Finite Group Databasebeta

← #13167494

Subgroups — #13167494

Order 1728.

570 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#88941C212yesmaximal, complemented
2#11723697C211yesmaximal, characteristic, complemented
3#11723685C211yesmaximal, characteristic, complemented
4C3×#63779C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593426—31nomaximal
7C2×C3×C4⋊(C3⋊D3)—21no—
8C2×#6295—21no—
9C2×#6295C2212yes—
10C2×#6295C2211yescharacteristic
11C2×#6294—21no—
12C2×C3×#622C2212yes—
13#88941—24no—
14C3×#3752—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16#88815C2211yescharacteristic
17C2×#3774—32no—
18#63914—31no—
19#63911—61no—
20#63902—31no—
21#63880—61no—
22#63779D311yescharacteristic, complemented
23#63739—32no—
24C22×C3×C3⋊D3—21no—
25C22×C3×C3⋊D3—22no—
26C2×C33×C4D411yescharacteristic
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4⋊(C3⋊D3)—42no—
31#6295—22no—
32#6295—21no—
33#6295—22no—
34#6295—42no—
35#6295D412yescomplemented
36#6294—42no—
37C3×#622—22no—
38#4843—91nomaximal
39C2×C4⋊(C3⋊D3)—21no—
40C2×C3×D12—61no—
41C2×C3×D12—61no—
42C2×C3×D12—121no—
43C2×#626—61no—
44C2×#626—62no—
45C2×#626—62no—
46C2×#625—31no—
47C2×#625—32no—
48C2×#625—61no—
49C2×#624—61no—
50C2×D3×Dic3—61no—
51C2×D3×Dic3—61no—
52C2×#622C2×D312yes—
53#3774—64no—
54#3752—21no—
55C4×C3⋊Dic3C2×D311yescharacteristic
56C3×(C2×D3)⋊C4—61no—
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—121no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61#3724—31no—
62Dic3⋊Dic3—61no—
63C22×C33C2×D411yescharacteristic
64C2×C3×C3⋊D3—22no—
65C2×C3×C3⋊D3—41no—
66C2×C3×C3⋊D3—41no—
67C2×C3×C3⋊D3—42no—
68C2×C3×C3⋊D3—42no—
69C2×C3×C3⋊D3—42no—
70C33×C4—21no—
71C3×C3⋊Dic3—41no—
72C3×C3⋊Dic3D812yescomplemented
73C3×C3⋊Dic3C2×D411yescharacteristic
74C3×C3⋊Dic3#4111yescharacteristic
75#940—181no—
76C2×D24—92no—
77#67⋊C4—181no—
78#874—91no—
79C4×D12—91no—
80C3×C4⋊C8—91no—
81#821—92no—
82C22×C3⋊D3—21no—
83C22×C3⋊D3—22no—
84C22×C3×D3—61no—
85C22×C3×D3—61no—
86C22×C3×D3—62no—
87C22×C3×D3—62no—
88C22×C3×D3—121no—
89C22×C3×D3—122no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4D1211yescharacteristic
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3D1211yescharacteristic
97C2×C3⋊Dic3C22×D311yescharacteristic
98C4⋊(C3⋊D3)—42no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—121no—
106C3×D12—122no—
107C3×D12—122no—
108C3×D12—124no—
109#626—62no—
110#626—62no—
111#626—122no—
112#626—122no—
113#626—124no—
114#625—32no—
115#625—61no—
116#625—62no—
117#625—62no—
118#625—122no—
119#624—122no—
120D3×Dic3—122no—
121D3×Dic3—122no—
122#622—22no—
123#459—271noSylow
124C2×C33#12111yescharacteristic
125C2×C33C2×D811yescharacteristic
126C2×C33AGL(1,ℤ/8)11yescharacteristic
127C3×C3⋊D3—42no—
128C3×C3⋊D3—82no—
129C2×C3×D4—181no—
130C2×#67—181no—
131C2×#67—182no—
132C2×#67—182no—
133C22×Dic3—181no—
134C2×D12—61no—
135C2×D12—61no—
136C2×D12—91no—
137C2×D12—92no—
138C2×D12—121no—
139C2×D12—181no—
140C2×C4×D3—181no—
141C2×C4×D3—181no—
142C2×C3×C8—92no—
143C3×C22⋊C4—181no—
144C3×C42—91no—
145(C2×D3)⋊C4—61no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—121no—
148C4⋊Dic3—91no—
149Dic3⋊C4—181no—
150C4×Dic3—61no—
151C4×Dic3—61no—
152D24—184no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32C2×D1211yescharacteristic
158C2×C3⋊D3—22no—
159C2×C3⋊D3—42no—
160C2×C3⋊D3—41no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—42no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—122no—
167C2×C3×D3—121no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—124no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C2×C3×D3—241no—
179C2×C3×D3—241no—
180C2×C3×D3—242no—
181C2×C3×D3—242no—
