Finite Group Databasebeta

← #13137927

Subgroups — #13137927

Order 1728.

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

normal subgroups only

This group has 435 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
1C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2#11721141C211yesmaximal, characteristic
3#11720530C211yesmaximal, characteristic, complemented
4#63636⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590511—31nomaximal
9#10590423—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6198—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16C2×#6180C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21C4⋊C4×C3⋊D3D311yescharacteristic, complemented
22C3×D3×C4⋊C4—31no—
23#63916—31no—
24#63909—31no—
25#63636D311yescharacteristic, complemented
26C3×#809—31no—
27C3×#808—31no—
28C3×#808—31no—
29C3×#808—61no—
30#63574—31no—
31#63573—31no—
32#63564—31no—
33#63563—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—42no—
48#6198—42no—
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4968—91no—
52#4863—91no—
53#4777—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×C4×D3—31no—
59C2×C3×C4×D3—61no—
60C2×#626—61no—
61C2×#626—61no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—61no—
72C3×Dic3⋊C4—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
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)⋊C3SD1612yes—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89#1084—41no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×C4⋊C4—91no—
93#938—91no—
94#930—91no—
95#895—91no—
96D3×C4⋊C4—31no—
97D3×C4⋊C4—91no—
98C3×#106—91no—
99C3×#102—91no—
100#832—91no—
101#830—91no—
102#821—91no—
103(C2×D3)⋊C8—91no—
104(C2×D3)⋊C8—91no—
105#817—91no—
106#809—31no—
107#809—91no—
108#809—91no—
109#808—31no—
110#808—31no—
111#808—61no—
112#808—91no—
113#808—91no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3#6711yescharacteristic
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)—21no—
128C4×C3⋊D3—21no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—42no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×D12—61no—
137C3×D12—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—122no—
141#626—122no—
142#626—122no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—61no—
148#481—271noSylow
149C2×H(3,𝔽3)#12311yescharacteristic
150C2×H(3,𝔽3)C2×SD1611yescharacteristic
151C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
152(C3⋊D3)⋊C3—22no—
153(C3⋊D3)⋊C3—81no—
154C2×C3×D4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—181no—
157C2×#67—181no—
158C2×#67—181no—
159C2×D12—31no—
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—91no—
167C2×C3×C8—91no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—61no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—91no—
173C3×C4⋊C4—181no—
174C3×C4⋊C4D3211yescharacteristic
175#217—181no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—31no—
181C4⋊Dic3—61no—
182C4⋊Dic3—91no—
183Dic3⋊C4—61no—
184Dic3⋊C4—181no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—61no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—21no—
194C2×C3⋊D3—21no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3#21512yes—
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—81no—
210C32×C4#23511yescharacteristic
211C32×C4#23511yescharacteristic
212C32×C4#23511yescharacteristic
213C32×C4#23711yescharacteristic
214C3⋊Dic3—21no—
215C3⋊Dic3—41no—
216C3×Dic3—61no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C3×Dic3—121no—
220C3×Dic3—241no—
221#121—271no—
222C2×C4⋊C4—91no—
223#106—91no—
224#106—271no—
225#102—91no—
226#102—271no—
227C22⋊C8—271no—
228H(3,𝔽3)#48111yescharacteristic, 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—
237C2×C3×C4—21no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—121no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—362no—
250C2×C3×C4C2×D3211yescharacteristic
251#67—362no—
252#67—362no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—91no—
258C2×Dic3—121no—
259C2×Dic3—181no—
260D12—61no—
261D12—61no—
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—181no—
272C3×C8—181no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—62no—
276C3⋊C8—61no—
277C3⋊C8—181no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32#88311yescharacteristic
282C2×C32#91111yescharacteristic
283C2×C32D3×SD1611yescharacteristic
284C2×C32#92911yescharacteristic
285C2×C32#93211yescharacteristic
286C2×C32C2×#21511yescharacteristic
287C3⋊D3—22no—
288C3⋊D3—81no—
289C3×D3—62no—
290C3×D3—241no—
291C2×D4—91no—
292C2×D4—541no—
293C22×C4—91no—
294C22×C4—181no—
295C2×C8—91no—
296C2×C8—91no—
297C4⋊C4—91no—
298C4⋊C4—91no—
299C4⋊C4—181no—
300C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
301C22⋊C4—541no—
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×C3D3×#6711yescharacteristic
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—61no—
330C3×C4—61no—
331C3×C4—121no—
332C3×C4—121no—
333C3×C4—181no—
334C3×C4—241no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4#379511yescharacteristic
338C3×C4#379811yescharacteristic
339Dic3—61no—
340Dic3—121no—
341Dic3—121no—
342Dic3—121no—
343Dic3—181no—
344Dic3—241no—
345Dic3—361no—
346C32—21no—
347C32#477711yescharacteristic, complemented
348C32#496811yescharacteristic, complemented
349C23—91no—
350C23—181no—
351D4—181no—
352D4—181no—
353D4—1082no—
354C2×C4—21no—
355C2×C4—91no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—362no—
361C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
362C8—181no—
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×C3#6390511yescharacteristic
376C2×C3#6393211yescharacteristic
377C2×C3D3×#21511yescharacteristic
378D3—62no—
379D3—182no—
380D3—241no—
381D3—721no—
382C22—92no—
383C22—181no—
384C22—181no—
385C22—361no—
386C22—361no—
387C22—361no—
388C22#8867611yescharacteristic
389C4—41no—
390C4—181no—
391C4—361no—
392C4—361no—
393C4#8864811yescharacteristic
394C4#8865011yescharacteristic
395C3—31no—
396C3—31no—
397C3—61no—
398C3#1059050011yescharacteristic
399C2—182no—
400C2—721no—
401C2#1172113911yescharacteristic
402C2#1172120011yescharacteristic
403C2#1172123211yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radical#11721141C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#481
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
3. #11720530C2semidirect
4. #63636⋊C3C2semidirect
5. #11720318C2semidirect
6. #11720317C2semidirect
7. #11720307C2semidirect
21. C4⋊C4×C3⋊D3D3semidirect
25. #63636D3semidirect
228. H(3,𝔽3)#481semidirect
347. C32#4777semidirect
348. C32#4968semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×(C3⋊D3)⋊C32
#117211412
#117205302
#63636⋊C32
#117203182
#117203172
#117203072
#105905113
#105904233