Finite Group Databasebeta

← C2×#14554331 #72244205

Subgroups — C2×#14554331 #72244205

Order 4374.

4048 conjugacy classes of subgroups; only the 622 normal subgroups are shown.

normal subgroups only

This group has 4048 conjugacy classes of subgroups — too many to draw. Only its 622 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#14554331C211yesmaximal, characteristic, direct product factor, Sylow
2C2×C3×3−1+4C311yesmaximal, characteristic, complemented
3C2×C3×#6912C313yesmaximal, complemented
4C2×C3×#6910C313yesmaximal, complemented
5C2×C3×#6909C313yesmaximal, complemented
6C2×C3×#6909C313yesmaximal, complemented
7C2×#10624755C3127yesmaximal, complemented
8C3×3−1+4C2×C311yescharacteristic, complemented
9C3×#6912C2×C313yescomplemented
10C3×#6910C2×C313yescomplemented
11C3×#6909C2×C313yescomplemented
12C3×#6909C2×C313yescomplemented
13#10624755C2×C3127yescomplemented
14C2×3−1+4—39no—
15C2×3−1+4—39no—
16C2×3−1+4—39no—
17C2×C3×#737—39no—
18C2×C32×3−1+2C3213yes—
19C2×C32×3−1+2C3213yes—
20C2×C32×3−1+2C3213yes—
21C2×C32×3−1+2C3219yescomplemented
22C2×C32×H(3,𝔽3)C3213yes—
23C2×C32×H(3,𝔽3)C3211yescharacteristic
24C2×#6912—39no—
25C2×#6912—39no—
26C2×#6912C32127yescomplemented
27C2×#6910—39no—
28C2×#6910—39no—
29C2×#6910C32127yescomplemented
30C2×#6909—39no—
31C2×#6909—39no—
32C2×#6909—39no—
33C2×#6909—39no—
34C2×#6909—39no—
35C2×#6909C32127yescomplemented
36C2×C3×#733—39no—
37C2×C3×#731—39no—
38C2×C3×AΣL(1,𝔽27)—39no—
39C2×C3×AΣL(1,𝔽27)C32127yescomplemented
403−1+4—39no—
413−1+4—39no—
423−1+4—39no—
43C3×#737—39no—
44C32×3−1+2C2×C3213yes—
45C32×3−1+2C2×C3213yes—
46C32×3−1+2C2×C3213yes—
47C32×3−1+2C2×C3219yescomplemented
48C32×H(3,𝔽3)C2×C3213yes—
49C32×H(3,𝔽3)C2×C3211yescharacteristic
50#6912—39no—
51#6912—39no—
52#6912C2×C32127yescomplemented
53#6910—39no—
54#6910—39no—
55#6910C2×C32127yescomplemented
56#6909—39no—
57#6909—39no—
58#6909—39no—
59#6909—39no—
60#6909—39no—
61#6909C2×C32127yescomplemented
62C3×#733—39no—
63C3×#731—39no—
64C3×AΣL(1,𝔽27)—39no—
65C3×AΣL(1,𝔽27)C2×C32127yescomplemented
66C2×C34—33no—
67C2×C34C3313yes—
68C2×C34C3311yescharacteristic
69C2×#737—381no—
70C2×#737—381no—
71C2×#737—381no—
72C2×C3×3−1+2—33no—
73C2×C3×3−1+2—33no—
74C2×C3×3−1+2—33no—
75C2×C3×3−1+2—33no—
76C2×C3×3−1+2—33no—
77C2×C3×3−1+2—33no—
78C2×C3×3−1+2—33no—
79C2×C3×3−1+2—33no—
80C2×C3×3−1+2—318no—
81C2×C3×3−1+2—381no—
82C2×C3×3−1+2—33no—
83C2×C3×3−1+2—33no—
84C2×C3×3−1+2—33no—
85C2×C3×3−1+2—33no—
86C2×C3×3−1+2—33no—
87C2×C3×3−1+2—33no—
88C2×C3×3−1+2—33no—
89C2×C3×3−1+2—33no—
90C2×C3×3−1+2—33no—
91C2×C3×3−1+2—33no—
92C2×C3×3−1+2—33no—
93C2×C3×3−1+2—33no—
94C2×C3×3−1+2—33no—
95C2×C3×3−1+2—33no—
96C2×C3×3−1+2—33no—
97C2×C3×3−1+2—33no—
98C2×C3×3−1+2—33no—
99C2×C3×3−1+2—33no—
100C2×C3×3−1+2—33no—
101C2×C3×3−1+2—33no—
102C2×C3×3−1+2—33no—
103C2×C3×3−1+2—33no—
104C2×C3×3−1+2H(3,𝔽3)19yes—
