Finite Group Databasebeta

← #13137974

Subgroups — #13137974

Order 1728.

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

normal subgroups only

This group has 456 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
1C4⋊C4×#272C211yesmaximal, characteristic
2#11721094C211yesmaximal, characteristic
3#63637⋊C3C211yesmaximal, characteristic, complemented
4#63635⋊C3C211yesmaximal, characteristic, complemented
5#11720323C211yesmaximal, characteristic
6#11720309C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590516—31nomaximal
9#10590426—31nomaximal
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88448—21no—
21C3×D3×C4⋊C4—31no—
22C3×D3×C4⋊C4—31no—
23C3×D3×C4⋊C4—61no—
24#63857—31no—
25#63856—31no—
26#63637D311yescharacteristic, complemented
27#63635D311yescharacteristic, complemented
28C3×#810—31no—
29C3×#808—31no—
30#63574—31no—
31#63573—31no—
32#63567—31no—
33#63565—31no—
34(C2×D3)⋊(C3⋊C8)—31no—
35(C2×D3)⋊(C3⋊C8)—31no—
36C22×#272D411yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C4×#272—21no—
42C4×#272—21no—
43C4×#272—42no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46#634⋊C3—21no—
47#634⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#4971—91no—
51#4862—91no—
52#4805—91no—
53C2×#634C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×C3×C4×D3—31no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—31no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4C2×D311yescharacteristic
63C32×C4⋊C4C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×Dic3⋊C4—61no—
72C3×Dic3⋊C4—61no—
73C3×Dic3⋊C4—62no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76#3724—61no—
77Dic3⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×#272—21no—
82C2×#272—21no—
83C2×#272SD1612yes—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)#4111yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—21no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C2×C3×C4⋊C4—91no—
92#946—91no—
93#924—91no—
94#897—91no—
95D3×C4⋊C4—31no—
96D3×C4⋊C4—31no—
97D3×C4⋊C4—61no—
98C3×#106—91no—
99C3×#103—91no—
100#835—91no—
101#830—91no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#817—91no—
105#816—91no—
106#810—31no—
107#810—91no—
108#810—91no—
109#808—31no—
110#808—91no—
111#808—91no—
112#808—91no—
113C22×C3×D3—31no—
114C22×C3×D3—31no—
115C22×C3×D3—61no—
116C2×C32×C4—22no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124#634—21no—
125#634—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—122no—
141C3×C4×D3—122no—
142C3×C4×D3—124no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)—21no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149#483—271noSylow
150C2×H(3,𝔽3)#12311yescharacteristic
151C2×H(3,𝔽3)C2×SD1611yescharacteristic
152C2×H(3,𝔽3)#13711yescharacteristic
153#272—22no—
154C2×C3×Q8—91no—
155C22×C3×C4—91no—
156C22×C3×C4—181no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—61no—
160C2×C4×D3—61no—
161C2×C4×D3—61no—
162C2×C4×D3—62no—
163C2×Dic6—31no—
164C2×Dic6—91no—
165C2×C3×C8—91no—
166C2×C3×C8—91no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—61no—
170C3×C4⋊C4—91no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—181no—
173C3×C4⋊C4D3211yescharacteristic
174#217—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180C4⋊Dic3—61no—
181C4⋊Dic3—91no—
182C4⋊Dic3—181no—
183Dic3⋊C4—61no—
184Dic3⋊C4—61no—
185Dic3⋊C4—62no—
186Dic3⋊C4—181no—
187Dic3⋊C4—181no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—91no—
191C2×C3⋊C8—91no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3×D3—32no—
196C2×C3×D3—32no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—42no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4#23511yescharacteristic
210C32×C4#23511yescharacteristic
211C32×C4#23711yescharacteristic
