Finite Group Databasebeta

← #10714314

Subgroups — #10714314

Order 768.

704 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#2683C211yesmaximal, characteristic, complemented
2#69662C211yesmaximal, characteristic
3#69598C212yesmaximal
4#67641C211yesmaximal, characteristic, complemented
5#67637C212yesmaximal, complemented
6#13375—31nomaximal, Sylow
7C2×#1004C2211yescharacteristic
8C2×C3×C22⋊Q8—22no—
9C3×#396—22no—
10C3×#395—21no—
11C3×#390C2211yescharacteristic
12C3×#389—21no—
13C3×#386C2212yes—
14C2×#834C2212yes—
15C2×#833C2211yescharacteristic
16#5213—22no—
17#4966—21no—
18C2×#806—22no—
19C2×#805—21no—
20#2683D311yescharacteristic, complemented
21#1792—31no—
22#1790—32no—
23#1393—31no—
24#1390—32no—
25C22×C3×Q8—21no—
26C22×C3×D4C2311yescharacteristic
27C23×C3×C4D412yes—
28C22×#67—22no—
29#1004—21no—
30#1004—22no—
31#1004—42no—
32#1004Q812yescomplemented
33C2×#237—41no—
34C2×D3×D4—41no—
35C2×#235—21no—
36C3×C22⋊Q8—48no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4D412yes—
45C2×C3×C42D411yescharacteristic
46C2×#217—22no—
47C2×(C2×D3)⋊C4—42no—
48C2×#208—21no—
49C3×#95—21no—
50C3×#95—22no—
51#834—22no—
52#834—22no—
53#833—22no—
54C3⋊#95—42no—
55#810—41no—
56#809—41no—
57#806—44no—
58#805—42no—
59C2×H(5,𝔽2)—31no—
60C2×C22⋊Q8—22no—
61C2×#104—61no—
62#417—61no—
63C2×#100—31no—
64C2×#99—32no—
65C2×#99—62no—
66#396—22no—
67#395—21no—
68#390C2×D311yescharacteristic
69#389—21no—
70#386—62no—
71#386C2×D312yes—
72C24×C3C2×D412yes—
73C23×D3—22no—
74C2×C3×Q8—44no—
75C2×C3×Q8—81no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4C2×Q811yescharacteristic
79C2×C3×D4#4111yescharacteristic
80C2×C3×D4#4112yes—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—44no—
86C22×C3×C4—42no—
87C22×C3×C4C2×D411yescharacteristic
88C2×#67—24no—
89C2×#67—24no—
90C2×#67—42no—
91C2×#67—42no—
92C2×#67—41no—
93C2×#67—42no—
94C22×Dic3—22no—
95C22×Dic3—42no—
96#237—42no—
97#237—42no—
98#237—44no—
99D3×D4—42no—
100D3×D4—42no—
101D3×D4—44no—
102#235—22no—
103#235—22no—
104#235—41no—
105#235—41no—
106C2×D12—21no—
107C2×C4×D3—41no—
108C2×Dic6—21no—
109C3×C4⋊C4—24no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—44no—
112C3×C4⋊C4—44no—
113C3×C4⋊C4—81no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—44no—
116C3×C22⋊C4—42no—
117C3×C42—41no—
118#217—42no—
119#217—42no—
120(C2×D3)⋊C4—84no—
121#208—41no—
122#208—41no—
123C2×C3⋊C8—41no—
124H(5,𝔽2)—32no—
125H(5,𝔽2)—62no—
126H(5,𝔽2)—61no—
127H(5,𝔽2)—122no—
128C2×#41—61no—
129C2×#41—121no—
130C22×Q8—21no—
131C22×D4—62no—
132C22×D4—121no—
133C22×D4C22×D311yescharacteristic
134C23×C4#6712yes—
135C2×M16—61no—
136C22⋊Q8—48no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4#6711yescharacteristic
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4—122no—
146C2×C22⋊C4#6712yes—
147C2×C42#6711yescharacteristic
148#104—122no—
149#103—121no—
150#102—121no—
151#100—62no—
152#99—62no—
153#99—62no—
154#99—124no—
155#95—21no—
156#95—22no—
157#95—122no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3#12011yescharacteristic
163C23×C3C22⋊Q812yes—
164C23×C3#12412yes—
165C22×D3—24no—
166C22×D3—41no—
167C22×D3—42no—
168C22×D3—82no—
169C22×D3—82no—
170C3×Q8—84no—
171C3×D4—41no—
172C3×D4—42no—
173C3×D4—41no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—22no—
