Finite Group Databasebeta

← #13138898

Subgroups — #13138898

Order 1728.

435 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 85 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×#6199C211yesmaximal, characteristic
2#11720553C211yesmaximal, characteristic
3#63642⋊C3C211yesmaximal, characteristic, complemented
4#63641⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic
6#11720312C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590661—31nomaximal
9#10590660—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#6199C411yescharacteristic
12C2×#6199C411yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88447C411yescharacteristic
20#88447C411yescharacteristic
21C4×D3×Dic3—31no—
22C4×D3×Dic3—31no—
23#63642D311yescharacteristic, complemented
24#63641D311yescharacteristic, complemented
25C3×(C2×D3)⋊C8—31no—
26C3×(C2×D3)⋊C8—31no—
27C3×(C2×D3)⋊C8—61no—
28C3×#815—31no—
29C3×Dic3⋊C8—31no—
30#63572—31no—
31Dic3⋊(C3⋊C8)—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33(C2×D3)⋊(C3⋊C8)—31no—
34#63555—31no—
35#63554—31no—
36C22×#272C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4×#272—21no—
40C4×#272—21no—
41C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45#6199—22no—
46#6199—22no—
47C8×H(3,𝔽3)—21no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51#4835—91no—
52#4719—91no—
53#4705—91no—
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—61no—
57C2×D3×Dic3—31no—
58C2×D3×Dic3—31no—
59C2×D3×Dic3—31no—
60C2×D3×Dic3—31no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C4×C3⋊Dic3C2×D311yescharacteristic
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C3×C4×Dic3—31no—
69C3×C4×Dic3—31no—
70C2×C3×C3⋊C8—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75Dic32—31no—
76Dic32—31no—
77Dic32—31no—
78Dic32—31no—
79C22×H(3,𝔽3)C22×C411yescharacteristic
80C2×#272—21no—
81C2×#272—21no—
82C2×#272M1612yes—
83C4×H(3,𝔽3)#4111yescharacteristic
84C4×H(3,𝔽3)#4111yescharacteristic
85#1084—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—21no—
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3M1612yes—
90C2×C4×Dic3—91no—
91C42×D3—91no—
92C42×D3—91no—
93C3×C4⋊C8—91no—
94C3×C22⋊C8—91no—
95C3×#97—91no—
96#830—91no—
97(C2×D3)⋊C8—31no—
98(C2×D3)⋊C8—31no—
99(C2×D3)⋊C8—61no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—91no—
102#815—31no—
103#815—91no—
104#815—91no—
105#815—91no—
106Dic3⋊C8—31no—
107Dic3⋊C8—91no—
108Dic3⋊C8—91no—
109Dic3⋊C8—91no—
110C4⋊(C3⋊C8)—91no—
111#803—91no—
112C22×C3×D3—31no—
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3C4×D311yescharacteristic
119C2×C3⋊Dic3C4×D311yescharacteristic
120C2×C3⋊Dic3C4×D311yescharacteristic
121C2×C3⋊Dic3C4×D311yescharacteristic
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—121no—
134C3×C4×D3—121no—
135D3×Dic3—62no—
136D3×Dic3—62no—
137D3×Dic3—62no—
138D3×Dic3—62no—
139C32×C8—21no—
140C32×C8—21no—
141C32×C8—41no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—121no—
149#432—271noSylow
150C2×H(3,𝔽3)#11711yescharacteristic
151C2×H(3,𝔽3)C2×M1611yescharacteristic
152C2×H(3,𝔽3)C2×M1611yescharacteristic
153#272—22no—
154C22×C3×C4—91no—
155C22×Dic3—91no—
156C22×Dic3—91no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—61no—
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3—91no—
163C2×C4×D3—91no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169C2×C3×C8—91no—
170C2×C3×C8D3211yescharacteristic
171C3×C42—91no—
172C3×C42—91no—
173C4×Dic3—31no—
174C4×Dic3—31no—
175C4×Dic3—91no—
176C4×Dic3—91no—
177C4×Dic3—91no—
178C4×Dic3—91no—
179C4×Dic3—91no—
180C4×Dic3—91no—
181C4×Dic3—91no—
182C4×Dic3—91no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—31no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32C2×C4×D311yescharacteristic
192C22×C32C2×C4×D311yescharacteristic
193C2×C3×D3—32no—
194C2×C3×D3—32no—
195C2×C3×D3—62no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—61no—
