Finite Group Databasebeta

← #13140966

Subgroups — #13140966

Order 1728.

516 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 83 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×#88549C211yesmaximal, characteristic, complemented
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#11721158C211yesmaximal, characteristic
5#11720324C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591622—31nomaximal
9#10591617—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C2×#6200—21no—
15#88549—22no—
16C2×#6180C2211yescharacteristic
17#3755⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C4×(C3⋊Dic3)⋊C3—21no—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23Q8×C3⋊Dic3D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic, complemented
25C3×Q8×Dic3—31no—
26C2×C3×#215—31no—
27C2×C3×#215—31no—
28C2×C3×#215—31no—
29C2×C3×#215—61no—
30#63914—31no—
31#63911—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63564—31no—
35#63563—31no—
36#63563—31no—
37#63562—31no—
38C22×#272—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)D412yescomplemented
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C4⋊#272—21no—
45C4⋊#272—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51#6200—42no—
52#6200—42no—
53#6180—21no—
54(C3⋊(C3⋊C8))⋊C3—21no—
55#5163—91no—
56#5163—91no—
57#5027—91no—
58C2×C32×Q8—21no—
59C2×C32×Q8C2×D311yescharacteristic
60C2×C32×Q8C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×#625—61no—
69C2×#625—61no—
70#3755—22no—
71C4⋊(C3⋊Dic3)—21no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×#215—64no—
77C3×#215—62no—
78C3×#215—62no—
79C3×#215—62no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4×Dic3—31no—
83C3×C4×Dic3—61no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—61no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×#272—41no—
90C2×#272—41no—
91C2×#272—41no—
92C2×(C3⋊D3)⋊C3—41no—
93C2×(C3⋊D3)⋊C3—41no—
94C2×(C3⋊D3)⋊C3—41no—
95C4×H(3,𝔽3)—22no—
96C4×H(3,𝔽3)C2×D411yescharacteristic
97C4×H(3,𝔽3)#4111yescharacteristic
98(C3⋊Dic3)⋊C3—21no—
99(C3⋊Dic3)⋊C3—41no—
100(C3⋊Dic3)⋊C3SD1612yescomplemented
101C2×C3×SD16—91no—
102C2×C3×SD16—91no—
103C3×C4×Q8—91no—
104Q8×Dic3—31no—
105Q8×Dic3—91no—
106C2×#215—31no—
107C2×#215—31no—
108C2×#215—31no—
109C2×#215—61no—
110C2×#215—91no—
111#940—91no—
112#940—91no—
113#874—91no—
114#832—91no—
115#832—91no—
116#821—91no—
117#821—91no—
118Dic3⋊C8—91no—
119Dic3⋊C8—91no—
120#809—91no—
121#809—91no—
122C4⋊(C3⋊C8)—91no—
123C22×C3⋊D3—21no—
124C22×C3×D3—61no—
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C22×C3×D3—121no—
128C32×Q8—21no—
129C32×Q8—21no—
130C32×Q8—22no—
131C32×Q8—41no—
132C32×Q8D1212yescomplemented
133C32×Q8#6712yescomplemented
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C22×D311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3#6711yescharacteristic
142C2×C3⋊Dic3#6711yescharacteristic
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C2×C3×Dic3—31no—
146C2×C3×Dic3—31no—
147C2×C3×Dic3—61no—
148C3×D12—62no—
149C3×D12—61no—
150C3×D12—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×D12—61no—
154C3×D12—61no—
155C3×D12—121no—
156#626—122no—
157#626—122no—
158#625—122no—
159#625—122no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—61no—
165#460—271noSylow
166C2×H(3,𝔽3)#12111yescharacteristic
167C2×H(3,𝔽3)C2×SD1611yescharacteristic
168C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
169#272—81no—
170(C3⋊D3)⋊C3—81no—
171C2×C3×Q8—31no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8D3211yescharacteristic
175C2×C3×D4—91no—
176C2×C3×D4—91no—
177C2×#67—181no—
178C2×#67—181no—
179C2×#67—181no—
180C2×#67—181no—
181C2×D12—31no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—61no—
185C2×D12—91no—
186C2×D12—181no—
187C2×D12—181no—
188C3×SD16—182no—
189C3×SD16—182no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—181no—
194C3×C42—91no—
195C3×C42—181no—
196#215—62no—
197#215—62no—
198#215—64no—
199#215—62no—
200#215—182no—
201C4⋊Dic3—31no—
202C4⋊Dic3—61no—
203C4⋊Dic3—91no—
204C4⋊Dic3—181no—
205C4×Dic3—31no—
206C4×Dic3—61no—
207C4×Dic3—91no—
208C4×Dic3—181no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—91no—
214C22×C32—21no—
215C22×C32D3×D411yescharacteristic
216C22×C32C2×#6711yescharacteristic
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3⋊D3—41no—
220C2×C3×D3—122no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—241no—
231C2×C3×D3—241no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—42no—
237C32×C4C2×D1211yescharacteristic