105C2×C3×3−1+2H(3,𝔽3)19yes—
106C2×C3×3−1+2H(3,𝔽3)19yes—
107C2×C3×3−1+2C3319yes—
108C2×C3×3−1+2C3319yes—
109C2×C3×3−1+2C3319yes—
110C2×C3×H(3,𝔽3)—33no—
111C2×C3×H(3,𝔽3)—33no—
112C2×C3×H(3,𝔽3)—33no—
113C2×C3×H(3,𝔽3)—33no—
114C2×C3×H(3,𝔽3)—33no—
115C2×C3×H(3,𝔽3)—33no—
116C2×C3×H(3,𝔽3)—33no—
117C2×C3×H(3,𝔽3)—33no—
118C2×C3×H(3,𝔽3)—33no—
119C2×C3×H(3,𝔽3)—33no—
120C2×C3×H(3,𝔽3)—33no—
121C2×C3×H(3,𝔽3)—33no—
122C2×C3×H(3,𝔽3)—33no—
123C2×C3×H(3,𝔽3)—33no—
124C2×C3×H(3,𝔽3)H(3,𝔽3)19yes—
125C2×C3×H(3,𝔽3)C3319yes—
126C2×C32×C9—33no—
127C2×C32×C9—39no—
128C2×C32×C9—33no—
129C2×C32×C9—33no—
130C2×#733—381no—
131C2×#731—381no—
132C2×AΣL(1,𝔽27)—381no—
133C2×AΣL(1,𝔽27)—381no—
134C34—33no—
135C34C2×C3313yes—
136C34C2×C3311yescharacteristic
137#737—381no—
138#737—381no—
139#737—381no—
140C3×3−1+2—33no—
141C3×3−1+2—33no—
142C3×3−1+2—33no—
143C3×3−1+2—33no—
144C3×3−1+2—33no—
145C3×3−1+2—33no—
146C3×3−1+2—33no—
147C3×3−1+2—33no—
148C3×3−1+2—318no—
149C3×3−1+2—381no—
150C3×3−1+2—33no—
151C3×3−1+2—33no—
152C3×3−1+2—33no—
153C3×3−1+2—33no—
154C3×3−1+2—33no—
155C3×3−1+2—33no—
156C3×3−1+2—33no—
157C3×3−1+2—33no—
158C3×3−1+2—33no—
159C3×3−1+2—33no—
160C3×3−1+2—33no—
161C3×3−1+2—33no—
162C3×3−1+2—33no—
163C3×3−1+2—33no—
164C3×3−1+2—33no—
165C3×3−1+2—33no—
166C3×3−1+2—33no—
167C3×3−1+2—33no—
168C3×3−1+2—33no—
169C3×3−1+2—33no—
170C3×3−1+2—33no—
171C3×3−1+2—33no—
172C3×3−1+2C2×H(3,𝔽3)19yes—
173C3×3−1+2C2×H(3,𝔽3)19yes—
174C3×3−1+2C2×H(3,𝔽3)19yes—
175C3×3−1+2C2×C3319yes—
176C3×3−1+2C2×C3319yes—
177C3×3−1+2C2×C3319yes—
178C3×H(3,𝔽3)—33no—
179C3×H(3,𝔽3)—33no—
180C3×H(3,𝔽3)—33no—
181C3×H(3,𝔽3)—33no—
182C3×H(3,𝔽3)—33no—
183C3×H(3,𝔽3)—33no—
184C3×H(3,𝔽3)—33no—
185C3×H(3,𝔽3)—33no—
186C3×H(3,𝔽3)—33no—
187C3×H(3,𝔽3)—33no—
188C3×H(3,𝔽3)—33no—
189C3×H(3,𝔽3)—33no—
190C3×H(3,𝔽3)—33no—
191C3×H(3,𝔽3)—33no—
192C3×H(3,𝔽3)C2×H(3,𝔽3)19yes—
193C3×H(3,𝔽3)C2×C3319yes—
194C32×C9—33no—
195C32×C9—39no—
196C32×C9—33no—
197C32×C9—33no—
198#733—381no—
199#731—381no—
200AΣL(1,𝔽27)—381no—
201AΣL(1,𝔽27)—381no—
202C2×C33—39no—
203C2×C33—39no—
204C2×C33—93no—
205C2×C33—93no—
206C2×C33—93no—
207C2×C33—271no—
208C2×C33—273no—
209C2×C33—279no—
210C2×C33C3×H(3,𝔽3)19yes—
211C2×C33C3×H(3,𝔽3)13yes—
212C2×C33C3×H(3,𝔽3)13yes—
213C2×C33C3×H(3,𝔽3)13yes—
214C2×C33C3×H(3,𝔽3)13yes—
215C2×C33C3411yescharacteristic
216C2×3−1+2—3162no—
217C2×3−1+2—39no—
218C2×3−1+2—39no—
219C2×3−1+2—39no—
220C2×3−1+2—39no—
221C2×3−1+2—39no—
222C2×3−1+2—39no—
223C2×3−1+2—39no—
224C2×3−1+2—39no—
225C2×3−1+2—39no—
226C2×3−1+2—39no—
227C2×3−1+2—39no—
228C2×3−1+2—39no—
229C2×3−1+2—39no—
230C2×3−1+2—39no—
231C2×3−1+2—93no—
232C2×3−1+2—93no—
233C2×3−1+2—93no—
234C2×3−1+2—93no—
235C2×3−1+2—93no—
236C2×3−1+2—93no—
237C2×3−1+2—93no—
238C2×3−1+2—93no—
239C2×3−1+2—93no—
240C2×3−1+2—93no—
241C2×3−1+2—93no—
242C2×3−1+2—93no—
243C2×3−1+2C3×H(3,𝔽3)127yes—
244C2×H(3,𝔽3)—39no—
