Finite Group Databasebeta

← #10592581⋊C3 #13142223

Subgroups — #10592581⋊C3 #13142223

Order 1728.

435 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 90 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#64104⋊C3C211yesmaximal, characteristic
2C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63647⋊C3C211yesmaximal, characteristic, complemented
6#63636⋊C3C211yesmaximal, characteristic, complemented
7#63634⋊C3C211yesmaximal, characteristic, complemented
8#10592581C311yesmaximal, characteristic, complemented
9C3×#4882—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#64104C2×C311yescharacteristic
22C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
23C3×#895—31no—
24C3×D3×C4⋊C4—31no—
25C32×#107—21no—
26C32×#107D311yescharacteristic, complemented
27C32×#107C2×C311yescharacteristic, complemented
28#63648C2×C311yescharacteristic, complemented
29#63647C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C3×#821—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#809—31no—
35C3×#807—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)—42no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—21no—
48C4×(C3⋊D3)⋊C3—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#480—91no—
52#4882—31no—
53#4882—91no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—61no—
62C2×C32×C8—21no—
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C22×C311yescharacteristic
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69C32×C4⋊C4C22×C311yescharacteristic
70C32×C4⋊C4C22×C311yescharacteristic
71#3752—21no—
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73#3750—21no—
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×Dic3⋊C4—61no—
78C2×C3×C3⋊C8—31no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3D812yes—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)—41no—
88C4×H(3,𝔽3)#4111yescharacteristic
89C4×H(3,𝔽3)#4111yescharacteristic
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C3×#121—91no—
93C2×C3×C4⋊C4—91no—
94#895—31no—
95#895—91no—
96D3×C4⋊C4—31no—
97D3×C4⋊C4—91no—
98C3×#107—31no—
99C3×#107—31no—
100C3×#107—61no—
101C3×#107—91no—
102C3×#107C3×D311yescharacteristic
103C3×#102—91no—
104C3×#102—91no—
105C3×C22⋊C8—91no—
106#821—31no—
107#821—91no—
108(C2×D3)⋊C8—31no—
109(C2×D3)⋊C8—91no—
110#809—31no—
111#809—91no—
112#807—31no—
113#807—91no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3C3×D411yescharacteristic
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3C3×D411yescharacteristic
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)—42no—
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C4×C3⋊D3—42no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×D12—122no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—122no—
143C32×C8—22no—
144C32×C8—21no—
145C32×C8—21no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—61no—
148#480—91noSylow
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)C2×D811yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
152(C3⋊D3)⋊C3—22no—
153(C3⋊D3)⋊C3—81no—
154C2×C3×D4—91no—
155C2×C3×D4—181no—
156C22×C3×C4—91no—
157C22×C3×C4—181no—
158C2×D12—31no—
159C2×D12—61no—
160C2×D12—91no—
161C2×D12—181no—
162C2×C4×D3—31no—
163C2×C4×D3—61no—
164C2×C4×D3—91no—
165C2×C4×D3—181no—
166C2×C3×C8—31no—
167C2×C3×C8—31no—
168C2×C3×C8—61no—
169C2×C3×C8—91no—
170C2×C3×C8C2×C3×D311yescharacteristic
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—31no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C4⋊C4C2×C3×D311yescharacteristic
180C3×C4⋊C4C2×C3×D311yescharacteristic
181C3×C22⋊C4—181no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3—91no—
186Dic3⋊C4—61no—
187Dic3⋊C4—181no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×C3×D411yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3C3×D812yes—
199C2×C3×D3—32no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—41no—
210C32×C4—41no—
211C32×C4—81no—
212C32×C4—81no—
213C32×C4#23511yescharacteristic
214C32×C4#23911yescharacteristic
215C32×C4C3×#4111yescharacteristic
216C32×C4C3×#4111yescharacteristic
217C3⋊Dic3—21no—
218C3⋊Dic3—41no—
219C3×Dic3—61no—
220C3×Dic3—121no—
