Finite Group Databasebeta

← #13138943

Subgroups — #13138943

Order 1728.

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

normal subgroups only

This group has 449 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
1#11721159C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic, complemented
3#11720553C211yesmaximal, characteristic, complemented
4#63648⋊C3C211yesmaximal, characteristic, complemented
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720315C211yesmaximal, characteristic, complemented
8#10590680—31nomaximal
9#10590679—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6200—21no—
13C2×#6199—21no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#88450—21no—
20#88449—21no—
21#63909—31no—
22#63908—31no—
23D3×C4⋊Dic3—31no—
24D3×C4⋊Dic3—31no—
25#63648D311yescharacteristic, complemented
26#63645D311yescharacteristic, complemented
27C3×#821—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—31no—
30C3×(C2×D3)⋊C8—61no—
31C3×#818—31no—
32#63575—31no—
33#63575—31no—
34#63564—31no—
35#63563—31no—
36C22×#272D411yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C4×#272—21no—
41C4×#272—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6200—42no—
48#6199—42no—
49C8×H(3,𝔽3)—21no—
50#6180—21no—
51#4882—91no—
52#4774—91no—
53#4736—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×#626—61no—
60C2×#625—61no—
61C2×D3×Dic3—61no—
62C2×D3×Dic3—61no—
63C2×C32×C8—21no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75Dic3⋊Dic3—61no—
76Dic3⋊Dic3—61no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×#272—21no—
81C2×#272—21no—
82C2×#272D812yes—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C2×(C3⋊D3)⋊C3—41no—
86C4×H(3,𝔽3)#4111yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88#1084—21no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92#938—91no—
93#930—91no—
94C2×C4⋊Dic3—91no—
95#895—91no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—91no—
98C3×#107—91no—
99C3×#102—91no—
100C3×C22⋊C8—91no—
101#832—91no—
102#821—31no—
103#821—91no—
104#821—91no—
105(C2×D3)⋊C8—31no—
106(C2×D3)⋊C8—31no—
107(C2×D3)⋊C8—61no—
108#818—31no—
109#818—91no—
110#818—91no—
111#809—91no—
112#807—91no—
113#807—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—31no—
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)—21no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3×D12—61no—
134C3×D12—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—121no—
140C3×C4×D3—121no—
141#626—122no—
142#625—122no—
143D3×Dic3—122no—
144D3×Dic3—122no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—41no—
148C3×C3⋊C8—61no—
149C3×C3⋊C8—61no—
150C3×C3⋊C8—121no—
151#480—271noSylow
152C2×H(3,𝔽3)#12311yescharacteristic
153C2×H(3,𝔽3)C2×D811yescharacteristic
154C2×H(3,𝔽3)#13711yescharacteristic
155#272—22no—
156(C3⋊D3)⋊C3—81no—
157C2×C3×D4—91no—
158C22×C3×C4—91no—
159C2×#67—181no—
160C2×#67—181no—
161C22×Dic3—181no—
162C2×D12—31no—
163C2×D12—91no—
164C2×D12—181no—
165C2×C4×D3—31no—
166C2×C4×D3—31no—
167C2×C4×D3—61no—
168C2×C4×D3—181no—
169C2×C4×D3—181no—
170C2×C3×C8—31no—
171C2×C3×C8—31no—
172C2×C3×C8—61no—
173C2×C3×C8—91no—
174C2×C3×C8D3211yescharacteristic
175C3×C4⋊C4—91no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—91no—
178#217—181no—
179(C2×D3)⋊C4—181no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—31no—
184C4⋊Dic3—91no—
185C4⋊Dic3—91no—
186C4⋊Dic3—91no—
187C4⋊Dic3—181no—
188Dic3⋊C4—181no—
189Dic3⋊C4—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8—31no—
192C2×C3⋊C8—61no—
193C22×C32—21no—
194C22×C32D3×D411yescharacteristic
195C22×C32D3×D411yescharacteristic
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3×D3—32no—
200C2×C3×D3—32no—
201C2×C3×D3—62no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4#23511yescharacteristic
214C32×C4#23511yescharacteristic
215C32×C4#23711yescharacteristic
216C32×C4#23911yescharacteristic
217C3⋊Dic3—41no—