200C2×C3×D3—121no—
201C2×C3×D3—121no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4#23511yescharacteristic
205C32×C4#23511yescharacteristic
206C32×C4#23711yescharacteristic
207C32×C4#23911yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—21no—
210C3⋊Dic3—21no—
211C3⋊Dic3ΓL(1,𝔽25)12yes—
212C3×Dic3—32no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—121no—
219C2×C42—271no—
220C4⋊C8—91no—
221C4⋊C8—271no—
222C22⋊C8—91no—
223C22⋊C8—271no—
224#97—91no—
225#97—271no—
226H(3,𝔽3)#43211yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—31no—
229C22×D3—31no—
230C22×D3—61no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4C2×D3211yescharacteristic
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249C2×Dic3—91no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—91no—
253C2×Dic3—182no—
254C2×Dic3—182no—
255C4×D3—61no—
256C4×D3—61no—
257C4×D3—61no—
258C4×D3—61no—
259C4×D3—121no—
260C4×D3—121no—
261C4×D3—182no—
262C4×D3—182no—
263C4×D3—182no—
264C4×D3—182no—
265C3×C8—21no—
266C3×C8—61no—
267C3×C8—61no—
268C3×C8—121no—
269C3×C8—181no—
270C3×C8—181no—
271C3×C8—181no—
272C3⋊C8—61no—
273C3⋊C8—61no—
274C3⋊C8—61no—
275C3⋊C8—61no—
276C3⋊C8—121no—
277C3⋊C8—181no—
278C3⋊C8—181no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32#87211yescharacteristic
283C2×C32#89211yescharacteristic
284C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
285C2×C32D3×M1611yescharacteristic
286C2×C32D3×M1611yescharacteristic
287C2×C32D3×M1611yescharacteristic
288C3×D3—62no—
289C3×D3—62no—
290C3×D3—122no—
291C22×C4—91no—
292C22×C4—271no—
293C22×C4—271no—
294C2×C8—91no—
295C2×C8—91no—
296C2×C8—91no—
297C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C42—91no—
299C42—91no—
300C42—271no—
301C42—271no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—61no—
305C22×C3—92no—
306C22×C3—181no—
307C22×C3—181no—
308C22×C3C4×D3211yescharacteristic
309C2×D3—32no—
310C2×D3—32no—
311C2×D3—62no—
312C2×D3—61no—
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—61no—
316C2×D3—121no—
317C2×D3—121no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—61no—
323C3×C4—61no—
324C3×C4—92no—
325C3×C4—181no—
326C3×C4—181no—
327C3×C4—181no—
328C3×C4—181no—
329C3×C4#379911yescharacteristic
330C3×C4#380411yescharacteristic
331Dic3—32no—
332Dic3—61no—
333Dic3—61no—
334Dic3—61no—
335Dic3—61no—
336Dic3—61no—
337Dic3—92no—
338Dic3—121no—
339Dic3—181no—
340Dic3—181no—
341Dic3—181no—
342C32—21no—
343C32#470511yescharacteristic, complemented
344C32#483511yescharacteristic, complemented
345C23—91no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—542no—
354C2×C4—542no—
355C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
356C8—21no—
357C8—181no—
358C8—181no—
359C8—181no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—61no—
369C2×C3—182no—
370C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
371C2×C3#6379111yescharacteristic
372C2×C3#6383611yescharacteristic
373D3—62no—
374D3—62no—
375D3—122no—
376C22—92no—
377C22—181no—
378C22—181no—
379C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
380C4—92no—
381C4—181no—
382C4—181no—
383C4—181no—
384C4—181no—
385C4#8865211yescharacteristic
386C4#8867111yescharacteristic
387C3—31no—
388C3—31no—
389C3—61no—
390C3#1059066511yescharacteristic
391C2—182no—
392C2#1172083511yescharacteristic
393C2#1172083611yescharacteristic
394C2#1172107611yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#432
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63642⋊C3C2semidirect
4. #63641⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720311C2semidirect
23. #63642D3semidirect
24. #63641D3semidirect
226. H(3,𝔽3)#432semidirect
343. C32#4705semidirect
344. C32#4835semidirect

Maximal subgroups

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

subgroupindex
C4×#61992
#117205532
#63642⋊C32
#63641⋊C32
#117203232
#117203122
#117203112
#105906613
#105906603