Finite Group Databasebeta

← #13140989

Subgroups — #13140989

Order 1728.

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

normal subgroups only

This group has 515 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×#88550C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C211yesmaximal, characteristic, complemented
4#11721110C211yesmaximal, characteristic, complemented
5C2×#88438C211yesmaximal, characteristic, complemented
6C2×#88432C211yesmaximal, characteristic
7#11720310C211yesmaximal, characteristic, complemented
8#10591610—31nomaximal
9#10591609—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14#88550—22no—
15C2×#6180C2211yescharacteristic
16#3755⋊C3—22no—
17#3752⋊C3—22no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—22no—
21#88438—22no—
22#88432—22no—
23#64156D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic, complemented
25C3×#947—31no—
26C2×C3×#216—31no—
27C2×C3×#216—31no—
28C2×C3×#216—61no—
29C2×C3×#215—31no—
30#63864—31no—
31#63863—31no—
32C2×#3717—31no—
33C2×#3715—31no—
34C2×#3714—31no—
35C2×#3714—31no—
36#63553—31no—
37Dic3×C3⋊C8—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×#1084—21no—
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3D412yes—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50C4⋊((C3⋊D3)⋊C3)—21no—
51C4⋊((C3⋊D3)⋊C3)—21no—
52#6180D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54#5191—91no—
55#5168—91no—
56#5059—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×D12—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6—61no—
67#3755—22no—
68#3752—22no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#216—62no—
72C3×#216—62no—
73C3×#216—64no—
74C3×#215—62no—
75C3×(C2×D3)⋊C4—62no—
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#3722—62no—
82(C2×D3)⋊Dic3—62no—
83#3717—62no—
84#3715—62no—
85#3714—62no—
86#3714—62no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—42no—
91C2×(C3⋊D3)⋊C3—42no—
92C4×H(3,𝔽3)—41no—
93C4×H(3,𝔽3)C2×D411yescharacteristic
94C4×H(3,𝔽3)C2×D411yescharacteristic
95#1084—41no—
96(C3⋊Dic3)⋊C3—22no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C3×#124—91no—
100#947—31no—
101#947—91no—
102#947—91no—
103C2×#216—31no—
104C2×#216—31no—
105C2×#216—61no—
106C2×#215—31no—
107C2×#215—91no—
108C2×#215—91no—
109#936—91no—
110C2×#214—91no—
111C2×#213—91no—
112C2×#213—91no—
113C2×D24—91no—
114C2×#204—91no—
115#875—91no—
116C8×Dic3—91no—
117C8×Dic3—91no—
118C4×C3⋊C8—91no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3—21no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
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#6712yes—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)—21no—
139C4⋊(C3⋊D3)—21no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—121no—
144C3×D12—61no—
145C3×D12—61no—
146C3×D12—61no—
147C3×D12—61no—
148C3×Dic6—62no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3⋊(C3⋊C8)#6712yes—
155C3×C3⋊C8—32no—
156C3×C3⋊C8—32no—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—62no—
159#495—271noSylow
160C2×H(3,𝔽3)#12711yescharacteristic
161C2×H(3,𝔽3)#13512yes—
162(C3⋊D3)⋊C3—81no—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8D3211yescharacteristic
169C2×C3×D4—91no—
170C2×C3×D4—91no—
171C2×D12—31no—
172C2×D12—31no—
173C2×D12—91no—
174C2×D12—91no—
175C2×Dic6—31no—
176C2×Dic6—31no—
177C2×Dic6—61no—
178C3×Q16—182no—
179C3×SD16—182no—
180C2×C3×C8—91no—
181C2×C3×C8—91no—
182C3×C22⋊C4—182no—
183C3×C42—91no—
184#217—182no—
185#216—62no—
186#216—62no—
187#216—64no—
188#215—62no—
189#215—182no—
190#215—182no—
191#214—182no—
192#213—182no—
193#213—182no—
194(C2×D3)⋊C4—62no—
195(C2×D3)⋊C4—182no—
196(C2×D3)⋊C4—182no—
197(C2×D3)⋊C4—182no—
198C4×Dic3—31no—
199C4×Dic3—91no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—91no—
205D24—182no—
206#204—182no—
207C22×C32—21no—
208C22×C32D3×D411yescharacteristic
209C22×C32C2×#6711yescharacteristic
210C2×C3⋊D3—42no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—42no—
