Finite Group Databasebeta

← #13164507

Subgroups — #13164507

Order 1728.

688 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 64 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
12−1+4⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11722393C211yesmaximal, characteristic, complemented
3#11722392C211yesmaximal, characteristic, complemented
4#11722388C212yesmaximal, complemented
5C2×#88610C212yesmaximal, complemented
6C3×#5927—33nomaximal
7C3×#5838—31nomaximal
8#88744—41nomaximal
9#41⋊H(3,𝔽3)C2211yescharacteristic
10C2×Q8⋊H(3,𝔽3)C2212yes—
11#88611C2211yescharacteristic
12#88610C2211yescharacteristic
13#88610C2212yes—
14C32×2−1+4D311yescharacteristic, complemented
15C3×#1010—31no—
16C3×2−1+4⋊C3D313yescomplemented
17C3×#985—33no—
18C3×#984—33no—
19C3×#983—36no—
20C2×C3×#226—36no—
21C3×#951—32no—
22C3×#951—31no—
23C3×#950—32no—
24C3×#950—31no—
25C3×#948—31no—
26C2×C3×#216—31no—
27C2×C3×#216—32no—
28C3×#942—31no—
29C3×#942—32no—
30D4×H(3,𝔽3)—41no—
31#6235—42no—
32C2×#1084—42no—
33C4×#272—41no—
34#6229—41no—
35Q8⋊H(3,𝔽3)C2311yescharacteristic
36#5927—33no—
37C3×#578—91no—
38#5838—31no—
39C32×#41—31no—
40C32×#41—32no—
41C32×#41C2×D311yescharacteristic
42C2×C32×Q8—31no—
43C2×C32×Q8C2×D312yes—
44C3×D3×Q8—61no—
45C3×#237—62no—
46C3×#237—61no—
47C3×#237—123no—
48C3×#235—32no—
49C3×#235—31no—
50C2×C3×Dic6—32no—
51C2×C3×Dic6—31no—
52C3×#41⋊C3C2×D313yes—
53C2×C3×SL(2,𝔽3)C2×D316yes—
54C3×GL(2,𝔽3)—33no—
55C3×#226—33no—
56C3×#226—36no—
57C3×#216—32no—
58C3×#216—32no—
59C3×#216—32no—
60C3×#216—32no—
61C3×#216—31no—
62C3×#215—32no—
63C3×#215—31no—
64C3×#214—32no—
65C3×#214—31no—
66C3×#213—31no—
67C3×#208—32no—
68C3×#208—31no—
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—31no—
71C22×H(3,𝔽3)—42no—
72C2×#272—41no—
73C4×H(3,𝔽3)—41no—
74#1084—42no—
75#1084—41no—
76C3×2−1+4—31no—
77C3×2−1+4—91no—
78C3×2−1+4C3⋊D311yescharacteristic
79#1010—31no—
802−1+4⋊C3—33no—
81#985—33no—
82#984—33no—
83#983—36no—
84C2×#226—36no—
85C3×#137—91no—
86C3×#137—92no—
87C3×#137—91no—
88C3×#137—92no—
89C3×#135—92no—
90C3×#135—91no—
91C2×C3×Q16—91no—
92C2×C3×Q16—92no—
93C3×#131—91no—
94#951—32no—
95#951—31no—
96#950—32no—
97#950—31no—
98#948—31no—
99C2×#216—31no—
100C2×#216—32no—
101#942—31no—
102#942—32no—
103C32×Q8—32no—
104C32×Q8—31no—
105C32×Q8C22×D311yescharacteristic
106C32×D4—31no—
107C32×D4—32no—
108C32×D4—43no—
109C32×D4S411yescharacteristic
110C2×C32×C4—32no—
111C2×C32×C4—31no—
112C2×C32×C4—61no—
113C3×#67—61no—
114C3×#67—62no—
115C3×#67—126no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—126no—
119C3×D12—31no—
