Finite Group Databasebeta

← #13137982

Subgroups — #13137982

Order 1728.

491 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 69 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×#272C211yesmaximal, characteristic, complemented
2#11721159C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic, complemented
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590500—31nomaximal
9#10590423—31nomaximal
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×#6200—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊#1084C2211yescharacteristic
16#88540—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21C3×D3×C4⋊C4—31no—
22C3×D3×C4⋊C4—31no—
23C3×D3×C4⋊C4—61no—
24#63909—31no—
25#63908—31no—
26#63636D311yescharacteristic, complemented
27#63635D311yescharacteristic, complemented
28C3×#809—31no—
29C3×#808—31no—
30#63574—31no—
31#63573—31no—
32#63564—31no—
33#63562—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272—21no—
43C4×#272—21no—
44C4×#272—42no—
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48#6200—42no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4968—91no—
52#4863—91no—
53#4777—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×C4×D3—31no—
57C2×C3×C4×D3—31no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—62no—
62C2×#626—61no—
63C2×#625—61no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—61no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C3×Dic3⋊C4—62no—
77C2×C3×C3⋊C8—31no—
78C2×C3×C3⋊C8—31no—
79#3722—61no—
80(C2×D3)⋊Dic3—61no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×#272—21no—
83C2×#272—21no—
84C2×#272SD1612yes—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C2×(C3⋊D3)⋊C3—41no—
88C4×H(3,𝔽3)—41no—
89C4×H(3,𝔽3)#4111yescharacteristic
90C4×H(3,𝔽3)#4111yescharacteristic
91#1084—21no—
92#1084—41no—
93(C3⋊Dic3)⋊C3—41no—
94C2×C3×C4⋊C4—91no—
95#938—91no—
96#930—91no—
97#895—91no—
98D3×C4⋊C4—31no—
99D3×C4⋊C4—31no—
100D3×C4⋊C4—61no—
101C3×#106—91no—
102C3×#102—91no—
103#832—91no—
104#830—91no—
105#821—91no—
106(C2×D3)⋊C8—91no—
107(C2×D3)⋊C8—91no—
108#817—91no—
109#809—31no—
110#809—91no—
111#809—91no—
112#808—31no—
113#808—91no—
114#808—91no—
115#808—91no—
116C22×C3⋊D3—21no—
117C22×C3×D3—31no—
118C22×C3×D3—31no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—31no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—124no—
146C3×C4×D3—122no—
147C3×C4×D3—122no—
148#626—122no—
149#625—122no—
150C3⋊(C3⋊C8)—21no—
151C3⋊(C3⋊C8)—21no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—61no—
154#481—271noSylow
155C2×H(3,𝔽3)#12311yescharacteristic
156C2×H(3,𝔽3)C2×SD1611yescharacteristic
157C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
158#272—22no—
159(C3⋊D3)⋊C3—81no—
160C2×C3×D4—91no—
161C22×C3×C4—91no—
162C22×C3×C4—181no—
163C2×#67—181no—
164C2×#67—181no—
165C2×D12—31no—
166C2×D12—91no—
167C2×D12—181no—
168C2×C4×D3—31no—
169C2×C4×D3—31no—
170C2×C4×D3—61no—
171C2×C4×D3—61no—
172C2×C4×D3—61no—
173C2×C4×D3—62no—
174C2×C3×C8—91no—
175C2×C3×C8—91no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—91no—
181C3×C4⋊C4—181no—
182C3×C4⋊C4D3211yescharacteristic
183#217—181no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186C4⋊Dic3—31no—
187C4⋊Dic3—31no—
188C4⋊Dic3—31no—
189C4⋊Dic3—61no—
190C4⋊Dic3—91no—
191Dic3⋊C4—61no—
192Dic3⋊C4—61no—
193Dic3⋊C4—62no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—91no—
198C22×C32—21no—
199C22×C32D3×D411yescharacteristic
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—32no—
205C2×C3×D3—32no—
206C2×C3×D3—62no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—42no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4#23511yescharacteristic
222C32×C4#23511yescharacteristic
223C32×C4#23711yescharacteristic