212C32×C4#23911yescharacteristic
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3×Dic3—62no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—122no—
219C3×Dic3—121no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—121no—
223C22⋊Q8—271no—
224C2×C4⋊C4—91no—
225#106—91no—
226#106—271no—
227#103—91no—
228#103—271no—
229C22⋊C8—271no—
230H(3,𝔽3)#48311yescharacteristic, complemented, Sylow
231C23×C3—91no—
232C22×D3—31no—
233C22×D3—31no—
234C22×D3—61no—
235C3×Q8—181no—
236C3×Q8—181no—
237C2×C3×C4—21no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—62no—
243C2×C3×C4—61no—
244C2×C3×C4—91no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—362no—
251C2×C3×C4C2×D3211yescharacteristic
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—62no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—181no—
261C2×Dic3—181no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—61no—
265C4×D3—61no—
266C4×D3—62no—
267C4×D3—62no—
268C4×D3—122no—
269C4×D3—122no—
270C4×D3—124no—
271Dic6—61no—
272Dic6—61no—
273Dic6—181no—
274Dic6—181no—
275C3×C8—181no—
276C3×C8—181no—
277C3⋊C8—61no—
278C3⋊C8—61no—
279C3⋊C8—181no—
280C3⋊C8—181no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32#88311yescharacteristic
285C2×C32#89411yescharacteristic
286C2×C32D3×SD1611yescharacteristic
287C2×C32D3×SD1611yescharacteristic
288C2×C32#91511yescharacteristic
289C2×C32#91811yescharacteristic
290C3×D3—64no—
291C3×D3—62no—
292C3×D3—62no—
293C2×Q8—91no—
294C22×C4—91no—
295C22×C4—181no—
296C2×C8—91no—
297C2×C8—91no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C4⋊C4—181no—
301C4⋊C4—541no—
302C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C22⋊C4—541no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—61no—
307C22×C3—92no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3#381111yescharacteristic
311C2×D3—32no—
312C2×D3—32no—
313C2×D3—62no—
314C2×D3—61no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—61no—
318C2×D3—62no—
319C2×D3—62no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—41no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—121no—
328C3×C4—121no—
329C3×C4—122no—
330C3×C4—181no—
331C3×C4—361no—
332C3×C4—361no—
333C3×C4—361no—
334C3×C4#379611yescharacteristic
335C3×C4#379811yescharacteristic
336Dic3—61no—
337Dic3—61no—
338Dic3—62no—
339Dic3—121no—
340Dic3—121no—
341Dic3—122no—
342Dic3—121no—
343Dic3—121no—
344Dic3—361no—
345Dic3—361no—
346C32—21no—
347C32#480511yescharacteristic, complemented
348C32#486211yescharacteristic, complemented
349C23—91no—
350Q8—181no—
351Q8—181no—
352C2×C4—21no—
353C2×C4—91no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—362no—
360C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
361C8—181no—
362C8—181no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—182no—
373C2×C3#6385311yescharacteristic
374C2×C3#6393011yescharacteristic
375C2×C3#6395111yescharacteristic
376D3—62no—
377D3—62no—
378D3—64no—
379C22—92no—
380C22—181no—
381C22—181no—
382C22#8867711yescharacteristic
383C4—41no—
384C4—181no—
385C4—361no—
386C4—361no—
387C4—361no—
388C4#8864911yescharacteristic
389C4#8865011yescharacteristic
390C3—31no—
391C3—31no—
392C3—61no—
393C3#1059052911yescharacteristic
394C2—182no—
395C2#1172110411yescharacteristic
396C2#1172118811yescharacteristic
397C2#1172123111yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#483
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #63637⋊C3C2semidirect
4. #63635⋊C3C2semidirect
6. #11720309C2semidirect
7. #11720307C2semidirect
26. #63637D3semidirect
27. #63635D3semidirect
230. H(3,𝔽3)#483semidirect
347. C32#4805semidirect
348. C32#4862semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#117210942
#63637⋊C32
#63635⋊C32
#117203232
#117203092
#117203072
#105905163
#105904263