120C3×C4×D3—62no—
121C3×C4×D3—61no—
122C3×C4×D3—123no—
123C3×Dic6—31no—
124C3×Dic6—32no—
125C3×Dic6—62no—
126C3×Dic6—61no—
127C3×Dic6—123no—
128C3×SL(2,𝔽3)C22×D313yes—
129C3×C3⋊C8—31no—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—31no—
132#578—91noSylow
133C2×H(3,𝔽3)—41no—
134C2×H(3,𝔽3)—82no—
135#272—81no—
136C3×#41—31no—
137C3×#41—31no—
138C3×#41—32no—
139C3×#41—32no—
140C3×#41—31no—
141C3×#41—61no—
142C3×#41—62no—
143C3×#41—92no—
144C3×#41—91no—
145C3×#41—182no—
146C3×#41—181no—
147C3×#41C2×C3⋊D311yescharacteristic
148C2×C3×Q8—31no—
149C2×C3×Q8—31no—
150C2×C3×Q8—32no—
151C2×C3×Q8—61no—
152C2×C3×Q8—92no—
153C2×C3×Q8—91no—
154C2×C3×Q8—181no—
155C2×C3×Q8C2×C3⋊D312yes—
156D3×Q8—61no—
157#237—62no—
158#237—61no—
159#237—123no—
160#235—32no—
161#235—31no—
162C2×Dic6—32no—
163C2×Dic6—31no—
164#41⋊C3—33no—
165C2×SL(2,𝔽3)—36no—
166GL(2,𝔽3)—33no—
167#226—33no—
168#226—36no—
169C3×Q16—92no—
170C3×Q16—92no—
171C3×Q16—92no—
172C3×Q16—91no—
173C3×Q16—92no—
174C3×SD16—92no—
175C3×SD16—91no—
176C3×SD16—92no—
177C3×SD16—91no—
178C3×D8—91no—
179C3×M16—92no—
180C3×M16—91no—
181C2×C3×C8—92no—
182C2×C3×C8—91no—
183#216—32no—
184#216—32no—
185#216—32no—
186#216—32no—
187#216—31no—
188#215—32no—
189#215—31no—
190#214—32no—
191#214—31no—
192#213—31no—
193#208—32no—
194#208—31no—
195C2×C3⋊C8—32no—
196C2×C3⋊C8—31no—
197C22×C32—31no—
198C22×C32—46no—
199C22×C32C2×S412yes—
200C2×C3×D3—61no—
201C2×C3×D3—123no—
202C32×C4—31no—
203C32×C4—32no—
204C32×C4—43no—
205C32×C4C2×S411yescharacteristic
206C3×Dic3—62no—
207C3×Dic3—61no—
208C3×Dic3—126no—
209C3×Dic3—123no—
2102−1+4—91no—
2112−1+4#27211yescharacteristic, complemented
212#137—91no—
213#137—92no—
214#137—91no—
215#137—92no—
216#135—92no—
217#135—91no—
218C2×Q16—91no—
219C2×Q16—92no—
220#131—91no—
221H(3,𝔽3)—41noSylow
222C3×Q8—32no—
223C3×Q8—31no—
224C3×Q8—31no—
225C3×Q8—32no—
226C3×Q8—31no—
227C3×Q8—62no—
228C3×Q8—61no—
229C3×Q8—92no—
230C3×Q8—91no—
231C3×Q8—182no—
232C3×Q8—181no—
233C3×Q8C22×C3⋊D311yescharacteristic
234C3×D4—31no—
235C3×D4—32no—
236C3×D4—32no—
237C3×D4—31no—
238C3×D4—31no—
239C3×D4—61no—
240C3×D4—62no—
241C3×D4—91no—
242C3×D4—123no—
243C3×D4—181no—
244C3×D4—182no—
245C3×D4C3⋊S411yescharacteristic
246C2×C3×C4—32no—
247C2×C3×C4—31no—
248C2×C3×C4—32no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4—62no—
254C2×C3×C4—61no—
255C2×C3×C4—182no—
256C2×C3×C4—181no—
257C2×C3×C4—182no—
258C2×C3×C4—181no—
259#67—61no—
260#67—62no—
261#67—126no—
262C2×Dic3—61no—
263C2×Dic3—62no—
264C2×Dic3—126no—
265D12—31no—
266C4×D3—62no—
267C4×D3—61no—
268C4×D3—123no—
269Dic6—32no—