177C2×C3×C4—24no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—44no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—82no—
189C2×C3×C4C22⋊Q811yescharacteristic
190C2×C3×C4#12411yescharacteristic
191#67—44no—
192#67—44no—
193#67—42no—
194#67—42no—
195C2×Dic3—24no—
196C2×Dic3—42no—
197C2×Dic3—41no—
198C2×Dic3—82no—
199C2×Dic3—82no—
200C2×Dic3—82no—
201D12—22no—
202D12—41no—
203C4×D3—42no—
204C4×D3—42no—
205Dic6—22no—
206Dic6—41no—
207C3⋊C8—81no—
208C24—62no—
209C24C2×#6712yes—
210#41—62no—
211#41—62no—
212#41—121no—
213#41—122no—
214#41—124no—
215#41—122no—
216#41—121no—
217C2×Q8—44no—
218C2×Q8—61no—
219C2×Q8—81no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—61no—
223C2×D4—64no—
224C2×D4—64no—
225C2×D4—122no—
226C2×D4—122no—
227C2×D4—124no—
228C2×D4—122no—
229C2×D4—122no—
230C2×D4—122no—
231C2×D4—121no—
232C2×D4D3×Q811yescharacteristic
233C2×D4#23911yescharacteristic
234C2×D4#23912yes—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—62no—
242C22×C4—121no—
243C22×C4—122no—
244C22×C4C2×#6711yescharacteristic
245M16—121no—
246M16—121no—
247C2×C8—121no—
248C4⋊C4—24no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—81no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22⋊C4—244no—
259C42—41no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—82no—
269C22×C3#39311yescharacteristic
270C22×C3B(4,𝔽2)11yescharacteristic
271C22×C3B(4,𝔽2)11yescharacteristic
272C2×D3—42no—
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—84no—
276C2×D3—162no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—42no—
280C3×C4—44no—
281C3×C4—82no—
282C3×C4—81no—
283Dic3—42no—
284Dic3—162no—
285C23—22no—
286C23—22no—
287C23—42no—
288C23—42no—
289C23—64no—
290C23—122no—
291C23—121no—
292C23—242no—
293C23—242no—
294C23#94612yes—
295C23#94712yes—
296C23#95311yescharacteristic
297Q8—62no—
298Q8—84no—
299Q8—121no—
300D4—41no—
301D4—42no—
302D4—41no—
303D4—62no—
304D4—121no—
305D4—124no—
306D4—124no—
307D4—122no—
308D4—122no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—64no—
324C2×C4—82no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—121no—
329C2×C4—242no—
330C2×C4—242no—
331C2×C4—242no—
332C2×C4#94611yescharacteristic
333C2×C4#94711yescharacteristic
334C8—241no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—42no—
338C2×C3—42no—
339C2×C3#202911yescharacteristic
340C2×C3#219611yescharacteristic
341C2×C3#220111yescharacteristic
342D3—82no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—82no—
352C22—122no—
353C22—244no—
354C22—241no—
355C22—241no—
356C22—482no—
357C22#523111yescharacteristic
358C22#524411yescharacteristic
359C22#524411yescharacteristic
360C4—21no—
361C4—21no—
362C4—42no—
363C4—44no—
364C4—82no—
365C4—81no—
366C4—122no—
367C4—482no—
368C3#1337511yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—42no—
372C2—42no—
373C2—242no—
374C2#7189711yescharacteristic
375C2#7190911yescharacteristic
376C2#7195811yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#2683C2
socleC22×C3B(4,𝔽2)
prime radicalC2×#1004C22
hypercenter#2683D3
hypocenterC3#13375
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. C3×#2683C2semidirect
4. #67641C2semidirect
5. #67637C2semidirect
20. #2683D3semidirect
32. #1004Q8semidirect
368. C3#13375semidirect

Maximal subgroups

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

subgroupindex
C3×#26832
#696622
#695982
#676412
#676372
#133753