218C3⋊Dic3—41no—
219C3⋊Dic3—41no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C3×Dic3—121no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—121no—
226#121—271no—
227C2×C4⋊C4—271no—
228#107—91no—
229#107—271no—
230#102—91no—
231#102—271no—
232C22⋊C8—91no—
233H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
234C23×C3—91no—
235C23×C3—181no—
236C22×D3—31no—
237C22×D3—31no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—181no—
241C3×D4—181no—
242C3×D4—181no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—61no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—181no—
252C2×C3×C4C2×D3211yescharacteristic
253#67—362no—
254#67—362no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—362no—
265D12—61no—
266D12—61no—
267D12—181no—
268D12—181no—
269D12—362no—
270C4×D3—61no—
271C4×D3—61no—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—121no—
275C4×D3—121no—
276C4×D3—362no—
277C4×D3—362no—
278C3×C8—21no—
279C3×C8—61no—
280C3×C8—61no—
281C3×C8—121no—
282C3×C8—181no—
283C3⋊C8—61no—
284C3⋊C8—61no—
285C3⋊C8—121no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32#88311yescharacteristic
290C2×C32#89411yescharacteristic
291C2×C32D3×D811yescharacteristic
292C2×C32D3×D811yescharacteristic
293C2×C32#91511yescharacteristic
294C2×C32#91811yescharacteristic
295C3⋊D3—81no—
296C3×D3—62no—
297C3×D3—62no—
298C3×D3—122no—
299C3×D3—241no—
300C2×D4—91no—
301C2×D4—541no—
302C22×C4—91no—
303C22×C4—541no—
304C2×C8—91no—
305C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C4⋊C4—541no—
310C22⋊C4—541no—
311C22×C3—31no—
312C22×C3—31no—
313C22×C3—61no—
314C22×C3—92no—
315C22×C3—181no—
316C22×C3—181no—
317C22×C3—361no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3#380211yescharacteristic
321C2×D3—32no—
322C2×D3—32no—
323C2×D3—62no—
324C2×D3—61no—
325C2×D3—61no—
326C2×D3—61no—
327C2×D3—61no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—361no—
334C2×D3—361no—
335C2×D3—361no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—181no—
343C3×C4—361no—
344C3×C4—361no—
345C3×C4—361no—
346C3×C4#379911yescharacteristic
347C3×C4#380411yescharacteristic
348Dic3—61no—
349Dic3—61no—
350Dic3—121no—
351Dic3—121no—
352Dic3—121no—
353Dic3—121no—
354Dic3—361no—
355Dic3—361no—
356Dic3—361no—
357C32—21no—
358C32#477411yescharacteristic, complemented
359C32#488211yescharacteristic, complemented
360C23—91no—
361C23—181no—
362D4—181no—
363D4—181no—
364D4—1082no—
365C2×C4—91no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—1082no—
372C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—31no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—61no—
384C2×C3—182no—
385C2×C3—721no—
386C2×C3#6379711yescharacteristic
387C2×C3#6381111yescharacteristic
388C2×C3#6385411yescharacteristic
389D3—62no—
390D3—62no—
391D3—122no—
392D3—241no—
393D3—721no—
394C22—92no—
395C22—181no—
396C22—181no—
397C22—361no—
398C22—361no—
399C22—361no—
400C22#8865511yescharacteristic
401C4—181no—
402C4—361no—
403C4—361no—
404C4—361no—
405C4#8865211yescharacteristic
406C4#8867111yescharacteristic
407C3—31no—
408C3—31no—
409C3—61no—
410C3#1059065011yescharacteristic
411C2—182no—
412C2—721no—
413C2#1172083811yescharacteristic
414C2#1172084511yescharacteristic
415C2#1172110211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721159C2
hypercenterC2×C8((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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721152C2semidirect
3. #11720553C2semidirect
4. #63648⋊C3C2semidirect
5. #63645⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720315C2semidirect
25. #63648D3semidirect
26. #63645D3semidirect
233. H(3,𝔽3)#480semidirect
358. C32#4774semidirect
359. C32#4882semidirect

Maximal subgroups

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

subgroupindex
#117211592
#117211522
#117205532
#63648⋊C32
#63645⋊C32
#117203182
#117203152
#105906803
#105906793