Finite Group Databasebeta

← #13140540

Subgroups — #13140540

Order 1728.

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

normal subgroups only

This group has 400 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#64145⋊C3C211yesmaximal, characteristic
2#11721116C211yesmaximal, characteristic, complemented
3#11720578C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#11720313C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591496—31nomaximal
9#10591488—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#6196—21no—
13C4⋊#1084C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3757⋊C3—21no—
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88448—21no—
21#64145D311yescharacteristic
22C3×#936—31no—
23#63919—31no—
24#63918—31no—
25#63659D311yescharacteristic
26C3×#832—31no—
27C3×#832—31no—
28C3×#832—31no—
29C3×#832—61no—
30#63567—31no—
31#63567—31no—
32#63566—31no—
33#63565—31no—
34#63555—31no—
35#63554—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#634⋊C3—21no—
46#634⋊C3—21no—
47#6196—42no—
48#6180—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50#5055—91no—
51#4765—91no—
52#4765—91no—
53C2×C32×D4—21no—
54C2×C32×D4C2×D311yescharacteristic
55C2×C32×D4C2×D311yescharacteristic
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—31no—
58C2×#627—61no—
59C2×#627—61no—
60#3757—21no—
61#3757—21no—
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×#217—61no—
66C3×#217—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)—41no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)#4111yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—21no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×#124—91no—
90#936—31no—
91#936—91no—
92#888—91no—
93#888—91no—
94#869—91no—
95C3×#102—91no—
96C3×#102—91no—
97#835—91no—
98#835—91no—
99#832—31no—
100#832—31no—
101#832—31no—
102#832—61no—
103#832—91no—
104#816—91no—
105#816—91no—
106#815—91no—
107#815—91no—
108#810—91no—
109#810—91no—
110#803—91no—
111C23×C32—21no—
112C23×C32—21no—
113C23×C32—41no—
114C32×D4—22no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—21no—
118C32×D4—21no—
119C32×D4—41no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127#634—21no—
128#634—21no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—121no—
136C3×Dic6—61no—
137C3×Dic6—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)—81no—
147C2×H(3,𝔽3)#12411yescharacteristic
148C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
149C2×H(3,𝔽3)#13711yescharacteristic
150C2×C3×Q8—91no—
151C2×C3×D4—31no—
152C2×C3×D4—31no—
153C2×C3×D4—61no—
154C2×C3×D4D3211yescharacteristic
155C2×Dic6—31no—
156C2×Dic6—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#217—61no—
168#217—61no—
169#217—181no—
170#217—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×C32—42no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—81no—
195C22×C32—81no—
196C22×C32D3×D411yescharacteristic
197C22×C32C2×#6711yescharacteristic
198C32×C4—21no—
199C32×C4—21no—
200C32×C4#23511yescharacteristic
201C32×C4#23711yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4#23711yescharacteristic
204C3⋊Dic3—21no—
205C3⋊Dic3—21no—
206C3⋊Dic3—41no—
207C3⋊Dic3—41no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—121no—
211C3×Dic3—121no—
212C3×Dic3—121no—
213C3×Dic3—121no—
214C3×Dic3—241no—
215C4⋊Q8—271no—
216#124—91no—
217#103—271no—
218#103—271no—
219#102—91no—
220#102—91no—
221#97—271no—
222H(3,𝔽3)#48911yescharacteristic, complemented, Sylow
223C23×C3—21no—
224C23×C3—61no—
225C23×C3—61no—
226C23×C3—121no—
227C3×Q8—181no—
228C3×Q8—181no—
229C3×D4—21no—
230C3×D4—21no—
231C3×D4—61no—
232C3×D4—61no—
233C3×D4—61no—
234C3×D4—61no—
235C3×D4—62no—
236C3×D4—121no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—91no—
242C2×C3×C4—181no—
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—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—121no—
255C2×Dic3—181no—
256C2×Dic3—181no—
257Dic6—61no—
258Dic6—61no—
259Dic6—181no—
260Dic6—181no—
261Dic6—362no—
262Dic6—362no—
263Dic6—362no—
264C3×C8—181no—
265C3×C8—181no—
266C3⋊C8—61no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C3⋊C8—121no—
270C3⋊C8—181no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—82no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32#88511yescharacteristic
278C2×C32#91111yescharacteristic
279C2×C32#91511yescharacteristic
280C2×C32#93611yescharacteristic
281C2×C32#94911yescharacteristic
282C2×C32#95111yescharacteristic
283C2×Q8—91no—
284C2×Q8—541no—
285C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C2×C8—91no—
287C2×C8—91no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—541no—
291C22⋊C4—181no—
292C22⋊C4—181no—
293C42—91no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—61no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—122no—
307C22×C3—241no—
308C22×C3—241no—
309C22×C3D3×#6711yescharacteristic
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—61no—
315C3×C4—61no—
316C3×C4—181no—
317C3×C4—181no—
318C3×C4—361no—
319C3×C4—361no—
320C3×C4—361no—
321C3×C4#380411yescharacteristic
322C3×C4#380511yescharacteristic
323Dic3—61no—
324Dic3—61no—
325Dic3—121no—
326Dic3—121no—
327Dic3—121no—
328Dic3—121no—
329Dic3—181no—
330Dic3—181no—
331Dic3—241no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#476511yescharacteristic, complemented
336C32#505511yescharacteristic, complemented
337C23—21no—
338Q8—181no—
339Q8—181no—
340Q8—1082no—
341D4—21no—
342D4—21no—
343C2×C4—91no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
349C8—181no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—81no—
361C2×C3—241no—
362C2×C3—241no—
363C2×C3—242no—
364C2×C3#6392511yescharacteristic
365C2×C3#6392911yescharacteristic
366C2×C3#6395911yescharacteristic
367C22—41no—
368C22—41no—
369C22—41no—
370C22#8867611yescharacteristic
371C4—181no—
372C4—181no—
373C4—361no—
374C4—361no—
375C4—361no—
376C4#8867111yescharacteristic
377C4#8867211yescharacteristic
378C3—31no—
379C3—31no—
380C3—61no—
381C3#1059148811yescharacteristic
382C2—81no—
383C2#1172119511yescharacteristic
384C2#1172119911yescharacteristic
385C2#1172130411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting, prime radicalC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((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×D4

Complements

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

NH ≅ G/Ntype
2. #11721116C2semidirect
5. #11720313C2semidirect
6. #11720312C2semidirect
7. #11720309C2semidirect
222. H(3,𝔽3)#489semidirect
335. C32#4765semidirect
336. C32#5055semidirect

Maximal subgroups

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

subgroupindex
#64145⋊C32
#117211162
#117205782
#63659⋊C32
#117203132
#117203122
#117203092
#105914963
#105914883