245C2×H(3,𝔽3)—39no—
246C2×H(3,𝔽3)—39no—
247C2×H(3,𝔽3)—39no—
248C2×H(3,𝔽3)—39no—
249C2×H(3,𝔽3)—39no—
250C2×H(3,𝔽3)—93no—
251C2×H(3,𝔽3)—93no—
252C2×H(3,𝔽3)—93no—
253C2×H(3,𝔽3)—93no—
254C2×H(3,𝔽3)—93no—
255C2×H(3,𝔽3)—93no—
256C2×C3×C9—381no—
257C2×C3×C9—39no—
258C2×C3×C9—39no—
259C2×C3×C9—39no—
260C2×C3×C9—39no—
261C2×C3×C9—39no—
262C2×C3×C9—39no—
263C2×C3×C9—93no—
264C2×C3×C9—93no—
265C2×C3×C9—93no—
266C2×C3×C9—93no—
267C2×C3×C9—93no—
268C2×C3×C9—93no—
269C33—39no—
270C33—39no—
271C33—93no—
272C33—93no—
273C33—93no—
274C33—271no—
275C33—273no—
276C33—279no—
277C33C2×C3×H(3,𝔽3)19yes—
278C33C2×C3×H(3,𝔽3)13yes—
279C33C2×C3×H(3,𝔽3)13yes—
280C33C2×C3×H(3,𝔽3)13yes—
281C33C2×C3×H(3,𝔽3)13yes—
282C33C2×C3411yescharacteristic
2833−1+2—3162no—
2843−1+2—39no—
2853−1+2—39no—
2863−1+2—39no—
2873−1+2—39no—
2883−1+2—39no—
2893−1+2—39no—
2903−1+2—39no—
2913−1+2—39no—
2923−1+2—39no—
2933−1+2—39no—
2943−1+2—39no—
2953−1+2—39no—
2963−1+2—39no—
2973−1+2—39no—
2983−1+2—93no—
2993−1+2—93no—
3003−1+2—93no—
3013−1+2—93no—
3023−1+2—93no—
3033−1+2—93no—
3043−1+2—93no—
3053−1+2—93no—
3063−1+2—93no—
3073−1+2—93no—
3083−1+2—93no—
3093−1+2—93no—
3103−1+2C2×C3×H(3,𝔽3)127yes—
311H(3,𝔽3)—39no—
312H(3,𝔽3)—39no—
313H(3,𝔽3)—39no—
314H(3,𝔽3)—39no—
315H(3,𝔽3)—39no—
316H(3,𝔽3)—39no—
317H(3,𝔽3)—93no—
318H(3,𝔽3)—93no—
319H(3,𝔽3)—93no—
320H(3,𝔽3)—93no—
321H(3,𝔽3)—93no—
322H(3,𝔽3)—93no—
323C3×C9—381no—
324C3×C9—39no—
325C3×C9—39no—
326C3×C9—39no—
327C3×C9—39no—
328C3×C9—39no—
329C3×C9—39no—
330C3×C9—93no—
331C3×C9—93no—
332C3×C9—93no—
333C3×C9—93no—
334C3×C9—93no—
335C3×C9—93no—
336C2×C32—39no—
337C2×C32—93no—
338C2×C32—91no—
339C2×C32—93no—
340C2×C32—93no—
341C2×C32—93no—
342C2×C32—99no—
343C2×C32—99no—
344C2×C32—279no—
345C2×C32—279no—
346C2×C32—279no—
347C2×C32—279no—
348C2×C32—279no—
349C2×C32C32⋊H(3,𝔽3)13yes—
350C2×C32C32⋊H(3,𝔽3)13yes—
351C2×C32C32⋊H(3,𝔽3)13yes—
352C2×C32C32×H(3,𝔽3)11yescharacteristic
353C2×C32C32×H(3,𝔽3)13yes—
354C2×C9—327no—
355C2×C9—99no—
356C2×C9—99no—
357C32—39no—
358C32—93no—
359C32—91no—
360C32—93no—
361C32—93no—
362C32—93no—
363C32—99no—
364C32—99no—
365C32—279no—
366C32—279no—
367C32—279no—
368C32—279no—
369C32—279no—
370C32C2×C32⋊H(3,𝔽3)13yes—
371C32C2×C32⋊H(3,𝔽3)13yes—
372C32C2×C32⋊H(3,𝔽3)13yes—
373C32C2×C32×H(3,𝔽3)11yescharacteristic
374C32C2×C32×H(3,𝔽3)13yes—
375C9—327no—
376C9—99no—
377C9—99no—
378C2×C3—31no—
379C2×C3—33no—
380C2×C3—33no—
381C2×C3—33no—
382C2×C3—33no—
383C2×C3—99no—
384C2×C3—279no—
385C2×C3C3×C32⋊H(3,𝔽3)11yescharacteristic
386C3—31no—
387C3—33no—
388C3—33no—
389C3—33no—
390C3—33no—
391C3—99no—
392C3—279no—
393C3C2×C3×C32⋊H(3,𝔽3)11yescharacteristic
394C2#1455433111yescharacteristic, direct product factor, Sylow
subgroupG/H
center, socleC2×C3C3×C32⋊H(3,𝔽3)
commutator, Frattini, cosocleC33C2×C34
LCSG ▷ C33 ▷ C3 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C34 ◁ G

