Finite Group Databasebeta

← #13140969

Subgroups — #13140969

Order 1728.

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

normal subgroups only

This group has 488 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#11721861C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3#11721136C211yesmaximal, characteristic
4C2×#88438C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720304C211yesmaximal, characteristic, complemented
8#10591565—31nomaximal
9#10591564—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14#88540—21no—
15#88540—21no—
16C4×#1084C2211yescharacteristic
17#3755⋊C3—22no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88438—22no—
23C2×#3755D311yescharacteristic
24C3×#944—31no—
25C3×#944—31no—
26C3×#944—61no—
27C2×C3×#215—31no—
28#63894—31no—
29Dic3×D12—31no—
30C2×#3717—31no—
31C2×#3715—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34Dic3⋊(C3⋊C8)—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63566—31no—
37#63565—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084D411yescharacteristic
45C2×#1084D411yescharacteristic
46#6229—21no—
47#6229—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)D412yes—
51((C3⋊D3)⋊C3)⋊C4—42no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5162—91no—
55#5083—91no—
56#4818—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×#624—61no—
66C2×D3×Dic3—61no—
67#3755—22no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#215—62no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C3×Dic3⋊C4—61no—
77C3×Dic3⋊C4—62no—
78C3×Dic3⋊C4—62no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84#3722—61no—
85(C2×D3)⋊Dic3—61no—
86#3717—62no—
87#3715—62no—
88C22×H(3,𝔽3)C2×D411yescharacteristic
89C2×(C3⋊D3)⋊C3—22no—
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—41no—
92C4×H(3,𝔽3)—41no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)#4111yescharacteristic
95#1084—21no—
96#1084—41no—
97#1084SD1612yescomplemented
98(C3⋊Dic3)⋊C3—41no—
99C2×C3×SD16—91no—
100C3×C4⋊Q8—91no—
101#944—31no—
102#944—31no—
103#944—61no—
104C2×#215—31no—
105C2×#215—91no—
106C2×#215—91no—
107D4×Dic3—91no—
108C2×#214—91no—
109C2×#204—91no—
110#893—91no—
111C4×D12—91no—
112C3×#103—91no—
113#835—31no—
114#835—91no—
115#835—91no—
116#816—91no—
117Dic3⋊C8—91no—
118Dic3⋊C8—91no—
119#810—91no—
120C4⋊(C3⋊C8)—91no—
121C22×C3⋊D3—21no—
122C22×C3×D3—61no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—41no—
128C32×Q8—41no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)#6712yes—
138C2×C3×Dic3—31no—
139C2×C3×Dic3—31no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C3×D12—32no—
149C3×D12—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—61no—
154C3×Dic6—121no—
155C3×Dic6—121no—
156#624—122no—
157D3×Dic3—122no—
158C3⋊(C3⋊C8)—21no—
159C3⋊(C3⋊C8)—21no—
160C3×C3⋊C8—61no—
161C3×C3⋊C8—61no—
162#461—271noSylow
163C2×H(3,𝔽3)#12111yescharacteristic
164C2×H(3,𝔽3)C2×SD1611yescharacteristic
165C2×H(3,𝔽3)#13711yescharacteristic
166(C3⋊D3)⋊C3—42no—
167C2×C3×Q8—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—91no—
171C2×C3×Q8D3211yescharacteristic
172C2×C3×D4—91no—
173C22×Dic3—181no—
174C2×D12—31no—
175C2×D12—91no—
176C2×C4×D3—181no—
177C2×C4×D3—181no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180C2×Dic6—61no—
181C3×SD16—182no—
182C2×C3×C8—91no—
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185C3×C4⋊C4—181no—
186C3×C4⋊C4—181no—
187C3×C42—91no—
188#217—181no—
189#215—62no—
190#215—182no—
191#215—182no—
192#214—182no—
193(C2×D3)⋊C4—181no—
194(C2×D3)⋊C4—181no—
195C4⋊Dic3—31no—
196C4⋊Dic3—91no—
197Dic3⋊C4—61no—
198Dic3⋊C4—61no—
199Dic3⋊C4—61no—
200Dic3⋊C4—61no—
201Dic3⋊C4—62no—
202Dic3⋊C4—62no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—61no—
206C2×C3⋊C8—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—91no—
210#204—182no—
211C22×C32—21no—
212C22×C32D3×D411yescharacteristic
213C22×C32D3×D411yescharacteristic
