Finite Group Databasebeta

← #13140900

Subgroups — #13140900

Order 1728.

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

normal subgroups only

This group has 392 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#64156⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720583C211yesmaximal, characteristic, complemented
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#11720318C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic
8#10591632—31nomaximal
9#10591624—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#6196—21no—
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3752⋊C3—21no—
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#64156D311yescharacteristic
22C3×#947—31no—
23#63919—31no—
24#63918—31no—
25#63662D311yescharacteristic, complemented
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63564—31no—
31#63564—31no—
32#63563—31no—
33#63562—31no—
34#63555—31no—
35#63554—31no—
36C22×(C3⋊D3)⋊C3—21no—
37Q8×H(3,𝔽3)—21no—
38Q8×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47#6196—42no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5055—91no—
51#4794—91no—
52#4794—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C3×D12—31no—
58C2×#627—61no—
59C2×#627—61no—
60#3752—21no—
61#3752—21no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—61no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—61no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)—41no—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—41no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88(C3⋊Dic3)⋊C3—41no—
89C3×#124—91no—
90#947—31no—
91#947—91no—
92#888—91no—
93#888—91no—
94#869—91no—
95C3×#103—91no—
96C3×#103—91no—
97#835—31no—
98#835—31no—
99#835—31no—
100#835—61no—
101#835—91no—
102#832—91no—
103#832—91no—
104#821—91no—
105#821—91no—
106#815—91no—
107#815—91no—
108#809—91no—
109#809—91no—
110#803—91no—
111C22×C3⋊D3—21no—
112C22×C3×D3—61no—
113C32×Q8—22no—
114C32×Q8—21no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—21no—
118C32×Q8—41no—
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#6711yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C2×C3×Dic3—31no—
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—
138#627—122no—
139#627—122no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—121no—
145#489—271noSylow
146C2×H(3,𝔽3)#12411yescharacteristic
147C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
148C2×H(3,𝔽3)#13711yescharacteristic
149(C3⋊D3)⋊C3—81no—
150C2×C3×Q8—31no—
151C2×C3×Q8—31no—
152C2×C3×Q8—61no—
153C2×C3×Q8D3211yescharacteristic
154C2×C3×D4—91no—
155C2×D12—31no—
156C2×D12—91no—
157C2×Dic6—181no—
158C2×Dic6—181no—
159C2×Dic6—181no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C22⋊C4—181no—
165C3×C22⋊C4—181no—
166C3×C42—91no—
167(C2×D3)⋊C4—61no—
168(C2×D3)⋊C4—61no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174C4⋊Dic3—61no—
175C4⋊Dic3—91no—
176C4⋊Dic3—181no—
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C4×Dic3—31no—
180C4×Dic3—91no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—91no—
186C22×C32—21no—
187C22×C32D3×D411yescharacteristic
188C22×C32C2×#6711yescharacteristic
189C2×C3⋊D3—41no—
190C2×C3⋊D3—41no—
191C2×C3⋊D3—41no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4#23711yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—41no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—121no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—241no—
213C4⋊Q8—271no—
214#124—91no—
215#103—91no—
216#103—91no—
217#102—271no—
218#102—271no—
219#97—271no—
220H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
221C23×C3—181no—
222C22×D3—61no—
223C22×D3—181no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—61no—
227C3×Q8—61no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8—62no—
231C3×Q8—121no—
232C3×D4—181no—
233C3×D4—181no—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4C2×D3211yescharacteristic
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—91no—
252C2×Dic3—91no—
253C2×Dic3—121no—
254C2×Dic3—181no—
255D12—61no—
256D12—61no—
257D12—181no—
258D12—181no—
259Dic6—362no—
260Dic6—362no—
261Dic6—362no—
262C3×C8—181no—
263C3×C8—181no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C3⋊C8—121no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#88511yescharacteristic
273C2×C32#91411yescharacteristic
274C2×C32#91811yescharacteristic
275C2×C32#93611yescharacteristic
276C2×C32#94911yescharacteristic
277C2×C32#95111yescharacteristic
278C3⋊D3—81no—
279C3×D3—241no—
280C2×Q8—541no—
281C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C2×D4—91no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—541no—
288C22⋊C4—181no—
289C22⋊C4—181no—
290C42—91no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—61no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3—361no—
297C22×C3D3×#6711yescharacteristic
298C2×D3—121no—
299C2×D3—121no—
300C2×D3—121no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—361no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—41no—
309C3×C4—61no—
310C3×C4—61no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—122no—
314C3×C4—181no—
315C3×C4—181no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4#380411yescharacteristic
319C3×C4#380511yescharacteristic
320Dic3—61no—
321Dic3—61no—
322Dic3—121no—
323Dic3—121no—
324Dic3—121no—
325Dic3—181no—
326Dic3—181no—
327Dic3—241no—
328Dic3—361no—
329C32—21no—
330C32#479411yescharacteristic, complemented
331C32#505511yescharacteristic, complemented
332C23—181no—
333Q8—21no—
334Q8—21no—
335Q8—1082no—
336D4—181no—
337D4—181no—
338C2×C4—21no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—181no—
342C2×C4—181no—
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—61no—
353C2×C3—61no—
354C2×C3—61no—
355C2×C3—721no—
356C2×C3#6393911yescharacteristic
357C2×C3#6394311yescharacteristic
358C2×C3#6395911yescharacteristic
359D3—241no—
360D3—721no—
361C22—361no—
362C22—361no—
363C22—361no—
364C22#8867611yescharacteristic
365C4—41no—
366C4—181no—
367C4—181no—
368C4—361no—
369C4—361no—
370C4#8867111yescharacteristic
371C4#8867211yescharacteristic
372C3—31no—
373C3—31no—
374C3—61no—
375C3#1059162411yescharacteristic
376C2—721no—
377C2#1172123311yescharacteristic
378C2#1172123711yescharacteristic
379C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#489
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721116C2semidirect
3. #11720583C2semidirect
4. #63662⋊C3C2semidirect
6. #11720312C2semidirect
25. #63662D3semidirect
220. H(3,𝔽3)#489semidirect
330. C32#4794semidirect
331. C32#5055semidirect

Maximal subgroups

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

subgroupindex
#64156⋊C32
#117211162
#117205832
#63662⋊C32
#117203182
#117203122
#117203082
#105916323
#105916243