182C32×C4—22no—
183C32×C4—21no—
184C32×C4—41no—
185C32×C4—42no—
186C32×C4—81no—
187C3⋊Dic3—41no—
188C3⋊Dic3D2412yescomplemented
189C3⋊Dic3D3×D411yescharacteristic
190C3⋊Dic3#23911yescharacteristic
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—121no—
198C3×Dic3—121no—
199C3×Dic3—241no—
200C2×D8—272no—
201#127—271no—
202C4×D4—271no—
203C4⋊C8—91no—
204#102—272no—
205C33#45911yescharacteristic, complemented, Sylow
206C23×C3—181no—
207C23×C3—182no—
208C22×D3—61no—
209C22×D3—61no—
210C22×D3—62no—
211C22×D3—62no—
212C22×D3—121no—
213C22×D3—122no—
214C3×D4—362no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—181no—
222C2×C3×C4GO+(4,𝔽2)11yescharacteristic
223#67—182no—
224#67—182no—
225#67—362no—
226#67—362no—
227#67—364no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—121no—
235C2×Dic3—362no—
236D12—92no—
237D12—122no—
238D12—122no—
239D12—124no—
240D12—181no—
241D12—182no—
242D12—182no—
243D12—362no—
244C4×D3—362no—
245C4×D3—362no—
246C3×C8—182no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—41no—
259C2×C32#89511yescharacteristic
260C2×C32C2×D2411yescharacteristic
261C2×C32#90811yescharacteristic
262C3⋊D3—42no—
263C3⋊D3—82no—
264C3×D3—122no—
265C3×D3—122no—
266C3×D3—242no—
267C3×D3—242no—
268C3×D3—244no—
269C3×D3—242no—
270C2×D4—181no—
271C2×D4—271no—
272C2×D4—272no—
273C2×D4—541no—
274C22×C4—541no—
275D8—544no—
276C2×C8—92no—
277C4⋊C4—271no—
278C22⋊C4—181no—
279C42—91no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—182no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3—362no—
288C22×C3—362no—
289C22×C3—362no—
290C22×C3C2×GO+(4,𝔽2)11yescharacteristic
291C2×D3—62no—
292C2×D3—62no—
293C2×D3—122no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—122no—
299C2×D3—122no—
300C2×D3—122no—
301C2×D3—124no—
302C2×D3—122no—
303C2×D3—122no—
304C2×D3—122no—
305C2×D3—241no—
306C2×D3—241no—
307C2×D3—242no—
308C2×D3—242no—
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4—81no—
314C3×C4—92no—
315C3×C4—91no—
316C3×C4—91no—
317C3×C4—361no—
318Dic3—62no—
319Dic3—61no—
320Dic3—61no—
321Dic3—62no—
322Dic3—61no—
323Dic3—61no—
324Dic3—121no—
325Dic3—121no—
326Dic3—241no—
327C32—21no—
328C32—21no—
329C32—41no—
330C32—41no—
331C32#484311yescharacteristic, complemented
332C23—181no—
333C23—182no—
334D4—272no—
335D4—362no—
336D4—541no—
337D4—542no—
338D4—542no—
339D4—1082no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—181no—
343C2×C4—1082no—
344C2×C4#157⋊D311yescharacteristic
345C8—182no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—362no—
359C2×C3—722no—
360C2×C3#6422511yescharacteristic
361C2×C3#6423011yescharacteristic
362C2×C3#6423611yescharacteristic
363D3—122no—
364D3—122no—
365D3—242no—
366D3—242no—
367D3—244no—
368D3—242no—
369C22—182no—
370C22—361no—
371C22—361no—
372C22—362no—
373C22—362no—
374C22—362no—
375C22C2×#157⋊D311yescharacteristic
376C4—21no—
377C4—92no—
378C4—91no—
379C4—91no—
380C4—361no—
381C3—21no—
382C3—21no—
383C3—41no—
384C3—41no—
385C3#1059342611yescharacteristic, complemented
386C2—362no—
387C2—722no—
388C2#1172433611yescharacteristic
389C2#1172434311yescharacteristic
390C2#1172435011yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#459
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88941C2semidirect
2. #11723697C2semidirect
3. #11723685C2semidirect
4. C3×#63779C2semidirect
5. #11722793C2semidirect
22. #63779D3semidirect
35. #6295D4semidirect
72. C3×C3⋊Dic3D8semidirect
188. C3⋊Dic3D24semidirect
205. C33#459semidirect
331. C32#4843semidirect
385. C3#10593426semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
#117236972
#117236852
C3×#637792
#117227932
#105934263
#48439