214C2×C3⋊D3—22no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3×D3—62no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4#23711yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3—41no—
230C3×Dic3—32no—
231C3×Dic3—32no—
232C3×Dic3—62no—
233C3×Dic3—62no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—122no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C2×SD16—91no—
241C2×SD16—271no—
242C4⋊Q8—91no—
243C4×D4—271no—
244C4⋊C8—271no—
245#103—91no—
246#103—271no—
247H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—181no—
251C3×Q8—21no—
252C3×Q8—21no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—121no—
258C3×Q8—121no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×D4—92no—
262C3×D4—181no—
263C2×C3×C4—21no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
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—31no—
278C2×Dic3—31no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—62no—
284C2×Dic3—61no—
285C2×Dic3—181no—
286C2×Dic3—362no—
287D12—32no—
288D12—61no—
289D12—92no—
290D12—181no—
291C4×D3—362no—
292C4×D3—362no—
293Dic6—61no—
294Dic6—61no—
295Dic6—61no—
296Dic6—61no—
297Dic6—121no—
298Dic6—121no—
299C3×C8—181no—
300C3×C8—181no—
301C3⋊C8—61no—
302C3⋊C8—61no—
303C3⋊C8—181no—
304C3⋊C8—181no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32#88411yescharacteristic
309C2×C32D3×SD1611yescharacteristic
310C2×C32D3×SD1611yescharacteristic
311C2×C32#91511yescharacteristic
312C2×C32#91811yescharacteristic
313C2×C32#93911yescharacteristic
314C3⋊D3—42no—
315C3×D3—122no—
316C2×Q8—91no—
317C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C2×D4—91no—
319C22×C4—541no—
320SD16—182no—
321SD16—542no—
322C2×C8—91no—
323C2×C8—91no—
324C4⋊C4—91no—
325C4⋊C4—181no—
326C4⋊C4—181no—
327C22⋊C4—541no—
328C42—91no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—61no—
332C22×C3—182no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3#381111yescharacteristic
336C2×D3—62no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—182no—
340C2×D3—361no—
341C2×D3—361no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—41no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—92no—
350C3×C4—121no—
351C3×C4—121no—
352C3×C4—122no—
353C3×C4—181no—
354C3×C4—361no—
355C3×C4—361no—
356C3×C4#380411yescharacteristic
357C3×C4D3×#6711yescharacteristic
358Dic3—32no—
359Dic3—32no—
360Dic3—62no—
361Dic3—61no—
362Dic3—61no—
363Dic3—62no—
364Dic3—121no—
365Dic3—121no—
366Dic3—122no—
367Dic3—121no—
368Dic3—361no—
369C32—21no—
370C32#481811yescharacteristic, complemented
371C32#516211yescharacteristic, complemented
372C23—181no—
373Q8—21no—
374Q8—21no—
375Q8—181no—
376Q8—181no—
377D4—92no—
378D4—181no—
379C2×C4—21no—
380C2×C4—91no—
381C2×C4—91no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4—1082no—
385C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
386C8—181no—
387C8—181no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—61no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—362no—
398C2×C3#6394011yescharacteristic
399C2×C3#6395111yescharacteristic
400C2×C3#6397111yescharacteristic
401D3—122no—
402D3—362no—
403C22—182no—
404C22—361no—
405C22—361no—
406C22#8867711yescharacteristic
407C4—41no—
408C4—92no—
409C4—181no—
410C4—361no—
411C4—361no—
412C4#8867111yescharacteristic
413C4#8867611yescharacteristic
414C3—31no—
415C3—31no—
416C3—61no—
417C3#1059156911yescharacteristic
418C2—362no—
419C2#1172122811yescharacteristic
420C2#1172123111yescharacteristic
421C2#1172131211yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
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. #11721861C2semidirect
5. #63662⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720304C2semidirect
32. #63662D3semidirect
97. #1084SD16semidirect
247. H(3,𝔽3)#461semidirect
370. C32#4818semidirect
371. C32#5162semidirect

Maximal subgroups

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

subgroupindex
#117218612
C2×#3755⋊C32
#117211362
C2×#884382
#63662⋊C32
#117203262
#117203042
#105915653
#105915643