Finite Group Databasebeta

← #10592758⋊C3 #13142918

Subgroups — #10592758⋊C3 #13142918

Order 1728.

531 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 100 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic, complemented
3C2×#3756⋊C3C211yesmaximal, characteristic, complemented
4C2×#3753⋊C3C211yesmaximal, characteristic
5C2×#3754⋊C3C211yesmaximal, characteristic, complemented
6#64084⋊C3C211yesmaximal, characteristic, complemented
7C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
8#10592758C311yesmaximal, characteristic, complemented
9C3×#5059—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#634⋊C3C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—22no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—22no—
17#3756⋊C3—22no—
18#3753⋊C3—22no—
19#3754⋊C3—22no—
20#3752⋊C3—22no—
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23C32×#124—21no—
24C32×#124D311yescharacteristic, complemented
25C32×#124C2×C311yescharacteristic, complemented
26C2×#3756C2×C311yescharacteristic, complemented
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3754C2×C311yescharacteristic, complemented
29C2×#3753C2×C311yescharacteristic
30#64084C2×C311yescharacteristic, complemented
31C2×C3×#216—31no—
32C2×C3×#215—31no—
33C2×C3×#214—31no—
34C2×C3×#213—31no—
35C3×#875—31no—
36C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
37C3×C4×C3⋊C8—31no—
38C23×H(3,𝔽3)—21no—
39C22×(C3⋊D3)⋊C3—21no—
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)—21no—
43D4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)D412yes—
46C2×C4×H(3,𝔽3)C2311yescharacteristic
47C2×(C3⋊Dic3)⋊C3—21no—
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)—21no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53(C3⋊(C3⋊C8))⋊C3D412yes—
54C3×#495—91no—
55#5059—31no—
56#5059—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C311yescharacteristic
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
64C2×#634C22×C311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×Dic6—31no—
67C32×C22⋊C4—24no—
68C32×C22⋊C4—22no—
69C32×C22⋊C4—22no—
70C32×C42—21no—
71C32×C42C2×D311yescharacteristic
72C32×C42C22×C311yescharacteristic
73#3756—22no—
74#3755—22no—
75#3754—22no—
76#3753—22no—
77#3752—22no—
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#216—62no—
81C3×#215—62no—
82C3×#214—62no—
83C3×#213—62no—
84C3×(C2×D3)⋊C4—62no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)—42no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C2×(C3⋊D3)⋊C3—41no—
91C2×(C3⋊D3)⋊C3—42no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)—41no—
94C4×H(3,𝔽3)C2×D411yescharacteristic
95C4×H(3,𝔽3)C2×D411yescharacteristic
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×Q16—91no—
98C2×C3×SD16—91no—
99C2×C3×SD16—91no—
100C2×C3×D8—91no—
101C3×#124—31no—
102C3×#124—31no—
103C3×#124—61no—
104C3×#124—91no—
105C3×#124C3×D311yescharacteristic
106C2×#216—31no—
107C2×#216—91no—
108C2×#215—31no—
109C2×#215—91no—
110C2×#214—31no—
111C2×#214—91no—
112C2×#213—31no—
113C2×#213—91no—
114#875—31no—
115#875—91no—
116C3×C4×C8—91no—
117C4×C3⋊C8—31no—
118C4×C3⋊C8—91no—
119C23×C32—22no—
120C23×C32—21no—
121C23×C32—21no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—21no—
127C32×Q8—21no—
128C32×Q8—41no—
129C32×Q8—41no—
130C32×D4—21no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—21no—
134C32×D4—41no—
135C32×D4—41no—
136C2×C32×C4—22no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—21no—
141C2×C32×C4#6712yes—
142C2×C32×C4C3×D412yes—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)—21no—
148#634—21no—
149#634—21no—
150C2×C3×Dic3—61no—
151C3×D12—61no—
152C3×D12—61no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3⋊(C3⋊C8)C3×D412yes—
156C3⋊(C3⋊C8)C3×D412yes—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—32no—
159#495—91noSylow
160C2×H(3,𝔽3)—81no—
161C2×H(3,𝔽3)#12711yescharacteristic
162C2×H(3,𝔽3)#13512yes—
163(C3⋊D3)⋊C3—81no—
164C2×C3×Q8—31no—
165C2×C3×Q8—31no—
166C2×C3×Q8—61no—
167C2×C3×Q8—91no—
168C2×C3×Q8C2×C3×D311yescharacteristic
169C2×C3×D4—31no—
170C2×C3×D4—31no—
171C2×C3×D4—61no—
172C2×C3×D4—91no—
173C2×C3×D4C2×C3×D311yescharacteristic
174C2×D12—31no—
175C2×D12—91no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×Q16—182no—
179C3×SD16—182no—
180C3×SD16—182no—
181C3×D8—182no—
182C2×C3×C8—91no—
183C2×C3×C8—91no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—62no—
186C3×C22⋊C4—62no—
187C3×C22⋊C4—64no—
188C3×C22⋊C4—182no—
189C3×C42—31no—
190C3×C42—31no—
191C3×C42—61no—
192C3×C42C2×C3×D311yescharacteristic
193#216—62no—
194#216—182no—
195#215—62no—
196#215—182no—
197#214—62no—
198#214—182no—
199#213—62no—
200#213—182no—
201(C2×D3)⋊C4—62no—
202(C2×D3)⋊C4—182no—