238C32×C4#23511yescharacteristic
239C32×C4#23711yescharacteristic
240C32×C4C2×#6711yescharacteristic
241C3⋊Dic3—21no—
242C3⋊Dic3—41no—
243C3⋊Dic3#21512yescomplemented
244C3×Dic3—32no—
245C3×Dic3—61no—
246C3×Dic3—121no—
247C2×SD16—91no—
248C2×SD16—91no—
249#127—271no—
250C4×Q8—91no—
251C4⋊C8—271no—
252#102—271no—
253#102—271no—
254H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
255C23×C3—181no—
256C23×C3—181no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—61no—
260C22×D3—121no—
261C22×D3—181no—
262C3×Q8—21no—
263C3×Q8—32no—
264C3×Q8—32no—
265C3×Q8—61no—
266C3×Q8—61no—
267C3×Q8—62no—
268C3×Q8—121no—
269C3×Q8#62612yes—
270C3×D4—181no—
271C3×D4—181no—
272C3×D4—181no—
273C3×D4—181no—
274C2×C3×C4—21no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—91no—
281C2×C3×C4—91no—
282C2×C3×C4—121no—
283C2×C3×C4—181no—
284C2×C3×C4C2×D3211yescharacteristic
285#67—362no—
286#67—362no—
287#67—362no—
288#67—362no—
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3—61no—
292C2×Dic3—91no—
293C2×Dic3—91no—
294C2×Dic3—181no—
295D12—61no—
296D12—61no—
297D12—61no—
298D12—61no—
299D12—62no—
300D12—61no—
301D12—61no—
302D12—121no—
303D12—181no—
304D12—181no—
305D12—362no—
306D12—362no—
307C3×C8—181no—
308C3×C8—181no—
309C3⋊C8—61no—
310C3⋊C8—61no—
311C3⋊C8—62no—
312C3⋊C8—61no—
313C3⋊C8—181no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32D3×SD1611yescharacteristic
318C2×C32#91411yescharacteristic
319C2×C32#93011yescharacteristic
320C2×C32#93911yescharacteristic
321C2×C32C2×#21511yescharacteristic
322C2×C32#94911yescharacteristic
323C3⋊D3—81no—
324C3×D3—242no—
325C3×D3—241no—
326C3×D3—241no—
327C3×D3—241no—
328C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—541no—
332C2×D4—541no—
333SD16—182no—
334SD16—182no—
335C2×C8—91no—
336C2×C8—91no—
337C4⋊C4—91no—
338C4⋊C4—181no—
339C42—91no—
340C42—181no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—61no—
344C22×C3—361no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3—361no—
349C22×C3—361no—
350C22×C3D3×#6711yescharacteristic
351C2×D3—121no—
352C2×D3—121no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—122no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—241no—
362C2×D3—241no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C3×C4—22no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—62no—
372C3×C4—62no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—92no—
376C3×C4—122no—
377C3×C4—181no—
378C3×C4—361no—
379C3×C4#380411yescharacteristic
380C3×C4C2×#62611yescharacteristic
381Dic3—32no—
382Dic3—61no—
383Dic3—92no—
384Dic3—121no—
385Dic3—181no—
386Dic3—361no—
387C32—21no—
388C32#502711yescharacteristic, complemented
389C32#516311yescharacteristic, complemented
390C23—181no—
391C23—181no—
392Q8—21no—
393Q8#620012yescomplemented
394D4—181no—
395D4—181no—
396D4—181no—
397D4—181no—
398D4—1082no—
399D4—1082no—
400C2×C4—21no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—181no—
404C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
405C8—181no—
406C8—181no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—61no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—721no—
417C2×C3—721no—
418C2×C3D3×#21511yescharacteristic
419C2×C3#6393911yescharacteristic
420C2×C3#6397811yescharacteristic
421D3—241no—
422D3—241no—
423D3—242no—
424D3—241no—
425D3—721no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22—361no—
431C22—361no—
432C22#8867611yescharacteristic
433C4—22no—
434C4—92no—
435C4—181no—
436C4—361no—
437C4#8867111yescharacteristic
438C4C2×#620011yescharacteristic
439C3—31no—
440C3—31no—
441C3—61no—
442C3#1059161711yescharacteristic
443C2—721no—
444C2—721no—
445C2#1172123211yescharacteristic
446C2#1172123311yescharacteristic
447C2#1172132211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721158C2
hypercenterC2×Q8((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×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88549C2semidirect
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720318C2semidirect
7. #11720311C2semidirect
23. Q8×C3⋊Dic3D3semidirect
24. C2×#3755D3semidirect
41. Q8×H(3,𝔽3)D4semidirect
100. (C3⋊Dic3)⋊C3SD16semidirect
132. C32×Q8D12semidirect
133. C32×Q8#67semidirect
243. C3⋊Dic3#215semidirect
254. H(3,𝔽3)#460semidirect
388. C32#5027semidirect
389. C32#5163semidirect
393. Q8#6200semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
Q8×(C3⋊Dic3)⋊C32
C2×#3755⋊C32
#117211582
#117203242
#117203182
#117203112
#105916223
#105916173