204—182no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3—22no—
220C2×C3⋊D3—41no—
221C2×C3⋊D3—41no—
222C2×C3×D3—62no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4D3×D411yescharacteristic
228C32×C4#23711yescharacteristic
229C32×C4#23711yescharacteristic
230C32×C4C2×#6711yescharacteristic
231C3⋊Dic3—21no—
232C3⋊Dic3—21no—
233C3⋊Dic3—41no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C3×Dic3—241no—
240C2×SD16—271no—
241C2×D8—91no—
242#124—91no—
243C4×D4—271no—
244C4⋊C8—271no—
245#103—271no—
246#102—91no—
247H(3,𝔽3)#46311yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—121no—
252C23×C3—181no—
253C22×D3—61no—
254C22×D3—181no—
255C3×Q8—181no—
256C3×Q8—181no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—62no—
264C3×D4—92no—
265C3×D4—121no—
266C3×D4—181no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C2×D3211yescharacteristic
275C2×Dic3—31no—
276C2×Dic3—31no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—121no—
283C2×Dic3—181no—
284C2×Dic3—362no—
285D12—32no—
286D12—61no—
287D12—92no—
288D12—181no—
289C4×D3—362no—
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—82no—
306C2×C32—81no—
307C2×C32—81no—
308C2×C32#91111yescharacteristic
309C2×C32#91211yescharacteristic
310C2×C32#93811yescharacteristic
311C2×C32#93911yescharacteristic
312C2×C32#94911yescharacteristic
313C2×C32#95011yescharacteristic
314C3⋊D3—42no—
315C3×D3—122no—
316C2×Q8—91no—
317C2×D4—91no—
318C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
319C22×C4—541no—
320SD16—542no—
321D8—182no—
322C2×C8—91no—
323C2×C8—91no—
324C4⋊C4—91no—
325C22⋊C4—181no—
326C22⋊C4—181no—
327C22⋊C4—541no—
328C42—91no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—61no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—122no—
342C22×C3—182no—
343C22×C3—241no—
344C22×C3—241no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3D3×#6711yescharacteristic
348C2×D3—62no—
349C2×D3—121no—
350C2×D3—121no—
351C2×D3—182no—
352C2×D3—361no—
353C2×D3—361no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—31no—
357C3×C4—31no—
358C3×C4—61no—
359C3×C4—61no—
360C3×C4—181no—
361C3×C4—181no—
362C3×C4—361no—
363C3×C4—361no—
364C3×C4#380511yescharacteristic
365C3×C4D3×#6711yescharacteristic
366Dic3—61no—
367Dic3—61no—
368Dic3—121no—
369Dic3—121no—
370Dic3—121no—
371Dic3—181no—
372Dic3—181no—
373Dic3—241no—
374Dic3—361no—
375C32—21no—
376C32#505711yescharacteristic, complemented
377C32#515311yescharacteristic, complemented
378C23—21no—
379C23—181no—
380Q8—181no—
381Q8—181no—
382D4—21no—
383D4—21no—
384D4—92no—
385D4—181no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—1082no—
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—81no—
404C2×C3—241no—
405C2×C3—241no—
406C2×C3—242no—
407C2×C3—362no—
408C2×C3#6392511yescharacteristic
409C2×C3#6393311yescharacteristic
410C2×C3#6397011yescharacteristic
411D3—122no—
412D3—362no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—182no—
417C22—361no—
418C22—361no—
419C22#8867611yescharacteristic
420C4—181no—
421C4—181no—
422C4—361no—
423C4—361no—
424C4#8867211yescharacteristic
425C4#8867611yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3#1059143111yescharacteristic
430C2—81no—
431C2—362no—
432C2#1172119511yescharacteristic
433C2#1172120111yescharacteristic
434C2#1172131311yescharacteristic
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. #64145⋊C3C2semidirect
3. #11721132C2semidirect
4. C2×#88431C2semidirect
5. #11720578C2semidirect
6. #11720311C2semidirect
7. #11720309C2semidirect
23. #64145D3semidirect
247. H(3,𝔽3)#463semidirect
376. C32#5057semidirect
377. C32#5153semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
C2×#3753⋊C32
#117211322
C2×#884312
#117205782
#117203112
#117203092
#105914343
#105914323