Finite Group Databasebeta

← #13140198

Subgroups — #13140198

Order 1728.

473 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 89 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#11721824C211yesmaximal, characteristic
2#64133⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4C2×#88439C211yesmaximal, characteristic, complemented
5#11720326C211yesmaximal, characteristic, complemented
6#11720309C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10591188—31nomaximal
9#10591187—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×#6229C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13#88552—21no—
14C4⋊#1084C2211yescharacteristic
15#88540—21no—
16#3757⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3—22no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22#88439—22no—
23#64133D311yescharacteristic
24C22×C3⋊(C3⋊C8)D311yescharacteristic
25C3×#924—31no—
26C3×#924—31no—
27C3×#924—31no—
28C3×#924—61no—
29C22×C3×C3⋊C8—31no—
30C2×#3719—31no—
31C2×#3718—31no—
32#63576—31no—
33#63575—31no—
34#63567—31no—
35#63566—31no—
36#63565—31no—
37#63565—31no—
38C23×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×#1084—21no—
45#6229—21no—
46#6229—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3—21no—
52(C3⋊(C3⋊C8))⋊C3D412yes—
53#5116—91no—
54#5051—91no—
55#5051—91no—
56C22×C32×C4—21no—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—31no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64#3757—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C2×C3⋊(C3⋊C8)—22no—
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×#217—62no—
71C3×#217—61no—
72C3×#217—61no—
73C3×#217—61no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—61no—
78C3×Dic3⋊C4—62no—
79C3×Dic3⋊C4—61no—
80C3×Dic3⋊C4—61no—
81C3×Dic3⋊C4—61no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C2×C3×C3⋊C8—62no—
85#3719—62no—
86#3718—62no—
87C22×H(3,𝔽3)—21no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)Q1612yescomplemented
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)C2×D411yescharacteristic
93C4×H(3,𝔽3)#4111yescharacteristic
94#1084—41no—
95#1084—41no—
96(C3⋊Dic3)⋊C3—41no—
97(C3⋊Dic3)⋊C3—41no—
98C22×C3×C8—91no—
99C3×C22⋊Q8—91no—
100C3×C22⋊Q8—91no—
101C2×#216—91no—
102C2×#216—91no—
103#924—31no—
104#924—31no—
105#924—31no—
106#924—61no—
107#924—91no—
108C22×C3⋊C8—31no—
109C22×C3⋊C8—91no—
110C2×Dic12—91no—
111#835—91no—
112#835—91no—
113#818—91no—
114#816—91no—
115#816—91no—
116#810—91no—
117#810—91no—
118#807—91no—
119#807—91no—
120C23×C32—21no—
121C23×C32D1211yescharacteristic
122C23×C32#6711yescharacteristic
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4—41no—
131C2×C32×C4D1211yescharacteristic
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137#634—21no—
138#634—21no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—61no—
152C3×Dic6—61no—
153C3×Dic6—121no—
154C3×Dic6—121no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)#6712yes—
157C3×C3⋊C8—32no—
158C3×C3⋊C8—61no—
159#467—271noSylow
160C2×H(3,𝔽3)—22no—
161C2×H(3,𝔽3)#12111yescharacteristic
162C2×H(3,𝔽3)C2×Q1611yescharacteristic
163C2×H(3,𝔽3)#13511yescharacteristic
164C2×C3×Q8—91no—
165C2×C3×Q8—91no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4—61no—
169C22×C3×C4D3211yescharacteristic
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—31no—
173C2×Dic6—61no—
174C2×Dic6—91no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C2×C3×C8—182no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—181no—
181C3×C4⋊C4—181no—
182C3×C22⋊C4—181no—
183C3×C22⋊C4—181no—
184#217—61no—
185#217—61no—
186#217—62no—
187#217—61no—
188#217—181no—
189#216—182no—
190#216—182no—
191C4⋊Dic3—31no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—61no—
195C4⋊Dic3—91no—
196Dic3⋊C4—61no—
197Dic3⋊C4—61no—
198Dic3⋊C4—62no—
199Dic3⋊C4—61no—
200Dic3⋊C4—181no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—31no—
203C2×C3⋊C8—62no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—182no—
207Dic12—182no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32Dic1212yescomplemented