Complements

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

NH ≅ G/Ntype
1. #14554331C2direct (N × H)
2. C2×C3×3−1+4C3semidirect
3. C2×C3×#6912C3semidirect
4. C2×C3×#6910C3semidirect
5. C2×C3×#6909C3semidirect
6. C2×C3×#6909C3semidirect
7. C2×#10624755C3semidirect
8. C3×3−1+4C2×C3semidirect
9. C3×#6912C2×C3semidirect
10. C3×#6910C2×C3semidirect
11. C3×#6909C2×C3semidirect
12. C3×#6909C2×C3semidirect
13. #10624755C2×C3semidirect
21. C2×C32×3−1+2C32semidirect
26. C2×#6912C32semidirect
29. C2×#6910C32semidirect
35. C2×#6909C32semidirect
39. C2×C3×AΣL(1,𝔽27)C32semidirect
47. C32×3−1+2C2×C32semidirect
52. #6912C2×C32semidirect
55. #6910C2×C32semidirect
61. #6909C2×C32semidirect
65. C3×AΣL(1,𝔽27)C2×C32semidirect
394. C2#14554331direct (N × H)

Maximal subgroups

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

subgroupindex
#145543312
C2×C3×3−1+43
C2×C3×#69123
C2×C3×#69103
C2×C3×#69093
C2×#106247553