Finite Group Databasebeta

← #13140629

Subgroups — #13140629

Order 1728.

502 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 75 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
2#64147⋊C3C211yesmaximal, characteristic
3#11721107C211yesmaximal, characteristic, complemented
4C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720314C211yesmaximal, characteristic, complemented
8#10591499—31nomaximal
9#10591465—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#638⋊C3—21no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14#88546—22no—
15C2×#6180C2211yescharacteristic
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#88448—21no—
22((C3⋊D3)⋊C3)⋊C8—22no—
23#64147D311yescharacteristic
24C3×#938—31no—
25C2×C3×#214—31no—
26C2×C3×#214—31no—
27C2×C3×#214—61no—
28#63871—31no—
29#63856—31no—
30C2×#3710—31no—
31C2×D3×C3⋊C8—31no—
32#63659D311yescharacteristic, complemented
33C3×#832—31no—
34#63573—31no—
35#63573—31no—
36#63566—31no—
37#63565—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3D411yescharacteristic
40D4×H(3,𝔽3)—21no—
41D4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229—21no—
46#638⋊C3—42no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3—21no—
51C4×(C3⋊D3)⋊C3—21no—
52#6180D412yescomplemented
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5171—91no—
55#5041—91no—
56#4778—91no—
57C2×C32×D4—21no—
58C2×C32×D4C2×D311yescharacteristic
59C2×C32×D4C2×D311yescharacteristic
60C2×#638—21no—
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×#67—61no—
63C2×C3×C4×D3—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3757—21no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#217—61no—
72C3×#214—62no—
73C3×#214—62no—
74C3×#214—64no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—61no—
81#3724—61no—
82Dic3⋊Dic3—61no—
83#3722—61no—
84(C2×D3)⋊Dic3—61no—
85#3710—62no—
86D3×C3⋊C8—62no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3SD1612yescomplemented
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96#1084—41no—
97(C3⋊Dic3)⋊C3—21no—
98(C3⋊Dic3)⋊C3—41no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×SD16—91no—
101C3×#121—91no—
102#946—91no—
103#938—31no—
104#938—91no—
105C2×#214—31no—
106C2×#214—31no—
107C2×#214—61no—
108#924—91no—
109C22×C3⋊C8—91no—
110C2×D3×C8—91no—
111C2×D3×C8—91no—
112#897—91no—
113C3×#102—91no—
114#835—91no—
115#832—31no—
116#832—91no—
117#817—91no—
118#816—91no—
119#810—91no—
120#808—91no—
121#808—91no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—41no—
125C22×C3⋊D3#6711yescharacteristic
126C22×C3×D3—31no—
127C32×D4—22no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C22×D311yescharacteristic
136#638—42no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3#6711yescharacteristic
140C4×C3⋊D3—21no—
141C4×C3⋊D3—21no—
142C3×#67—122no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—121no—
149C3×C4×D3—61no—
150C3×C4×D3—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—32no—
159C3×C3⋊C8—32no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—61no—
162#465—271noSylow
163C2×H(3,𝔽3)—81no—
164C2×H(3,𝔽3)#12111yescharacteristic
165C2×H(3,𝔽3)C2×SD1611yescharacteristic
166C2×H(3,𝔽3)#13511yescharacteristic
167(C3⋊D3)⋊C3—22no—
168C2×C3×Q8—91no—
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—181no—
173C2×C3×D4D3211yescharacteristic
174C22×C3×C4—91no—
175C2×#67—61no—
176C2×#67—181no—
177C2×C4×D3—31no—
178C2×C4×D3—91no—
179C2×Dic6—31no—
180C2×Dic6—31no—
181C2×Dic6—61no—
182C3×SD16—182no—
183C2×C3×C8—91no—
184C2×C3×C8—91no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—91no—
187C3×C22⋊C4—181no—
188#217—61no—
189#217—181no—
190#217—181no—
191#214—62no—
192#214—62no—
193#214—64no—
194(C2×D3)⋊C4—181no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—91no—
199C4⋊Dic3—91no—
200C4⋊Dic3—181no—
201Dic3⋊C4—181no—
202Dic3⋊C4—181no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—182no—
209D3×C8—182no—
210D3×C8—182no—
211C22×C32—21no—
212C22×C32—42no—
213C22×C32—41no—
214C22×C32—41no—
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—21no—
224C2×C3⋊D3—21no—
225C2×C3⋊D3#21412yescomplemented
226C2×C3×D3—32no—
227C2×C3×D3—61no—
228C2×C3×D3—61no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4#23511yescharacteristic
232C32×C4D3×D411yescharacteristic
233C32×C4D3×D411yescharacteristic
234C32×C4#23711yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—41no—
