Finite Group Databasebeta

← #13140956

Subgroups — #13140956

Order 1728.

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

normal subgroups only

This group has 488 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×#88549C211yesmaximal, characteristic
2#64153⋊C3C211yesmaximal, characteristic, complemented
3#11721149C211yesmaximal, characteristic
4C2×#88437C211yesmaximal, characteristic
5#63662⋊C3C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10591564—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×#6199—21no—
14#88549—22no—
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88451—21no—
22#88437—22no—
23#64153D311yescharacteristic, complemented
24C3×#944—31no—
25C2×C3×#215—31no—
26C2×C3×#215—31no—
27C2×C3×#215—61no—
28#63898—31no—
29Dic3×D12—31no—
30C2×#3716—31no—
31C2×#3715—31no—
32#63662D311yescharacteristic, complemented
33C3×#835—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63566—31no—
38C22×#272—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C4⋊#272—21no—
44C4⋊#272D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C2×(C3⋊Dic3)⋊C3D411yescharacteristic
49#634⋊C3—21no—
50#634⋊C3—21no—
51#6199—42no—
52#6180—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×#634C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×Dic6—31no—
65C2×D3×Dic3—61no—
66C2×D3×Dic3—61no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—21no—
69#3750—21no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×#215—62no—
73C3×#215—62no—
74C3×#215—64no—
75C3×C4⋊Dic3—31no—
76C3×Dic3⋊C4—61no—
77C3×Dic3⋊C4—61no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—61no—
83(C2×D3)⋊Dic3—61no—
84(C2×D3)⋊Dic3—61no—
85#3716—62no—
86#3715—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×#272—22no—
89C2×#272—41no—
90C2×#272—41no—
91C4×H(3,𝔽3)—41no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3SD1612yescomplemented
98C2×C3×SD16—91no—
99C3×C4⋊Q8—91no—
100#944—31no—
101#944—91no—
102C2×#215—31no—
103C2×#215—31no—
104C2×#215—61no—
105C2×#215—91no—
106D4×Dic3—91no—
107C2×#214—91no—
108C2×#204—91no—
109#893—91no—
110C4×D12—91no—
111C3×#103—91no—
112#835—31no—
113#835—91no—
114#835—91no—
115#816—91no—
116Dic3⋊C8—91no—
117Dic3⋊C8—91no—
118#810—91no—
119C4⋊(C3⋊C8)—91no—
120C22×C3×D3—61no—
121C22×C3×D3—61no—
122C22×C3×D3—121no—
123C32×Q8—22no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—41no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3#6711yescharacteristic
138C2×C3⋊Dic3#6711yescharacteristic
139#634—21no—
140#634—21no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C3×D12—32no—
146C3×D12—32no—
147C3×D12—62no—
148C3×D12—61no—
149C3×D12—61no—
150C3×D12—121no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153D3×Dic3—122no—
154D3×Dic3—122no—
155C3⋊(C3⋊C8)—21no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—61no—
158C3×C3⋊C8—61no—
159C3×C3⋊C8—61no—
160#461—271noSylow
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)C2×SD1611yescharacteristic
163C2×H(3,𝔽3)#13711yescharacteristic
164#272—42no—
165C2×C3×Q8—31no—
166C2×C3×Q8—31no—
167C2×C3×Q8—61no—
168C2×C3×Q8—91no—
169C2×C3×Q8D3211yescharacteristic
170C2×C3×D4—91no—
171C22×Dic3—181no—
172C2×D12—31no—
173C2×D12—31no—
174C2×D12—61no—
175C2×C4×D3—181no—
176C2×C4×D3—181no—
177C2×Dic6—31no—
178C2×Dic6—91no—
179C3×SD16—182no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C3×C4⋊C4—91no—
183C3×C4⋊C4—181no—
184C3×C4⋊C4—181no—
185C3×C42—91no—
186#217—181no—
187#215—62no—
188#215—62no—
189#215—64no—
190#215—182no—
191#214—182no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—181no—
194C4⋊Dic3—31no—
195C4⋊Dic3—91no—
196Dic3⋊C4—61no—
197Dic3⋊C4—61no—
198Dic3⋊C4—181no—
199Dic3⋊C4—181no—
200C4×Dic3—31no—
201C4×Dic3—91no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—91no—
207#204—182no—
208C22×C32—21no—
209C22×C32D3×D411yescharacteristic
210C22×C32C2×#6711yescharacteristic
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—122no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—241no—