214C2×C3×D3—122no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—122no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—42no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4D3×D411yescharacteristic
224C32×C4D3×D411yescharacteristic
225C32×C4D3×D411yescharacteristic
226C32×C4C2×#6711yescharacteristic
227C3⋊Dic3—22no—
228C3×Dic3—62no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C2×Q16—91no—
233C2×SD16—91no—
234C2×SD16—271no—
235C2×D8—271no—
236#124—91no—
237#124—271no—
238C4×C8—271no—
239H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
240C23×C3—181no—
241C23×C3—181no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—181no—
245C22×D3—181no—
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—61no—
249C3×Q8—61no—
250C3×Q8—61no—
251C3×Q8—61no—
252C3×Q8—62no—
253C3×Q8—121no—
254C3×Q8—181no—
255C3×Q8—181no—
256C3×D4—181no—
257C3×D4—181no—
258C3×D4—181no—
259C3×D4—181no—
260C2×C3×C4—21no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—92no—
267C2×C3×C4—121no—
268C2×C3×C4—181no—
269C2×C3×C4C2×D3211yescharacteristic
270C2×Dic3—32no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—92no—
274C2×Dic3—121no—
275D12—61no—
276D12—61no—
277D12—61no—
278D12—61no—
279D12—181no—
280D12—181no—
281D12—181no—
282D12—181no—
283Dic6—61no—
284Dic6—61no—
285Dic6—61no—
286Dic6—61no—
287Dic6—62no—
288Dic6—121no—
289C3×C8—92no—
290C3×C8—92no—
291C3⋊C8—32no—
292C3⋊C8—32no—
293C3⋊C8—32no—
294C3⋊C8—62no—
295C3⋊C8—92no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32#91612yes—
299C2×C32#94011yescharacteristic
300C2×C32#94011yescharacteristic
301C2×C32#95012yes—
302C3⋊D3—81no—
303C3⋊D3—81no—
304C3×D3—241no—
305C3×D3—241no—
306C2×Q8—91no—
307C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C2×D4—91no—
309C2×D4—91no—
310Q16—182no—
311SD16—182no—
312SD16—542no—
313D8—542no—
314C2×C8—91no—
315C2×C8—91no—
316C22⋊C4—182no—
317C22⋊C4—542no—
318C42—91no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—361no—
323C22×C3—361no—
324C22×C3—362no—
325C22×C3—362no—
326C22×C3D3×#6711yescharacteristic
327C2×D3—122no—
328C2×D3—122no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—362no—
334C2×D3—362no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—41no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—121no—
343C3×C4—121no—
344C3×C4—122no—
345C3×C4—182no—
346C3×C4—361no—
347C3×C4D3×#6711yescharacteristic
348C3×C4#381111yescharacteristic
349Dic3—62no—
350Dic3—121no—
351Dic3—121no—
352Dic3—122no—
353Dic3—182no—
354C32—21no—
355C32#505911yescharacteristic, complemented
356C32#516811yescharacteristic, complemented
357C23—181no—
358C23—181no—
359Q8—21no—
360Q8—21no—
361Q8—181no—
362Q8—181no—
363D4—181no—
364D4—181no—
365D4—181no—
366D4—181no—
367C2×C4—21no—
368C2×C4—92no—
369C2×C4—181no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—92no—
372C8—92no—
373C2×C3—32no—
374C2×C3—32no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—62no—
378C2×C3—61no—
379C2×C3—721no—
380C2×C3—721no—
381C2×C3#6394712yes—
382C2×C3#6397911yescharacteristic
383D3—241no—
384D3—241no—
385D3—721no—
386D3—721no—
387C22—362no—
388C22—362no—
389C22—361no—
390C22—361no—
391C22#8867611yescharacteristic
392C4—41no—
393C4—182no—
394C4—361no—
395C4#8867611yescharacteristic
396C4#8867711yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3#1059160611yescharacteristic
401C2—721no—
402C2—721no—
403C2#1172123912yes—
404C2#1172131711yescharacteristic
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×#88432C2
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#495
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×#88550C2semidirect
2. #64156⋊C3C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #11721110C2semidirect
5. C2×#88438C2semidirect
7. #11720310C2semidirect
23. #64156D3semidirect
24. C2×#3755D3semidirect
239. H(3,𝔽3)#495semidirect
355. C32#5059semidirect
356. C32#5168semidirect

Maximal subgroups

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

subgroupindex
C2×#885502
#64156⋊C32
C2×#3755⋊C32
#117211102
C2×#884382
C2×#884322
#117203102
#105916103
#105916093