Finite Group Databasebeta

← #13138857

Subgroups — #13138857

Order 1728.

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

normal subgroups only

This group has 372 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#11721106C211yesmaximal, characteristic
2#11721081C211yesmaximal, characteristic
3#11720551C211yesmaximal, characteristic, complemented
4#63647⋊C3C211yesmaximal, characteristic
5#63643⋊C3C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590715—31nomaximal
9#10590708—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17#88450—21no—
18#88450—21no—
19#88449—21no—
20#88447—21no—
21#63871—31no—
22#63857—31no—
23#63849—31no—
24#63841—31no—
25#63647D311yescharacteristic
26#63643D311yescharacteristic, complemented
27C3×(C2×D3)⋊C8—31no—
28C3×#818—31no—
29C3×#818—31no—
30C3×#818—61no—
31C3×#816—31no—
32#63574—31no—
33#63573—31no—
34#63567—31no—
35#63565—31no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4×(C3⋊D3)⋊C3—21no—
44C4×(C3⋊D3)⋊C3—21no—
45#634⋊C3—21no—
46#634⋊C3—21no—
47C8×H(3,𝔽3)—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#4888—91no—
50#4814—91no—
51#4723—91no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×Dic6—31no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—61no—
67C3×C4⋊Dic3—61no—
68C2×C3×C3⋊C8—31no—
69Dic3⋊Dic3—61no—
70Dic3⋊Dic3—61no—
71#3722—61no—
72#3722—61no—
73(C2×D3)⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75Dic32—61no—
76Dic32—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—21no—
79C2×(C3⋊D3)⋊C3—21no—
80C2×(C3⋊D3)⋊C3—21no—
81C4×H(3,𝔽3)#4111yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83#1084—41no—
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3—41no—
88#946—91no—
89#926—91no—
90#924—91no—
91#897—91no—
92#892—91no—
93#892—91no—
94C3×#107—91no—
95C3×#103—91no—
96C3×C22⋊C8—91no—
97#835—91no—
98(C2×D3)⋊C8—31no—
99(C2×D3)⋊C8—91no—
100#818—31no—
101#818—31no—
102#818—61no—
103#817—91no—
104#816—31no—
105#816—91no—
106#816—91no—
107#810—91no—
108#808—91no—
109#808—91no—
110C22×C3⋊D3D1211yescharacteristic
111C22×C3×D3—31no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3D1211yescharacteristic
118C4×C3⋊D3—21no—
119C4×C3⋊D3—21no—
120#634—21no—
121#634—21no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130C2×C3×Dic3—121no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×Dic6—61no—
134C3×Dic6—61no—
135C32×C8—22no—
136C32×C8—21no—
137C32×C8—21no—
138C3⋊(C3⋊C8)—21no—
139C3×C3⋊C8—61no—
140#485—271noSylow
141C2×H(3,𝔽3)#12311yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—41no—
145C2×C3×Q8—91no—
146C22×C3×C4—91no—
147C2×C4×D3—31no—
148C2×C4×D3—91no—
149C2×Dic6—31no—
150C2×Dic6—91no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C2×C3×C8D3211yescharacteristic
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159#217—181no—
160#217—181no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164(C2×D3)⋊C4—181no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—181no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32C2×D1211yescharacteristic
183C22×C32D3×D411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3×D3—61no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4#23511yescharacteristic
193C32×C4#23711yescharacteristic
194C32×C4#23711yescharacteristic
195C32×C4#23711yescharacteristic
196C3⋊Dic3—21no—
197C3⋊Dic3—41no—
198C3⋊Dic3—41no—
199C3×Dic3—61no—
200C3×Dic3—121no—
201C3×Dic3—121no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—241no—
207C3×Dic3—241no—
208C22⋊Q8—271no—
209#117—271no—
210#107—91no—
211#106—271no—
212#103—91no—
213#103—271no—
214C22⋊C8—91no—
215H(3,𝔽3)#48511yescharacteristic, complemented, Sylow
216C23×C3—91no—
217C22×D3—31no—
218C22×D3—91no—
219C3×Q8—181no—
220C3×Q8—181no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—61no—
224C2×C3×C4—91no—
225C2×C3×C4—181no—
226C2×C3×C4—181no—
227C2×C3×C4—181no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—31no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—91no—
240C2×Dic3—121no—
241C2×Dic3—121no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244C4×D3—61no—
245C4×D3—61no—
246C4×D3—181no—
247C4×D3—181no—
248Dic6—61no—
249Dic6—61no—
250Dic6—181no—
251Dic6—181no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—62no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#88311yescharacteristic
263C2×C32#88611yescharacteristic
264C2×C32#90411yescharacteristic
265C2×C32#90911yescharacteristic
266C2×C32#91511yescharacteristic
267C2×C32#91911yescharacteristic
268C3⋊D3—41no—
269C3×D3—121no—
270C2×Q8—91no—
271C22×C4—91no—
272C2×C8—91no—
273C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
274C4⋊C4—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—541no—
278C22⋊C4—541no—
279C22⋊C4—541no—
280C42—541no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—61no—
284C22×C3—181no—
285C22×C3—181no—
286C22×C3—181no—
287C22×C3D3×D1211yescharacteristic
288C2×D3—61no—
289C2×D3—61no—
290C2×D3—61no—
291C2×D3—181no—
292C2×D3—181no—
293C2×D3—181no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—181no—
301C3×C4—361no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4#379511yescharacteristic
306C3×C4#380511yescharacteristic
307Dic3—61no—
308Dic3—121no—
309Dic3—121no—
310Dic3—121no—
311Dic3—121no—
312Dic3—121no—
313Dic3—121no—
314Dic3—181no—
315Dic3—241no—
316Dic3—241no—
317Dic3—361no—
318Dic3—361no—
319C32—21no—
320C32#472311yescharacteristic, complemented
321C32#481411yescharacteristic, complemented
322C23—91no—
323Q8—181no—
324Q8—181no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—21no—
334C8—181no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—361no—
345C2×C3#6380011yescharacteristic
346C2×C3#6380811yescharacteristic
347C2×C3#6389111yescharacteristic
348D3—121no—
349D3—361no—
350C22—181no—
351C22—181no—
352C22—181no—
353C22#8865411yescharacteristic
354C4—181no—
355C4—361no—
356C4—361no—
357C4—361no—
358C4—361no—
359C4#8864811yescharacteristic
360C4#8867211yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3#1059071011yescharacteristic
365C2—361no—
366C2#1172085011yescharacteristic
367C2#1172085211yescharacteristic
368C2#1172113111yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#485
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
3. #11720551C2semidirect
5. #63643⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720307C2semidirect
26. #63643D3semidirect
215. H(3,𝔽3)#485semidirect
320. C32#4723semidirect
321. C32#4814semidirect

Maximal subgroups

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

subgroupindex
#117211062
#117210812
#117205512
#63647⋊C32
#63643⋊C32
#117203092
#117203072
#105907153
#105907083