Finite Group Databasebeta

← #13137892

Subgroups — #13137892

Order 1728.

418 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721756C211yesmaximal, characteristic, complemented
2#11721113C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic
4#63637⋊C3C211yesmaximal, characteristic, complemented
5#11720309C211yesmaximal, characteristic, complemented
6#11720308C211yesmaximal, characteristic
7#11720303C211yesmaximal, characteristic, complemented
8#10590535—31nomaximal
9#10590531—31nomaximal
10C2×#6229—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16C4×#1084C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88450—21no—
20#88450—21no—
21C3×#888—31no—
22C3×#888—31no—
23C3×#888—61no—
24#63864—31no—
25#63860—31no—
26#63637D311yescharacteristic, complemented
27#63636D311yescharacteristic
28C3×#810—31no—
29C3×#809—31no—
30#63567—31no—
31#63565—31no—
32#63564—31no—
33#63562—31no—
34#63554—31no—
35#63554—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084D411yescharacteristic
41C2×#1084D411yescharacteristic
42#6229—42no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46#634⋊C3—21no—
47#634⋊C3—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5089—91no—
51#4794—91no—
52#4765—91no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×#634C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—61no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—62no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C4⋊C4C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×Dic3⋊C4—61no—
69C3×Dic3⋊C4—61no—
70C3×Dic3⋊C4—62no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—31no—
73C3×C4×Dic3—61no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76#3722—61no—
77#3722—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×(C3⋊D3)⋊C3—41no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)#4111yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—21no—
88#1084—21no—
89#1084—41no—
90(C3⋊Dic3)⋊C3—41no—
91C3×C4⋊Q8—91no—
92#947—91no—
93#936—91no—
94#888—31no—
95#888—31no—
96#888—61no—
97#875—91no—
98C3×#103—91no—
99C3×#102—91no—
100#835—91no—
101#832—91no—
102#821—91no—
103#816—91no—
104#815—91no—
105#815—91no—
106#810—31no—
107#810—91no—
108#810—91no—
109#809—31no—
110#809—91no—
111#809—91no—
112#803—91no—
113C22×C3⋊D3—21no—
114C22×C3×D3—61no—
115C2×C32×C4—22no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—21no—
124#634—21no—
125#634—21no—
126C2×C3×Dic3—31no—
127C2×C3×Dic3—31no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—61no—
137C3×D12—61no—
138C3×Dic6—61no—
139C3×Dic6—61no—
140C3×Dic6—124no—
141C3×Dic6—122no—
142C3×Dic6—122no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#489—271noSylow
148C2×H(3,𝔽3)#12411yescharacteristic
149C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
150C2×H(3,𝔽3)#13711yescharacteristic
151(C3⋊D3)⋊C3—81no—
152C2×C3×Q8—91no—
153C2×C3×Q8—181no—
154C2×C3×D4—91no—
155C2×D12—31no—
156C2×D12—91no—
157C2×Dic6—31no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×Dic6—62no—
161C2×Dic6—91no—
162C2×C3×C8—91no—
163C2×C3×C8—91no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4—181no—
169C3×C4⋊C4D3211yescharacteristic
170C3×C42—91no—
171#217—181no—
172#217—181no—
173(C2×D3)⋊C4—181no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180Dic3⋊C4—61no—
181Dic3⋊C4—61no—
182Dic3⋊C4—62no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—61no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190C22×C32—21no—
191C22×C32D3×D411yescharacteristic
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—42no—
202C32×C4—41no—
203C32×C4—41no—
204C32×C4#23511yescharacteristic
205C32×C4#23511yescharacteristic
206C32×C4#23711yescharacteristic
207C32×C4#23711yescharacteristic
208C3⋊Dic3—41no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—122no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—121no—
219C4⋊Q8—91no—
220#124—271no—
221#103—91no—
222#103—271no—
223#102—91no—
224#102—271no—
225#97—271no—
226H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C22×D3—181no—
230C3×Q8—181no—
231C3×Q8—181no—
232C3×Q8—362no—
233C3×D4—181no—
234C3×D4—181no—
235C2×C3×C4—21no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4C2×D3211yescharacteristic
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—181no—
258D12—61no—
259D12—61no—
260D12—181no—
261D12—181no—
262Dic6—61no—
263Dic6—61no—
264Dic6—122no—
265Dic6—122no—
266Dic6—124no—
267Dic6—181no—
268Dic6—181no—
269C3×C8—181no—
270C3×C8—181no—
271C3⋊C8—61no—
272C3⋊C8—61no—
273C3⋊C8—181no—
274C3⋊C8—181no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32#88511yescharacteristic
279C2×C32#88511yescharacteristic
280C2×C32#91111yescharacteristic
281C2×C32#91411yescharacteristic
282C2×C32#91511yescharacteristic
283C2×C32#91811yescharacteristic
284C3⋊D3—81no—
285C3×D3—241no—
286C2×Q8—91no—
287C2×Q8—181no—
288C2×D4—91no—
289C2×C8—91no—
290C2×C8—91no—
291C4⋊C4—91no—
292C4⋊C4—181no—
293C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C22⋊C4—541no—
295C22⋊C4—541no—
296C42—91no—
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—61no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3—361no—
303C22×C3#381111yescharacteristic
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—361no—
308C2×D3—361no—
309C2×D3—361no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—41no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—121no—
318C3×C4—121no—
319C3×C4—122no—
320C3×C4—181no—
321C3×C4—181no—
322C3×C4—361no—
323C3×C4—361no—
324C3×C4#379511yescharacteristic
325C3×C4#379511yescharacteristic
326Dic3—61no—
327Dic3—61no—
328Dic3—61no—
329Dic3—61no—
330Dic3—62no—
331Dic3—62no—
332Dic3—121no—
333Dic3—121no—
334Dic3—122no—
335Dic3—121no—
336Dic3—361no—
337C32—21no—
338C32#476511yescharacteristic, complemented
339C32#479411yescharacteristic, complemented
340C23—181no—
341Q8—181no—
342Q8—181no—
343Q8—362no—
344D4—181no—
345D4—181no—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—91no—
349C2×C4—181no—
350C2×C4—181no—
351C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
352C8—181no—
353C8—181no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—721no—
364C2×C3#6391311yescharacteristic
365C2×C3#6393711yescharacteristic
366C2×C3#6395111yescharacteristic
367D3—241no—
368D3—721no—
369C22—361no—
370C22—361no—
371C22—361no—
372C22#8867711yescharacteristic
373C4—41no—
374C4—181no—
375C4—181no—
376C4—361no—
377C4—361no—
378C4#8864811yescharacteristic
379C4#8864811yescharacteristic
380C3—31no—
381C3—31no—
382C3—61no—
383C3#1059054111yescharacteristic
384C2—721no—
385C2#1172114311yescharacteristic
386C2#1172119311yescharacteristic
387C2#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 radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((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 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721756C2semidirect
4. #63637⋊C3C2semidirect
5. #11720309C2semidirect
7. #11720303C2semidirect
26. #63637D3semidirect
226. H(3,𝔽3)#489semidirect
338. C32#4765semidirect
339. C32#4794semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117211132
#63636⋊C32
#63637⋊C32
#117203092
#117203082
#117203032
#105905353
#105905313