Finite Group Databasebeta

← #13138001

Subgroups — #13138001

Order 1728.

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

normal subgroups only

This group has 449 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#11721156C211yesmaximal, characteristic
3#11720531C211yesmaximal, characteristic, complemented
4#63634⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10590510—31nomaximal
9#10590498—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88451—21no—
21C4⋊C4×C3⋊D3D311yescharacteristic, complemented
22C3×D3×C4⋊C4—31no—
23#63916—31no—
24#63908—31no—
25#63634D311yescharacteristic, complemented
26C3×#809—31no—
27C3×#809—31no—
28C3×#809—61no—
29C3×#807—31no—
30#63575—31no—
31#63575—31no—
32#63563—31no—
33#63562—31no—
34#63557—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272—21no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4×(C3⋊D3)⋊C3—21no—
46C4×(C3⋊D3)⋊C3—21no—
47C4×(C3⋊D3)⋊C3—42no—
48#6200—42no—
49#6180—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4967—91no—
52#4882—91no—
53#4774—91no—
54C2×C4×C3⋊D3—21no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×C3×D12—31no—
57C2×C3×D12—31no—
58C2×C3×D12—61no—
59C2×C3×C4×D3—31no—
60C2×C3×C4×D3—61no—
61C2×#626—61no—
62C2×#625—61no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—21no—
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
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(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—41no—
81C2×#272—41no—
82C2×#272—41no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—21no—
85C2×(C3⋊D3)⋊C3D812yes—
86C4×H(3,𝔽3)—41no—
87C4×H(3,𝔽3)#4111yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—41no—
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×#107—91no—
99C3×#102—91no—
100#832—91no—
101#830—91no—
102#821—91no—
103(C2×D3)⋊C8—91no—
104(C2×D3)⋊C8—91no—
105#818—91no—
106#809—31no—
107#809—31no—
108#809—61no—
109#809—91no—
110#807—31no—
111#807—91no—
112#807—91no—
113#807—91no—
114C22×C3⋊D3#6711yescharacteristic
115C22×C3×D3—31no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—121no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3#6711yescharacteristic
128C4×C3⋊D3—21no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—42no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×D12—62no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×D12—121no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—122no—
143#626—122no—
144#625—122no—
145C3⋊(C3⋊C8)—21no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—121no—
150#480—271noSylow
151C2×H(3,𝔽3)#12311yescharacteristic
152C2×H(3,𝔽3)C2×D811yescharacteristic
153C2×H(3,𝔽3)#13711yescharacteristic
154#272—81no—
155(C3⋊D3)⋊C3—22no—
156C2×C3×D4—91no—
157C22×C3×C4—91no—
158C22×C3×C4—181no—
159C2×#67—181no—
160C2×#67—181no—
161C2×D12—31no—
162C2×D12—31no—
163C2×D12—61no—
164C2×D12—181no—
165C2×C4×D3—31no—
166C2×C4×D3—61no—
167C2×C4×D3—91no—
168C2×C4×D3—181no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4—181no—
177C3×C4⋊C4D3211yescharacteristic
178#217—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185Dic3⋊C4—61no—
186Dic3⋊C4—181no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×#6711yescharacteristic
195C2×C3⋊D3—21no—
196C2×C3⋊D3—21no—
197C2×C3⋊D3#21312yes—
198C2×C3×D3—32no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C2×C3×D3—122no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—241no—
209C2×C3×D3—241no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—81no—
215C32×C4#23511yescharacteristic
216C32×C4#23511yescharacteristic
217C32×C4#23511yescharacteristic
218C32×C4#23911yescharacteristic
219C3⋊Dic3—21no—
220C3⋊Dic3—41no—
221C3⋊Dic3—41no—
222C3×Dic3—61no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225#121—271no—
226C2×C4⋊C4—91no—
227#107—91no—
228#107—271no—
229#102—91no—
230#102—271no—
231C22⋊C8—271no—
232H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
233C23×C3—91no—
234C23×C3—181no—
235C22×D3—31no—
236C22×D3—61no—
237C22×D3—61no—
238C22×D3—91no—
239C22×D3—121no—
240C3×D4—181no—
241C3×D4—181no—
242C2×C3×C4—21no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—91no—
249C2×C3×C4—121no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4—362no—
255C2×C3×C4C2×D3211yescharacteristic
256#67—362no—
257#67—362no—
258C2×Dic3—31no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—91no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264D12—61no—
265D12—61no—
266D12—61no—
267D12—61no—
268D12—62no—
269D12—121no—
270D12—362no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—122no—
274C4×D3—181no—
275C4×D3—181no—
276C4×D3—362no—
277C3×C8—181no—
278C3×C8—181no—
279C3⋊C8—61no—
280C3⋊C8—61no—
281C3⋊C8—61no—
282C3⋊C8—121no—
283C3⋊C8—181no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32#89411yescharacteristic
288C2×C32D3×D811yescharacteristic
289C2×C32#91811yescharacteristic
290C2×C32#92911yescharacteristic
291C2×C32C2×#21311yescharacteristic
292C2×C32#94211yescharacteristic
293C3⋊D3—22no—
294C3×D3—62no—
295C3×D3—242no—
296C3×D3—241no—
297C3×D3—241no—
298C2×D4—91no—
299C2×D4—541no—
300C22×C4—91no—
301C22×C4—181no—
302C2×C8—91no—
303C2×C8—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C22⋊C4—541no—
309C22×C3—31no—
310C22×C3—31no—
311C22×C3—61no—
312C22×C3—92no—
313C22×C3—181no—
314C22×C3—181no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3D3×#6711yescharacteristic
319C2×D3—32no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—92no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—122no—
330C2×D3—181no—
331C2×D3—181no—
332C2×D3—241no—
333C2×D3—241no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—41no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—181no—
344C3×C4—241no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4#379811yescharacteristic
348C3×C4#379911yescharacteristic
349Dic3—61no—
350Dic3—121no—
351Dic3—121no—
352Dic3—181no—
353Dic3—361no—
354Dic3—361no—
355C32—21no—
356C32#488211yescharacteristic, complemented
357C32#496711yescharacteristic, complemented
358C23—91no—
359C23—181no—
360D4—181no—
361D4—181no—
362D4—1082no—
363C2×C4—21no—
364C2×C4—91no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—362no—
370C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
371C8—181no—
372C8—181no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—182no—
383C2×C3—721no—
384C2×C3#6385011yescharacteristic
385C2×C3D3×#21311yescharacteristic
386C2×C3#6394811yescharacteristic
387D3—62no—
388D3—182no—
389D3—241no—
390D3—241no—
391D3—242no—
392C22—92no—
393C22—181no—
394C22—181no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22#8867611yescharacteristic
399C4—41no—
400C4—181no—
401C4—361no—
402C4—361no—
403C4#8865011yescharacteristic
404C4#8865211yescharacteristic
405C3—31no—
406C3—31no—
407C3—61no—
408C3#1059042211yescharacteristic
409C2—182no—
410C2—721no—
411C2#1172110011yescharacteristic
412C2#1172119411yescharacteristic
413C2#1172123411yescharacteristic
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#11721156C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#480
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. #11720531C2semidirect
4. #63634⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720317C2semidirect
7. #11720315C2semidirect
21. C4⋊C4×C3⋊D3D3semidirect
25. #63634D3semidirect
232. H(3,𝔽3)#480semidirect
356. C32#4882semidirect
357. C32#4967semidirect

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
#117211562
#117205312
#63634⋊C32
#117203242
#117203172
#117203152
#105905103
#105904983