219C2×C3×D3—241no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—42no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4D3×D411yescharacteristic
228C32×C4#23711yescharacteristic
229C3⋊Dic3—21no—
230C3⋊Dic3—41no—
231C3⋊Dic3—41no—
232C3⋊Dic3#21512yescomplemented
233C3×Dic3—32no—
234C3×Dic3—61no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C2×SD16—91no—
238C2×SD16—271no—
239C4⋊Q8—91no—
240C4×D4—271no—
241C4⋊C8—271no—
242#103—91no—
243#103—271no—
244H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
245C23×C3—181no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—121no—
249C3×Q8—21no—
250C3×Q8—21no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—62no—
256C3×Q8—121no—
257C3×Q8—181no—
258C3×Q8—181no—
259C3×D4—92no—
260C3×D4—181no—
261C2×C3×C4—21no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—121no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C2×D3211yescharacteristic
273C2×Dic3—31no—
274C2×Dic3—31no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—91no—
278C2×Dic3—91no—
279C2×Dic3—181no—
280C2×Dic3—181no—
281C2×Dic3—362no—
282D12—32no—
283D12—32no—
284D12—62no—
285D12—61no—
286D12—61no—
287D12—121no—
288C4×D3—362no—
289C4×D3—362no—
290Dic6—61no—
291Dic6—61no—
292Dic6—181no—
293Dic6—181no—
294C3×C8—181no—
295C3×C8—181no—
296C3⋊C8—61no—
297C3⋊C8—61no—
298C3⋊C8—62no—
299C3⋊C8—61no—
300C3⋊C8—181no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32#88411yescharacteristic
305C2×C32D3×SD1611yescharacteristic
306C2×C32#91811yescharacteristic
307C2×C32#93811yescharacteristic
308C2×C32C2×#21511yescharacteristic
309C2×C32#94211yescharacteristic
310C3×D3—122no—
311C3×D3—122no—
312C3×D3—242no—
313C2×Q8—91no—
314C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C2×D4—91no—
316C22×C4—541no—
317SD16—182no—
318SD16—542no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4—181no—
324C22⋊C4—541no—
325C42—91no—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—182no—
330C22×C3—361no—
331C22×C3—361no—
332C22×C3D3×#6711yescharacteristic
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—122no—
336C2×D3—121no—
337C2×D3—121no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—241no—
341C2×D3—241no—
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×C4#381111yescharacteristic
358Dic3—32no—
359Dic3—61no—
360Dic3—92no—
361Dic3—121no—
362Dic3—121no—
363Dic3—181no—
364Dic3—361no—
365Dic3—361no—
366C32—21no—
367C32#481811yescharacteristic, complemented
368C32#508311yescharacteristic, complemented
369C23—181no—
370Q8—21no—
371Q8—21no—
372Q8—181no—
373Q8—181no—
374D4—92no—
375D4—181no—
376C2×C4—21no—
377C2×C4—91no—
378C2×C4—91no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—1082no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—181no—
384C8—181no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—362no—
395C2×C3D3×#21511yescharacteristic
396C2×C3#6394811yescharacteristic
397C2×C3#6395811yescharacteristic
398D3—122no—
399D3—122no—
400D3—242no—
401C22—182no—
402C22—361no—
403C22—361no—
404C22#8867611yescharacteristic
405C4—41no—
406C4—92no—
407C4—181no—
408C4—361no—
409C4—361no—
410C4#8867111yescharacteristic
411C4#8867711yescharacteristic
412C3—31no—
413C3—31no—
414C3—61no—
415C3#1059156511yescharacteristic
416C2—362no—
417C2#1172123211yescharacteristic
418C2#1172123411yescharacteristic
419C2#1172130311yescharacteristic
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 radicalC2×C4⋊#272C22
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
2. #64153⋊C3C2semidirect
5. #63662⋊C3C2semidirect
6. #11720313C2semidirect
7. #11720311C2semidirect
23. #64153D3semidirect
32. #63662D3semidirect
97. (C3⋊Dic3)⋊C3SD16semidirect
232. C3⋊Dic3#215semidirect
244. H(3,𝔽3)#461semidirect
367. C32#4818semidirect
368. C32#5083semidirect

Maximal subgroups

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

subgroupindex
C2×#885492
#64153⋊C32
#117211492
C2×#884372
#63662⋊C32
#117203132
#117203112
#105915693
#105915643