Finite Group Databasebeta

← #13139970

Subgroups — #13139970

Order 1728.

386 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 75 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#64135⋊C3C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3#11720573C211yesmaximal, characteristic
4#11720315C211yesmaximal, characteristic, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#11720313C211yesmaximal, characteristic, complemented
7#11720313C211yesmaximal, characteristic, complemented
8#10591134—31nomaximal
9#10591127—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#3757⋊C3—21no—
15#3757⋊C3—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#64135D311yescharacteristic
22#64133D311yescharacteristic
23C3×#926—31no—
24C3×#924—31no—
25C3×#830—31no—
26C3×#830—31no—
27C3×#830—61no—
28#63575—31no—
29#63575—31no—
30#63573—31no—
31#63573—31no—
32#63567—31no—
33#63567—31no—
34#63566—31no—
35#63566—31no—
36C23×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47#6180—21no—
48#6180—21no—
49#5045—91no—
50#4996—91no—
51#4723—91no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×Dic6—31no—
57#3757—21no—
58#3757—21no—
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C4×C3⋊Dic3—21no—
64C3×#217—61no—
65C3×#217—61no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×Dic3⋊C4—61no—
70C3×C4×Dic3—61no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)—21no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C4×H(3,𝔽3)—21no—
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88C3×C22⋊Q8—91no—
89C3×#117—91no—
90#926—31no—
91#926—91no—
92#924—31no—
93#924—91no—
94C3×C22⋊C8—91no—
95#835—91no—
96#835—91no—
97#830—31no—
98#830—31no—
99#830—61no—
100#818—91no—
101#817—91no—
102#816—91no—
103#816—91no—
104#810—91no—
105#810—91no—
106#808—91no—
107#808—91no—
108#807—91no—
109#807—91no—
110C23×C32—21no—
111C23×C32#6711yescharacteristic
112C23×C32#6711yescharacteristic
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4#6711yescharacteristic
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129#634—21no—
130#634—21no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—61no—
141C3×C3⋊C8—121no—
142C3×C3⋊C8—121no—
143#485—271noSylow
144C2×H(3,𝔽3)—41no—
145C2×H(3,𝔽3)#12311yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×H(3,𝔽3)#13711yescharacteristic
148C2×C3×Q8—91no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4D3211yescharacteristic
153C2×Dic6—31no—
154C2×Dic6—91no—
155C2×C3×C8—91no—
156C2×C3×C8—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—181no—
161C3×C22⋊C4—181no—
162C3×C22⋊C4—181no—
163C3×C42—181no—
164#217—61no—
165#217—61no—
166#217—181no—
167#217—181no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174Dic3⋊C4—61no—
175Dic3⋊C4—181no—
176C4×Dic3—61no—
177C4×Dic3—181no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—61no—
183C2×C3⋊C8—61no—
184C22×C32—22no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32C2×#6711yescharacteristic
195C22×C32C2×#6711yescharacteristic
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—41no—
201C32×C4#23511yescharacteristic
202C32×C4#23511yescharacteristic
203C32×C4#23711yescharacteristic
204C32×C4#23711yescharacteristic
205C3⋊Dic3—41no—
206C3⋊Dic3—41no—
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C22⋊Q8—91no—
214#117—91no—
215#107—271no—
216#106—271no—
217#103—271no—
218#103—271no—
219C22⋊C8—91no—
220H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
221C23×C3—31no—
222C23×C3—31no—
223C23×C3—61no—
224C23×C3#62511yescharacteristic
225C3×Q8—181no—
226C3×Q8—181no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4#62511yescharacteristic
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—181no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255Dic6—61no—
256Dic6—61no—
257Dic6—181no—
258Dic6—181no—
259C3×C8—181no—
260C3×C8—181no—
261C3⋊C8—61no—
262C3⋊C8—61no—
263C3⋊C8—61no—
264C3⋊C8—61no—
265C3⋊C8—121no—
266C3⋊C8—121no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—42no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32#92911yescharacteristic
274C2×C32#93511yescharacteristic
275C2×C32#94211yescharacteristic
276C2×C32#95011yescharacteristic
277C2×C32#95011yescharacteristic
278C2×C32#95111yescharacteristic
279C2×Q8—91no—
280C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
281C2×C8—91no—
282C2×C8—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C22⋊C4—181no—
288C22⋊C4—181no—
289C42—181no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3—62no—
302C22×C3—61no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3C2×#62511yescharacteristic
306C3×C4—21no—
307C3×C4—31no—
308C3×C4—31no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—61no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—121no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4—361no—
319C3×C4—361no—
320C3×C4#380411yescharacteristic
321C3×C4#380411yescharacteristic
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—121no—
326Dic3—361no—
327Dic3—361no—
328Dic3—361no—
329Dic3—361no—
330C32—21no—
331C32#499611yescharacteristic, complemented
332C32#504511yescharacteristic, complemented
333C23#619811yescharacteristic
334Q8—181no—
335Q8—181no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4#619811yescharacteristic
343C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
344C8—181no—
345C8—181no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—41no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—121no—
357C2×C3—121no—
358C2×C3—122no—
359C2×C3#6382211yescharacteristic
360C2×C3#6382711yescharacteristic
361C2×C3#6396111yescharacteristic
362C22—21no—
363C22—21no—
364C22—21no—
365C22C2×#619811yescharacteristic
366C4—21no—
367C4—361no—
368C4—361no—
369C4—361no—
370C4—361no—
371C4#8867111yescharacteristic
372C4#8867111yescharacteristic
373C3—31no—
374C3—31no—
375C3—61no—
376C3#1059110211yescharacteristic
377C2—41no—
378C2#1172105811yescharacteristic
379C2#1172106011yescharacteristic
380C2#1172130611yescharacteristic
subgroupG/H
centerC22C2×#6198
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#625
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#485
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720315C2semidirect
5. #11720314C2semidirect
6. #11720313C2semidirect
7. #11720313C2semidirect
220. H(3,𝔽3)#485semidirect
331. C32#4996semidirect
332. C32#5045semidirect

Maximal subgroups

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

subgroupindex
#64135⋊C32
#64133⋊C32
#117205732
#117203152
#117203142
#117203132
#105911343
#105911273