221#121—91no—
222C2×C4⋊C4—91no—
223#107—91no—
224#107(C3⋊D3)⋊C311yescharacteristic, complemented
225#102—91no—
226#102—91no—
227C22⋊C8—91no—
228H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C23×C3—181no—
231C22×D3—31no—
232C22×D3—61no—
233C22×D3—91no—
234C22×D3—181no—
235C3×D4—181no—
236C3×D4—181no—
237C3×D4—362no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—91no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—362no—
254C2×C3×C4C22×C3×D311yescharacteristic
255C2×Dic3—31no—
256C2×Dic3—61no—
257C2×Dic3—91no—
258C2×Dic3—181no—
259D12—61no—
260D12—61no—
261D12—122no—
262D12—181no—
263D12—181no—
264D12—362no—
265C4×D3—61no—
266C4×D3—61no—
267C4×D3—122no—
268C4×D3—181no—
269C4×D3—181no—
270C4×D3—362no—
271C3×C8—21no—
272C3×C8—61no—
273C3×C8—61no—
274C3×C8—62no—
275C3×C8—181no—
276C3⋊C8—61no—
277C3⋊C8—181no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32#89411yescharacteristic
282C2×C32D3×D811yescharacteristic
283C2×C32#91811yescharacteristic
284C2×C32C3×#12311yescharacteristic
285C2×C32C2×C3×D811yescharacteristic
286C2×C32C3×#13711yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—81no—
289C3×D3—62no—
290C3×D3—241no—
291C2×D4—91no—
292C2×D4—181no—
293C22×C4—91no—
294C22×C4—181no—
295C2×C8—91no—
296C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
297C4⋊C4—91no—
298C4⋊C4—181no—
299C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
300C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
301C22⋊C4—181no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3—92no—
306C22×C3—181no—
307C22×C3—181no—
308C22×C3—361no—
309C22×C3—361no—
310C22×C3—361no—
311C22×C3C3×D3×D411yescharacteristic
312C2×D3—32no—
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—92no—
316C2×D3—121no—
317C2×D3—121no—
318C2×D3—121no—
319C2×D3—181no—
320C2×D3—181no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—361no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—61no—
331C3×C4—61no—
332C3×C4—121no—
333C3×C4—121no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—181no—
337C3×C4—241no—
338C3×C4—241no—
339C3×C4—361no—
340C3×C4C3×#23511yescharacteristic
341C3×C4C3×#23911yescharacteristic
342Dic3—61no—
343Dic3—121no—
344Dic3—181no—
345Dic3—361no—
346C32—21no—
347C32#488211yescharacteristic, complemented
348C32C3×#48011yescharacteristic, complemented
349C23—91no—
350C23—181no—
351D4—181no—
352D4—181no—
353D4—362no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—91no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—362no—
361C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
362C8—21no—
363C8—181no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—61no—
373C2×C3—182no—
374C2×C3—721no—
375C2×C3C3×#89411yescharacteristic
376C2×C3C3×D3×D811yescharacteristic
377C2×C3C3×#91811yescharacteristic
378D3—62no—
379D3—182no—
380D3—241no—
381D3—721no—
382C22—92no—
383C22—181no—
384C22—181no—
385C22—361no—
386C22—361no—
387C22—361no—
388C22D4×(C3⋊D3)⋊C311yescharacteristic
389C4—41no—
390C4—41no—
391C4—181no—
392C4—361no—
393C4#3828⋊C311yescharacteristic
394C4#8872211yescharacteristic
395C3—31no—
396C3—31no—
397C3—61no—
398C3C3×#488211yescharacteristic
399C2—182no—
400C2—721no—
401C2#64103⋊C311yescharacteristic
402C2D8×(C3⋊D3)⋊C311yescharacteristic
403C2#64127⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radical#64104⋊C3C2
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#480
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#107C2semidirect
4. #63648⋊C3C2semidirect
5. #63647⋊C3C2semidirect
6. #63636⋊C3C2semidirect
7. #63634⋊C3C2semidirect
8. #10592581C3semidirect
22. C4⋊C4×C3⋊D3C2×C3semidirect
26. C32×#107D3semidirect
27. C32×#107C2×C3semidirect
28. #63648C2×C3semidirect
29. #63647C2×C3semidirect
30. #63636C2×C3semidirect
31. #63634C2×C3semidirect
224. #107(C3⋊D3)⋊C3semidirect
228. H(3,𝔽3)#480semidirect
347. C32#4882semidirect
348. C32C3×#480semidirect

Maximal subgroups

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

subgroupindex
#64104⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1072
#63648⋊C32
#63647⋊C32
#63636⋊C32
#63634⋊C32
#105925813
C3×#48823