237C3⋊Dic3—41no—
238C3×Dic3—61no—
239C3×Dic3—122no—
240C3×Dic3—121no—
241C3×Dic3—121no—
242C3×Dic3—121no—
243C3×Dic3—121no—
244C2×SD16—91no—
245C22×C8—271no—
246C22⋊Q8—271no—
247#121—91no—
248#106—271no—
249#103—271no—
250#102—91no—
251H(3,𝔽3)#46511yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—61no—
254C23×C3—61no—
255C23×C3—91no—
256C23×C3—121no—
257C22×D3—31no—
258C22×D3—91no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—21no—
262C3×D4—21no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—61no—
266C3×D4—61no—
267C3×D4—62no—
268C3×D4—121no—
269C3×D4—362no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—61no—
273C2×C3×C4—91no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4C2×D3211yescharacteristic
280#67—122no—
281#67—362no—
282C2×Dic3—31no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—91no—
288C2×Dic3—121no—
289C2×Dic3—181no—
290C2×Dic3—181no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—181no—
294C4×D3—181no—
295Dic6—61no—
296Dic6—61no—
297Dic6—61no—
298Dic6—61no—
299Dic6—62no—
300Dic6—121no—
301C3×C8—92no—
302C3×C8—181no—
303C3⋊C8—32no—
304C3⋊C8—32no—
305C3⋊C8—62no—
306C3⋊C8—61no—
307C3⋊C8—181no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—82no—
312C2×C32—81no—
313C2×C32—81no—
314C2×C32#88411yescharacteristic
315C2×C32#91211yescharacteristic
316C2×C32D3×SD1611yescharacteristic
317C2×C32C2×#21411yescharacteristic
318C2×C32#93911yescharacteristic
319C2×C32#95011yescharacteristic
320C3⋊D3—22no—
321C3×D3—62no—
322C2×Q8—91no—
323C2×D4—181no—
324C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C22×C4—91no—
326SD16—182no—
327C2×C8—91no—
328C2×C8—91no—
329C2×C8—542no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—541no—
333C22⋊C4—181no—
334C22⋊C4—541no—
335C22×C3—31no—
336C22×C3—31no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—61no—
341C22×C3—92no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—121no—
345C22×C3—121no—
346C22×C3—121no—
347C22×C3—121no—
348C22×C3—122no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—241no—
352C22×C3—241no—
353C22×C3D3×#6711yescharacteristic
354C2×D3—32no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—92no—
358C2×D3—181no—
359C2×D3—181no—
360C3×C4—31no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—61no—
365C3×C4—61no—
366C3×C4—181no—
367C3×C4—361no—
368C3×C4—361no—
369C3×C4—361no—
370C3×C4#380411yescharacteristic
371C3×C4#381111yescharacteristic
372Dic3—61no—
373Dic3—121no—
374Dic3—121no—
375Dic3—122no—
376Dic3—121no—
377Dic3—121no—
378Dic3—181no—
379Dic3—361no—
380Dic3—361no—
381C32—21no—
382C32#477811yescharacteristic, complemented
383C32#504111yescharacteristic, complemented
384C23—21no—
385C23—91no—
386Q8—181no—
387Q8—181no—
388D4—21no—
389D4—21no—
390D4—362no—
391C2×C4—91no—
392C2×C4—181no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
398C8—92no—
399C8—181no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—31no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—61no—
407C2×C3—61no—
408C2×C3—61no—
409C2×C3—81no—
410C2×C3—182no—
411C2×C3—241no—
412C2×C3—241no—
413C2×C3—242no—
414C2×C3D3×#21411yescharacteristic
415C2×C3#6393311yescharacteristic
416C2×C3#6397111yescharacteristic
417D3—62no—
418D3—182no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—92no—
423C22—181no—
424C22—181no—
425C22#8867611yescharacteristic
426C4—181no—
427C4—361no—
428C4—361no—
429C4—361no—
430C4#8867111yescharacteristic
431C4#8867711yescharacteristic
432C3—31no—
433C3—31no—
434C3—61no—
435C3#1059146011yescharacteristic
436C2—81no—
437C2—182no—
438C2#1172119811yescharacteristic
439C2#1172120111yescharacteristic
440C2#1172131211yescharacteristic
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 radical#64147⋊C3C2
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#465
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
3. #11721107C2semidirect
4. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
5. #63659⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720314C2semidirect
32. #63659D3semidirect
52. #6180D4semidirect
93. C2×(C3⋊D3)⋊C3SD16semidirect
225. C2×C3⋊D3#214semidirect
251. H(3,𝔽3)#465semidirect
382. C32#4778semidirect
383. C32#5041semidirect

Maximal subgroups

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

subgroupindex
C2×#885462
#64147⋊C32
#117211072
C2×((C3⋊D3)⋊C3)⋊C82
#63659⋊C32
#117203262
#117203142
#105914993
#105914653