203C2×C3⋊C8—31no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—91no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—44no—
210C22×C32—41no—
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32C2×#6711yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C2×C3⋊D3—41no—
217C2×C3⋊D3—42no—
218C2×C3×D3—121no—
219C2×C3×D3—122no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—42no—
225C32×C4—42no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4D3×D411yescharacteristic
229C32×C4D3×D411yescharacteristic
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C3⋊Dic3—41no—
233C3×Dic3—121no—
234C2×Q16—91no—
235C2×SD16—91no—
236C2×SD16—91no—
237C2×D8—91no—
238#124—91no—
239#124(C3⋊D3)⋊C311yescharacteristic, complemented
240C4×C8—91no—
241H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
242C23×C3—21no—
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—62no—
246C23×C3—181no—
247C22×D3—61no—
248C22×D3—181no—
249C3×Q8—21no—
250C3×Q8—21no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—61no—
255C3×Q8—121no—
256C3×Q8—121no—
257C3×Q8—181no—
258C3×Q8—181no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—61no—
262C3×D4—61no—
263C3×D4—61no—
264C3×D4—61no—
265C3×D4—121no—
266C3×D4—121no—
267C3×D4—181no—
268C3×D4—181no—
269C2×C3×C4—21no—
270C2×C3×C4—32no—
271C2×C3×C4—32no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—61no—
279C2×C3×C4—181no—
280C2×C3×C4C3×#6712yes—
281C2×C3×C4C22×C3×D311yescharacteristic
282C2×Dic3—61no—
283C2×Dic3—181no—
284D12—61no—
285D12—61no—
286D12—181no—
287D12—181no—
288Dic6—61no—
289Dic6—61no—
290Dic6—181no—
291Dic6—181no—
292C3×C8—92no—
293C3×C8—92no—
294C3⋊C8—32no—
295C3⋊C8—32no—
296C3⋊C8—92no—
297C3⋊C8—92no—
298C2×C32—21no—
299C2×C32—22no—
300C2×C32—82no—
301C2×C32—81no—
302C2×C32—81no—
303C2×C32#94011yescharacteristic
304C2×C32#95012yes—
305C2×C32C3×#12711yescharacteristic
306C2×C32C3×#13512yes—
307C3⋊D3—81no—
308C3×D3—241no—
309C2×Q8—91no—
310C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
311C2×D4—91no—
312C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
313Q16—182no—
314SD16—182no—
315SD16—182no—
316D8—182no—
317C2×C8—91no—
318C2×C8—91no—
319C22⋊C4—22no—
320C22⋊C4—182no—
321C42C2×(C3⋊D3)⋊C311yescharacteristic
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—41no—
325C22×C3—42no—
326C22×C3—61no—
327C22×C3—121no—
328C22×C3—121no—
329C22×C3—122no—
330C22×C3—122no—
331C22×C3—122no—
332C22×C3—124no—
333C22×C3—361no—
334C22×C3—362no—
335C22×C3C2×C3×#6711yescharacteristic
336C2×D3—121no—
337C2×D3—122no—
338C2×D3—361no—
339C2×D3—362no—
340C3×C4—22no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—41no—
346C3×C4—62no—
347C3×C4—62no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—122no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—122no—
354C3×C4—361no—
355C3×C4C3×D3×D411yescharacteristic
356C3×C4C3×D3×D411yescharacteristic
357Dic3—121no—
358Dic3—361no—
359C32—21no—
360C32#505911yescharacteristic, complemented
361C32C3×#49511yescharacteristic, complemented
362C23—21no—
363C23—181no—
364Q8—21no—
365Q8—21no—
366Q8—181no—
367Q8—181no—
368D4—21no—
369D4—21no—
370D4—181no—
371D4—181no—
372C2×C4—21no—
373C2×C4—181no—
374C2×C4#638⋊C312yes—
375C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
376C8—92no—
377C8—92no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—32no—
381C2×C3—32no—
382C2×C3—61no—
383C2×C3—62no—
384C2×C3—81no—
385C2×C3—241no—
386C2×C3—241no—
387C2×C3—242no—
388C2×C3—721no—
389C2×C3C3×#94011yescharacteristic
390C2×C3C3×#95012yes—
391D3—241no—
392D3—721no—
393C22—41no—
394C22—42no—
395C22—361no—
396C22—362no—
397C22C2×#638⋊C311yescharacteristic
398C4—22no—
399C4—41no—
400C4—361no—
401C4D4×(C3⋊D3)⋊C311yescharacteristic
402C4D4×(C3⋊D3)⋊C311yescharacteristic
403C3—31no—
404C3—31no—
405C3—61no—
406C3C3×#505911yescharacteristic
407C2—81no—
408C2—721no—
409C2#64149⋊C311yescharacteristic
410C2#64159⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#124C2
socleC22×C3C2×C3×#67
prime radicalC2×#3753⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#495
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. C2×#3755⋊C3C2semidirect
3. C2×#3756⋊C3C2semidirect
5. C2×#3754⋊C3C2semidirect
6. #64084⋊C3C2semidirect
7. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
8. #10592758C3semidirect
24. C32×#124D3semidirect
25. C32×#124C2×C3semidirect
26. C2×#3756C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3754C2×C3semidirect
30. #64084C2×C3semidirect
36. C4×C3⋊(C3⋊C8)C2×C3semidirect
239. #124(C3⋊D3)⋊C3semidirect
241. H(3,𝔽3)#495semidirect
360. C32#5059semidirect
361. C32C3×#495semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#3755⋊C32
C2×#3756⋊C32
C2×#3753⋊C32
C2×#3754⋊C32
#64084⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#105927583
C3×#50593