217C22×C32#21612yescomplemented
218C22×C32C2×D1211yescharacteristic
219C22×C32C2×#6711yescharacteristic
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4#23511yescharacteristic
226C32×C4D3×D411yescharacteristic
227C32×C4#23711yescharacteristic
228C32×C4C2×#6711yescharacteristic
229C3⋊Dic3—41no—
230C3⋊Dic3—41no—
231C3×Dic3—122no—
232C3×Dic3—122no—
233C3×Dic3—121no—
234C3×Dic3—121no—
235C3×Dic3—121no—
236C3×Dic3—121no—
237C3×Dic3—121no—
238C3×Dic3—121no—
239C2×Q16—271no—
240C22×C8—91no—
241C22⋊Q8—91no—
242C22⋊Q8—91no—
243#107—271no—
244#103—271no—
245#103—271no—
246H(3,𝔽3)#46711yescharacteristic, complemented, Sylow
247C23×C3—31no—
248C23×C3—31no—
249C23×C3—61no—
250C23×C3#62611yescharacteristic
251C3×Q8—181no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8—181no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—31no—
260C2×C3×C4—31no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—121no—
268C2×C3×C4—121no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4#62611yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—181no—
284C2×Dic3—181no—
285Dic6—61no—
286Dic6—61no—
287Dic6—61no—
288Dic6—61no—
289Dic6—61no—
290Dic6—61no—
291Dic6—121no—
292Dic6—121no—
293Dic6—181no—
294Dic6—181no—
295C3×C8—92no—
296C3×C8—181no—
297C3⋊C8—32no—
298C3⋊C8—61no—
299C3⋊C8—92no—
300C3⋊C8—181no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—42no—
307C2×C32#90411yescharacteristic
308C2×C32C2×Dic1211yescharacteristic
309C2×C32#93011yescharacteristic
310C2×C32#93911yescharacteristic
311C2×C32C2×#21611yescharacteristic
312C2×C32#95011yescharacteristic
313C2×Q8—91no—
314C2×Q8—91no—
315C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
316Q16—542no—
317C2×C8—91no—
318C2×C8—91no—
319C2×C8—182no—
320C4⋊C4—91no—
321C4⋊C4—91no—
322C4⋊C4—181no—
323C4⋊C4—181no—
324C22⋊C4—181no—
325C22⋊C4—181no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—32no—
331C22×C3—32no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3—61no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3#371912yes—
341C22×C3C2×#62611yescharacteristic
342C3×C4—21no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—121no—
352C3×C4—361no—
353C3×C4—361no—
354C3×C4—361no—
355C3×C4—361no—
356C3×C4#380411yescharacteristic
357C3×C4D3×#6711yescharacteristic
358Dic3—121no—
359Dic3—121no—
360Dic3—121no—
361Dic3—121no—
362Dic3—122no—
363Dic3—122no—
364Dic3—121no—
365Dic3—121no—
366Dic3—361no—
367Dic3—361no—
368C32—21no—
369C32#505111yescharacteristic, complemented
370C32#511611yescharacteristic, complemented
371C23#620011yescharacteristic
372Q8—181no—
373Q8—181no—
374Q8—181no—
375Q8—181no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4#620011yescharacteristic
383C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C8—92no—
385C8—181no—
386C2×C3—22no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—31no—
391C2×C3—31no—
392C2×C3—31no—
393C2×C3—62no—
394C2×C3—62no—
395C2×C3—61no—
396C2×C3—61no—
397C2×C3—61no—
398C2×C3—122no—
399C2×C3#6382911yescharacteristic
400C2×C3C2×#371911yescharacteristic
401C2×C3#6397811yescharacteristic
402C22—21no—
403C22—21no—
404C22#8843912yescomplemented
405C22C2×#620011yescharacteristic
406C4—21no—
407C4—361no—
408C4—361no—
409C4—361no—
410C4—361no—
411C4#8867111yescharacteristic
412C4#8867611yescharacteristic
413C3—31no—
414C3—31no—
415C3—61no—
416C3#1059118711yescharacteristic
417C2—22no—
418C2#1172107311yescharacteristic
419C2C2×#8843911yescharacteristic
420C2#1172132211yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#467
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88439C2semidirect
5. #11720326C2semidirect
6. #11720309C2semidirect
7. #11720306C2semidirect
89. C22×H(3,𝔽3)Q16semidirect
216. C22×C32Dic12semidirect
217. C22×C32#216semidirect
246. H(3,𝔽3)#467semidirect
369. C32#5051semidirect
370. C32#5116semidirect
404. C22#88439semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64133⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884392
#117203262
#117203092
#117203062
#105911883
#105911873