270Dic6—31no—
271Dic6—62no—
272Dic6—61no—
273Dic6—123no—
274SL(2,𝔽3)—33no—
275C3×C8—91no—
276C3×C8—92no—
277C3×C8—91no—
278C3⋊C8—31no—
279C3⋊C8—32no—
280C3⋊C8—31no—
281C2×C32—22no—
282C2×C32—43no—
283C2×C32—61no—
284C2×C32—86no—
285C2×C32C22×S411yescharacteristic
286C3×D3—121no—
287C3×D3—243no—
288#41—31no—
289#41—32no—
290#41—92no—
291#41—91no—
292#41—182no—
293#41—181no—
294#41C2×#27211yescharacteristic
295C2×Q8—31no—
296C2×Q8—92no—
297C2×Q8—91no—
298C2×Q8—181no—
299C2×Q8C2×#27212yes—
300Q16—92no—
301Q16—92no—
302Q16—92no—
303Q16—91no—
304Q16—92no—
305SD16—92no—
306SD16—91no—
307SD16—92no—
308SD16—91no—
309D8—91no—
310M16—92no—
311M16—91no—
312C2×C8—92no—
313C2×C8—91no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—32no—
317C22×C3—61no—
318C22×C3—126no—
319C22×C3—181no—
320C22×C3C2×C3⋊S412yes—
321C2×D3—61no—
322C2×D3—123no—
323C3×C4—31no—
324C3×C4—32no—
325C3×C4—31no—
326C3×C4—32no—
327C3×C4—31no—
328C3×C4—61no—
329C3×C4—62no—
330C3×C4—123no—
331C3×C4—182no—
332C3×C4—181no—
333C3×C4C2×C3⋊S411yescharacteristic
334Dic3—62no—
335Dic3—61no—
336Dic3—126no—
337Dic3—123no—
338C32—43no—
339C32#592711yescharacteristic, complemented
340Q8—32no—
341Q8—31no—
342Q8—92no—
343Q8—91no—
344Q8—182no—
345Q8—181no—
346Q8C22×#27211yescharacteristic
347D4—31no—
348D4—32no—
349D4—91no—
350D4—181no—
351D4—182no—
352D4#625811yescharacteristic
353C2×C4—32no—
354C2×C4—31no—
355C2×C4—61no—
356C2×C4—182no—
357C2×C4—181no—
358C2×C4—182no—
359C2×C4—181no—
360C8—91no—
361C8—92no—
362C8—91no—
363C2×C3—22no—
364C2×C3—31no—
365C2×C3—62no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—121no—
369C2×C3—123no—
370C2×C3—246no—
371C2×C3—361no—
372C2×C3C22×C3⋊S411yescharacteristic
373D3—121no—
374D3—243no—
375C22—31no—
376C22—181no—
377C22C2×#625812yes—
378C4—31no—
379C4—32no—
380C4—182no—
381C4—181no—
382C4C2×#625811yescharacteristic
383C3—31no—
384C3—123no—
385C3#1059659011yescharacteristic
386C2—22no—
387C2—61no—
388C2—361no—
389C2C22×#625811yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C22×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FittingC32×2−1+4D3
hypercenterC3×D4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. 2−1+4⋊H(3,𝔽3)C2semidirect
2. #11722393C2semidirect
3. #11722392C2semidirect
4. #11722388C2semidirect
5. C2×#88610C2semidirect
14. C32×2−1+4D3semidirect
16. C3×2−1+4⋊C3D3semidirect
211. 2−1+4#272semidirect
339. C32#5927semidirect

Maximal subgroups

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

subgroupindex
2−1+4⋊H(3,𝔽3)2
#117223932
#117223922
#117223882
C2×#886102
C3×#59273
C3×#58383
#887444