Finite Group Databasebeta

← #13137445

Subgroups — #13137445

Order 1728.

703 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 79 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#11722491C211yesmaximal, characteristic
2C42×#272C211yesmaximal, characteristic, complemented
3#11721065C211yesmaximal, characteristic, complemented
4#63632⋊C3C212yesmaximal, complemented
5#11720305C212yesmaximal, complemented
6D3×#805—32nomaximal
7C2×C4×#272—21no—
8C2×C4×#272C2211yescharacteristic
9#3828⋊C3C2212yes—
10#88677—21no—
11#88672—21no—
12#88655C411yescharacteristic, complemented
13C4×((C3⋊D3)⋊C3)⋊C2—21no—
14#88651C411yescharacteristic, complemented
15#88650—21no—
16#88649C412yescomplemented
17C42×H(3,𝔽3)C2211yescharacteristic, complemented
18C4×#1084—21no—
19C4×#1084C2211yescharacteristic, complemented
20#3748⋊C3C2212yescomplemented
21#88435—21no—
22#88434—21no—
23D3×#235—32no—
24C3×C42×D3—32no—
25C3×C42×D3—61no—
26D3×#208—32no—
27#63632D312yescomplemented
28C3×#805—32no—
29#63570—32no—
30#63569—32no—
31C22×#272D411yescharacteristic
32C2×((C3⋊D3)⋊C3)⋊C2—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×#1084—21no—
36C2×#1084D411yescharacteristic
37C4×#272—24no—
38C4×#272C2×C411yescharacteristic
39C4×#272C2×C411yescharacteristic
40C4×#272D411yescharacteristic
41C4×#272D411yescharacteristic
42#638⋊C3—22no—
43C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
44C4×(C3⋊D3)⋊C3—22no—
45#634⋊C3C2×C412yescomplemented
46#6200—22no—
47#6199—21no—
48#6198—21no—
49((C3⋊D3)⋊C3)⋊C4—22no—
50#6196—21no—
51(C3⋊(C3⋊C8))⋊C3—22no—
52C2×#805—91no—
53D3×#104—92no—
54#3828C2×D312yes—
55C3×#235—32no—
56C2×C3×C4×D3—32no—
57C2×C3×C4×D3—62no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—62no—
60D3×#67—62no—
61#3804—62no—
62D3×D12—32no—
63C4×D32—62no—
64#3799—32no—
65#3798—62no—
66#3795—32no—
67D3×Dic6—32no—
68C32×C42—21no—
69C32×C42C2×D312yescomplemented
70#3748C2×D312yescomplemented
71C3×C4×Dic3—32no—
72C3×C4×Dic3—62no—
73C3×C4×Dic3—61no—
74C3×C4×Dic3—62no—
75C3×#208—32no—
76#3711—62no—
77D3×C3⋊C8—62no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×#272—22no—
80C2×#272C22⋊C411yescharacteristic
81C2×#272C22⋊C411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83((C3⋊D3)⋊C3)⋊C2—42no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)C2×D411yescharacteristic
87#1084—22no—
88#1084C22⋊C411yescharacteristic
89#1084C22⋊C411yescharacteristic
90(C3⋊Dic3)⋊C3—22no—
91D3×#41—92no—
92C2×#235—91no—
93C2×C3×C42—91no—
94C2×#208—91no—
95D3×M16—92no—
96C42×D3—32no—
97C42×D3—61no—
98C3×#104—92no—
99#837—92no—
100#825—92no—
101#805—32no—
102#805—92no—
103#805—92no—
104C22×C3×D3—32no—
105C22×C3×D3—61no—
106C2×D32—62no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D312yes—
111#638—22no—
112C4⋊(C3⋊D3)C4×D312yescomplemented
113C4×C3⋊D3—22no—
114#634C4×D312yescomplemented
115C3×#67—62no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C3×D12—32no—
121C3×C4×D3—32no—
122C3×C4×D3—32no—
123C3×C4×D3—32no—
124C3×C4×D3—32no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—68no—
130C3×C4×D3—68no—
131C3×C4×D3—62no—
132C3×Dic6—32no—
133#627—62no—
134#626—62no—
135#626—62no—
136#625—62no—
137#624—62no—
138D3×Dic3—62no—
139D3×Dic3—62no—
140C3⋊(C3⋊C8)—22no—
141C3×C3⋊C8—62no—
142C2×#104—271noSylow
143C2×H(3,𝔽3)—21no—
144C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
145#272—21no—
146#272#10412yescomplemented
147(C3⋊D3)⋊C3—42no—
148C3×#41—92no—
149C22×C3×C4—91no—
150C22×C3×C4—181no—
151C2×#67—181no—
152#239—92no—
153D3×Q8—92no—
154#237—92no—
155#237—182no—
156D3×D4—92no—
157D3×D4—182no—
158#235—32no—
159#235—92no—
160#235—92no—
161#235—182no—
162#235—181no—
163#235—181no—
164C2×D12—91no—
165C2×C4×D3—32no—
166C2×C4×D3—61no—
167C2×C4×D3—62no—
168C2×C4×D3—62no—
169C2×C4×D3—182no—
170C2×C4×D3—181no—
171C2×Dic6—91no—
172C3×M16—92no—
173C3×C42—32no—
174C3×C42—61no—
175C3×C42—91no—
176C3×C42—181no—
177C3×C42D3211yescharacteristic
178C4×Dic3—32no—
179C4×Dic3—61no—
180C4×Dic3—62no—
181C4×Dic3—62no—
182#208—32no—
183#208—92no—
184#208—181no—
185C2×C3⋊C8—181no—
186ΓL(1,𝔽25)—182no—
187D3×C8—182no—
188C22×C32—21no—
189C22×C32D3×D412yes—
190C2×C3⋊D3—22no—
191C2×C3×D3—32no—
192C2×C3×D3—32no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—62no—
198D32—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4C2×C4×D312yes—