224C32×C4#23911yescharacteristic
225C3⋊Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C3×Dic3—121no—
232C3×Dic3—121no—
233#121—271no—
234C2×C4⋊C4—91no—
235#106—91no—
236#106—271no—
237#102—91no—
238#102—271no—
239C22⋊C8—271no—
240H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C23×C3—181no—
243C22×D3—31no—
244C22×D3—31no—
245C22×D3—61no—
246C22×D3—61no—
247C22×D3—181no—
248C3×D4—181no—
249C3×D4—181no—
250C2×C3×C4—21no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—61no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4—362no—
263C2×C3×C4C2×D3211yescharacteristic
264#67—362no—
265#67—362no—
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—62no—
272C2×Dic3—61no—
273C2×Dic3—181no—
274D12—61no—
275D12—61no—
276D12—181no—
277D12—181no—
278D12—362no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—62no—
284C4×D3—62no—
285C4×D3—122no—
286C4×D3—122no—
287C4×D3—124no—
288C3×C8—181no—
289C3×C8—181no—
290C3⋊C8—61no—
291C3⋊C8—61no—
292C3⋊C8—181no—
293C3⋊C8—181no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32#88311yescharacteristic
298C2×C32#89411yescharacteristic
299C2×C32#91111yescharacteristic
300C2×C32D3×SD1611yescharacteristic
301C2×C32D3×SD1611yescharacteristic
302C2×C32#91411yescharacteristic
303C3⋊D3—81no—
304C3×D3—64no—
305C3×D3—62no—
306C3×D3—62no—
307C3×D3—241no—
308C2×D4—91no—
309C2×D4—541no—
310C22×C4—91no—
311C22×C4—181no—
312C2×C8—91no—
313C2×C8—91no—
314C4⋊C4—91no—
315C4⋊C4—91no—
316C4⋊C4—181no—
317C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C22⋊C4—541no—
319C22×C3—31no—
320C22×C3—31no—
321C22×C3—61no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3#381111yescharacteristic
329C2×D3—32no—
330C2×D3—32no—
331C2×D3—62no—
332C2×D3—61no—
333C2×D3—61no—
334C2×D3—61no—
335C2×D3—61no—
336C2×D3—62no—
337C2×D3—62no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—31no—
348C3×C4—41no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4—122no—
354C3×C4—181no—
355C3×C4—361no—
356C3×C4—361no—
357C3×C4#379611yescharacteristic
358C3×C4#379811yescharacteristic
359Dic3—61no—
360Dic3—61no—
361Dic3—62no—
362Dic3—121no—
363Dic3—121no—
364Dic3—122no—
365Dic3—121no—
366Dic3—361no—
367C32—21no—
368C32#477711yescharacteristic, complemented
369C32#486311yescharacteristic, complemented
370C23—91no—
371C23—181no—
372D4—181no—
373D4—181no—
374D4—1082no—
375C2×C4—21no—
376C2×C4—91no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—362no—
382C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
383C8—181no—
384C8—181no—
385C2×C3—31no—
386C2×C3—31no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—61no—
392C2×C3—61no—
393C2×C3—61no—
394C2×C3—182no—
395C2×C3—721no—
396C2×C3#6385311yescharacteristic
397C2×C3#6393711yescharacteristic
398C2×C3#6394011yescharacteristic
399D3—62no—
400D3—62no—
401D3—64no—
402D3—241no—
403D3—721no—
404C22—92no—
405C22—181no—
406C22—181no—
407C22—361no—
408C22—361no—
409C22—361no—
410C22#8867711yescharacteristic
411C4—41no—
412C4—181no—
413C4—361no—
414C4—361no—
415C4#8864911yescharacteristic
416C4#8865011yescharacteristic
417C3—31no—
418C3—31no—
419C3—61no—
420C3#1059051111yescharacteristic
421C2—182no—
422C2—721no—
423C2#1172110411yescharacteristic
424C2#1172119311yescharacteristic
425C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721159C2
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×#272C2semidirect
3. #63636⋊C3C2semidirect
4. #63635⋊C3C2semidirect
5. #11720323C2semidirect
6. #11720308C2semidirect
7. #11720307C2semidirect
26. #63636D3semidirect
27. #63635D3semidirect
240. H(3,𝔽3)#481semidirect
368. C32#4777semidirect
369. C32#4863semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#117211592
#63636⋊C32
#63635⋊C32
#117203232
#117203082
#117203072
#105905003
#105904233