204C32×C4D3×D412yes—
205C3⋊Dic3—22no—
206C3×Dic3—32no—
207C3×Dic3—32no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—64no—
211C3×Dic3—64no—
212C3×Dic3—62no—
213C2×#41—271no—
214C2×M16—271no—
215C2×C42—91no—
216#104—92no—
217#104—272no—
218H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—32no—
221C22×D3—61no—
222C22×D3—182no—
223C22×D3—181no—
224C3×Q8—92no—
225C3×D4—92no—
226C3×D4—182no—
227C2×C3×C4—21no—
228C2×C3×C4—32no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—61no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—91no—
237C2×C3×C4—184no—
238C2×C3×C4—182no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3211yescharacteristic
241#67—62no—
242#67—182no—
243#67—182no—
244#67—182no—
245#67—182no—
246C2×Dic3—32no—
247C2×Dic3—61no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3—182no—
251C2×Dic3—181no—
252D12—32no—
253D12—92no—
254D12—181no—
255D12—182no—
256C4×D3—32no—
257C4×D3—32no—
258C4×D3—32no—
259C4×D3—32no—
260C4×D3—61no—
261C4×D3—61no—
262C4×D3—61no—
263C4×D3—61no—
264C4×D3—68no—
265C4×D3—68no—
266C4×D3—62no—
267C4×D3—182no—
268C4×D3—182no—
269C4×D3—182no—
270C4×D3—182no—
271Dic6—32no—
272Dic6—92no—
273Dic6—182no—
274Dic6—181no—
275C3×C8—182no—
276C3⋊C8—62no—
277C3⋊C8—182no—
278C2×C32—22no—
279C2×C32—22no—
280C2×C32—21no—
281C2×C32D3×C22⋊C412yes—
282C3⋊D3—42no—
283C3×D3—34no—
284C3×D3—62no—
285C3×D3—62no—
286C3×D3—62no—
287C3×D3—122no—
288#41—92no—
289#41—272no—
290#41—541no—
291#41—541no—
292C2×Q8—271no—
293C2×D4—271no—
294C2×D4—541no—
295C22×C4—91no—
296C22×C4—181no—
297C22×C4—541no—
298M16—92no—
299M16—541no—
300C2×C8—541no—
301C42—91no—
302C42—181no—
303C42((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
304C22×C3—32no—
305C22×C3—61no—
306C22×C3—91no—
307C22×C3—91no—
308C22×C3—182no—
309C22×C3—182no—
310C22×C3#381111yescharacteristic
311C2×D3—32no—
312C2×D3—32no—
313C2×D3—61no—
314C2×D3—61no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—62no—
318C2×D3—182no—
319C2×D3—364no—
320C2×D3—362no—
321C3×C4—22no—
322C3×C4—32no—
323C3×C4—32no—
324C3×C4—64no—
325C3×C4—64no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—91no—
329C3×C4—91no—
330C3×C4—182no—
331C3×C4—182no—
332C3×C4C4×D3211yescharacteristic
333C3×C4#380211yescharacteristic
334Dic3—32no—
335Dic3—32no—
336Dic3—61no—
337Dic3—61no—
338Dic3—64no—
339Dic3—64no—
340Dic3—62no—
341Dic3—182no—
342C32—21no—
343C32D3×#10412yescomplemented
344C23—91no—
345C23—541no—
346Q8—92no—
347Q8—541no—
348D4—92no—
349D4—182no—
350D4—541no—
351D4—542no—
352C2×C4—21no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4—91no—
356C2×C4—91no—
357C2×C4—91no—
358C2×C4—184no—
359C2×C4—181no—
360C2×C4—182no—
361C2×C4—542no—
362C2×C4—541no—
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C8—182no—
365C2×C3—21no—
366C2×C3—32no—
367C2×C3—62no—
368C2×C3—62no—
369C2×C3—61no—
370C2×C3—92no—
371C2×C3—181no—
372C2×C3—362no—
373C2×C3#6392111yescharacteristic
374D3—34no—
375D3—62no—
376D3—62no—
377D3—62no—
378D3—122no—
379D3—362no—
380C22—91no—
381C22—91no—
382C22—182no—
383C22—182no—
384C22—1082no—
385C22#8867711yescharacteristic
386C4—22no—
387C4—91no—
388C4—91no—
389C4—182no—
390C4—182no—
391C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
392C4#8865511yescharacteristic
393C3—32no—
394C3—61no—
395C3#1059039311yescharacteristic
396C2—21no—
397C2—92no—
398C2—181no—
399C2—362no—
400C2#1172116111yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC2×C3#63921
prime radical#11722491C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×#272C2semidirect
3. #11721065C2semidirect
4. #63632⋊C3C2semidirect
5. #11720305C2semidirect
12. #88655C4semidirect
14. #88651C4semidirect
16. #88649C4semidirect
17. C42×H(3,𝔽3)C22semidirect
19. C4×#1084C22semidirect
20. #3748⋊C3C22semidirect
27. #63632D3semidirect
43. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
45. #634⋊C3C2×C4semidirect
69. C32×C42C2×D3semidirect
70. #3748C2×D3semidirect
112. C4⋊(C3⋊D3)C4×D3semidirect
114. #634C4×D3semidirect
146. #272#104semidirect
218. H(3,𝔽3)C2×#104semidirect
303. C42((C3⋊D3)⋊C3)⋊C2semidirect
343. C32D3×#104semidirect

Maximal subgroups

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

subgroupindex
#117224912
C42×#2722
#117210652
#63632⋊C32
#117203052
D3×#8053